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Math 261 - Calculus 1
Worksheets
Pierce College
Instructor: Bob Martinez
Linear Functions Review Worksheet
Show all work on your paper as described in class. Video links are included throughout for instruction on how to do the
various types of problems. Important: Work the problems to match everything that was shown in the videos. For example:
Suppose a video shows 3 ways to do a problem, (such as algebraically, graphically, and numerically), then your work should
show these 3 ways also. That is , each video is a model for the work I want to see on your paper.
TI-84 Calculator video - shows some common things we do with the calculator in our class:
http://youtu.be/I-BV9tQk9TsWatch and make sure you can do the methods shown in the video.
ESSAY.
1) Go to http://youtu.be/ZV4xnYXO88g and watch and take notes on the video "What is Calculus?"
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
2) The thickness of a glacier, in meters, is given by T = 50 - 0.4x, where x is the number of years since 1960.
Interpret the slope of the graph as a rate of change.
A) The glacier will disappear in 125 years
B) The glacier's thickness is decreasing by 0.4 meters per year.
C) The glacier's thickness decreases by 1 meter every 0.4 years
D) The glacier was 50 meters thick in 1960
3) In 1912, the glacier on Mount Kilimanjaro in Africa covered 5 acres. By 2002, this glacier melted to only 1 acre.
Assuming this glacier melted at a constant rate each year, write a linear equation that gives the acres, A, of this
glacier t years after 1912. http://youtu.be/HNJtksdcgsE
A) A = - 2
45t + 5 B) A = 5t + 1 C) A = -22.5t + 1912 D) A = .04t - 555
4) The thickness of a glacier, in meters, is given by T = 60 - 0.2x, where x is the number of years since 1960.
Interpret the slope of the graph as a rate of change.
A) The glacier's thickness decreases by 1 meter every 0.2 years
B) The glacier was 60 meters thick in 1960
C) The glacier's thickness is decreasing by 0.2 meters per year.
D) The glacier will disappear in 300 years
5) The fish population, P, in Crystal Lake t years after the year 2000 is given by P = 430 - 60t. What does the slope
of the graph tell you?
A) The fish population is 60.
B) The fish population is decreasing by 60 fish per year.
C) The fish population is 430.
D) The fish population is increasing by 60 fish per year.
6) Find an equation of the line that passes through (1, -7) and (-8, 8)
http://youtu.be/bhQbaQGZHW8
A) y = 3
5x -
38
5B) y =
5
3x +
26
3C) y = -
5
3x -
16
3D) y =-
3
5x -
32
5
1
7) Find the slope of the line joining (1, -3) and (-5, -8)
A)5
6B)
6
5C) -
5
6D)
11
4
8) Find the slope of the line joining (-2, 6) and (5, -6)
A) 0 B) - 7
12C)
12
7D) -
12
7
Find the slope-intercept form for the line satisfying the conditions.
9) Parallel to y = 3x - 2, passing through (1, -4)
A) y = -3x - 1 B) y = 3x + 7 C) y = -3x + 1 D) y = 3x - 7
10) Find an equation for the line that is perpendicular to y = 2
5x + 2,passing through (8, -6)
http://youtu.be/Yff1R6Oyxfo
A) y = - 5
2x + 14 B) y = -
2
5x + 16 C) y =
5
2x - 14 D) y = -
5
2x - 6
11) Find an equation of a line perpendicular to y = 4
5x + 3, passing through (16, -7)
A) y = 5
2x - 26 B) y = -
4
5x + 15 C) y = -
5
4x + 13 D) y = -
5
4x + 2
12) Solve the equation: -8b - 4 = -7 - 4b
A)3
4B)
4
3C) -
4
3D)
12
11
Solve the problem.
13) Decide whether the points in the table lie on a line. If they do, find the slope-intercept form of the line.
x -2 -1 0 1
f(x) -4 1 6 11
A) Yes; f(x) = 4x + 6 B) No C) Yes; f(x) = 5x + 8 D) Yes; f(x) = 5x + 6
The points in the table lie on a line. Find the equation of the line.
14) x 2 3 4 5
y -4 -7 -10 -13
A) y = -3x + 2 B) y = -4x + 3 C) y = -3x + 1 D) y = 3x - 8
2
Give the slope-intercept form of the line shown.
15)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
http://youtu.be/En3NtcW2BCI
A) y = 2
3x - 2 B) y = -
3
2x - 3 C) y = -
2
3x - 2 D) y = -
3
2x - 2
16)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
A) y = -4x - 3 B) y = 1
4x - 3 C) y =
1
5x - 3 D) y = -
1
4x - 4
17) Find an equation for the line that has a slope of - 3
4 and passes through (4, 2)
A) y = 3
4x - 5 B) y = -
3
4x + 5 C) y = -
3
4x + 4 D) y =
3
4x + 4
18) Find an equation for the line that has a slope of - 2
5 and passes through (4, 3)
A) y = - 2
5x +
23
5B) y =
2
5x -
23
5C) y =
2
5x + 4 D) y = -
2
5x + 4
3
ESSAY.
19) Office Outlet has 600 boxes of printer paper to sell. After 7 days, they had 460 boxes left. Suppose they continue
selling boxes at the same rate.
(a) Write a linear equation for the number of boxes, B, left after t days.
(b) Graph the equation for B. Remember to label the axes.
(c) State the slope of the graph, including units. What does the slope mean in this problem?
(d) What does the t-intercept tell us about this situation?
20) In 1960, a glacier in Arctic Ocean was 60 meters thick. The thickness of the glacier has been decreasing at a
constant rate of 0.2 meter per year.
a) Write an equation for the thickness, T, of the glacier in terms of x, the number of years since 1960.
b) How long will it take the glacier to be 45 meters thick?
c) Predict what year the glacier will disappear?
21) The fish population, P, in Crystal Lake t years after the year 2000 is shown on the graph below.
a) Find the t-intercept of the graph. What does it tell you about the problem?
b) Find the P-intercept of the graph. What does it tell you about the problem?
c) Find the slope of the line shown, including units.
d) What does the slope tell you about the problem?
e) Write P as a function of t.
t2 4 6 8 10 12 14 16
P
400
300
200
100
t2 4 6 8 10 12 14 16
P
400
300
200
100
4
22) A person is driving a car along a highway. The graph below shows the distance d in miles that the driver is
from home after t hours.
a) Find the t-intercept of the graph. What does it tell you about the problem?
b) Find the d-intercept of the graph. What does it tell you about the problem?
c) Find the slope of the line shown, including units.
d) What does the slope tell you about the problem?
e) Write d as a function of t.
x1 2 3 4 5 6
y350
300
250
200
150
100
50
x1 2 3 4 5 6
y350
300
250
200
150
100
50
23) At the Pierce Farm customers can pick their own tomatoes. There is a $3.00 entrance fee and the tomatoes cost
$1.50 per pound. At Winnetka Ranch, the entrance fee is only $1.00, but the tomatoes cost $2.00 per pound.
a) Write an equation for the total cost of buying x pounds of tomatoes at the Pierce Farm.
b) Write an equation for the total cost of buying x pounds of tomatoes at Winnetka Ranch.
c) How many pounds of tomatoes must be purchased in order for the total costs to be the same?
24) The number of female officers in the Marine Corps has been increasing at a constant rate since 1995. In 1995,
there were 690 female officers, and by 2003, this number had increased to 1,090.
a) Find a linear function that relates the number, N , of female officers in the Marine Corps to the number of
years, t, since 1995.
b) Estimate the number of female officers in the year 2000.
c) Interpret the slope of the linear function in part (a).
d) Interpret the N intercept.
25) The table below shows the amount of water in a tank w, in gallons, after t minutes.
t (minutes) 4 8 12 16
w (gallons) 200 160 120 80
(a) Write a linear equation for the amount of water in the tank, w, in terms of the time t.
(b) State the slope of the equation in part (a). What are the units of the slope? What does the slope mean in the
context of this problem?
(c) State the t-intercept of the equation in part (a). What does the t-intercept tell us about this situation?
5
Answer KeyTestname: LINEAR FUNCTIONS REVIEW WORKSHEET
1) watch the video
2) B
3) A
4) C
5) B
6) C
7) A
8) D
9) D
10) A
11) C
12) A
13) D
14) A
15) C
16) B
17) B
18) A
19) a) B=600-20t b) the graph c) -20 boxes/day means the number of boxes is decreasing at
20 boxes per day d) The number of days when there will be no boxes left (B = 0 at t = 30 days)
20) a) T = -0.2x + 60 b) 75 years (in the year 2035) c) 300 years (in the year 2260)
21) a) (14,0) in 14 years after the year 2000 (in 2014) there will be no fish left in the lake. b) (0,420) In the year 2000 there
were 420 fish in the lake. c) -30 fish/year d) The fish population in the lake is decreasing at the rate of 30 fish per
year. e) P = -30t + 420
22) a) (6,0) In 6 hours the driver will be home b) (0,300) At time 0 the driver was 300 miles away from home. c) -50
miles/hour d) The driver gets 50 miles closer to home every hour e) d = -50t + 300
23) a) P = 1.5x + 3 b) W = 2x + 1 c) 4 pounds
24) a) N = 50t + 690 b) 940 female officers c) slope = 50 female officers per year. The number of female officers in the
Marine Corps is increasing at the rate of 50 female officers per year. d) The N intercept is (0,690) In 1995 there were
690 female officers in the Marine Corps.
25) a) w = -10t + 240 b) -10 gallons/minute The amount of water in the tank is dropping at a rate of 10 gallons per minute.
c) (24,0) In 24 minutes all the water will be gone.
6
Quadratic Functions Review Worksheet
Show all work on your paper as described in class. Video links are included throughout for instruction on how to do the
various types of problems. Important: Work the problems to match everything that was shown in the videos. For example:
Suppose a video shows 3 ways to do a problem, (such as algebraically, graphically, and numerically), then your work should
show these 3 ways also. That is , each video is a model for the work I want to see on your paper.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) A projectile is thrown upward so that its distance above the ground after t seconds is h = -13t2 + 416t. After how
many seconds does it reach its maximum height? http://youtu.be/ZdCbfjfdP6MA) 32 sec B) 8 sec C) 24 sec D) 16 sec
2) A window washer accidentally drops a bucket from the top of a 144-foot building. The height h of the bucket
after t seconds is given by h = -16t2 + 144. When will the bucket hit the ground?
A) -3 sec B) 16 sec C) 3 sec D) 12 sec
Solve the problem.
3) An object is thrown upward with an initial velocity of 11 ft per second. Its height is given by h = -11t2 + 66t at
time t seconds. After how many seconds does it hit the ground?
A) 7 sec B) 3 sec C) 6 sec D) 9 sec
4) If an object is thrown upward with an initial velocity of 15 ft per second, its height is given by h = -15t2 + 90t
after time t seconds. What is its maximum height?
A) 45 ft B) 135 ft C) 225 ft D) 228 ft
Use the graph of y = ax2 + bx + c to solve the quadratic equation or inequality.
5) ax2 + bx + c < 0
x-10 10
y
10
-10
x-10 10
y
10
-10
http://youtu.be/-96ycSNVzdgA) x < -4 or x > 2 B) -4 ≤ x ≤ 2 C) -4 < x < 2 D) 2 < x < 4
1
6) ax2 + bx + c ≤ 0
x-10 10
y
10
-10
x-10 10
y
10
-10
A) x ≤ 2 or x ≥ 5 B) 2 < x < 5 or x > 5 C) 2 ≤ x ≤ 5 D) x < -2 or x < 5
Solve the quadratic inequality using your graphing calculator. Write your answer in interval notation.
7) x2 + 5x - 14 > 0 http://youtu.be/fWAqsyZnLioA) (-∞, -7) ∪ (2, ∞) B) (2, ∞) C) (-∞, -7) D) (-7, 2)
8) x2 - 2x - 3 < 0
A) (-∞, -1) B) (-∞, -1) ∪ (3, ∞) C) (-1, 3) D) (3, ∞)
9) Solve: x2 - 2x - 8 < 0
A) -4 < x < 2 B) x > -2 or x > 4 C) -2 < x < 4 D) x < -2 or x > 4
Solve the problem.
10) A flare fired from the bottom of a gorge is visible only when the flare is above the rim. If it is fired with an initial
velocity of 160 ft/sec, and the gorge is 336 ft deep, during what interval can the flare be seen?
(h = -16t2 + vot + ho.) http://youtu.be/-af_Bsd3Bb8A) 0 < t < 3 B) 6 < t < 10 C) 9 < t < 13 D) 3 < t < 7
11) A coin is tossed upward from a balcony 200 ft high with an initial velocity of 48 ft/sec. During what interval of
time will the coin be at a height of at least 40 ft?
(h = -16t2 + vot + ho.)
A) 0 ≤ t ≤ 1 B) 0 ≤ t ≤ 5 C) 5 ≤ t ≤ 10 D) 4 ≤ t ≤ 5
12) If a rocket is propelled upward from ground level, its height in meters after t seconds is given by
h = -9.8t2 + 78.4t. During what interval of time will the rocket be higher than 147 m?
A) 0 < t < 3 B) 3 < t < 5 C) 6 < t < 8 D) 5 < t < 6
2
Answer KeyTestname: QUADRATIC FUNCTIONS REVIEW WORKSHEET
1) D
2) C
3) C
4) B
5) C
6) A
7) A
8) C
9) C
10) D
11) B
12) B
3
Polynomial Functions Review Worksheet
Show all work on your paper as described in class. Video links are included throughout for instruction on how to do the
various types of problems. Important: Work the problems to match everything that was shown in the videos. For example:
Suppose a video shows 3 ways to do a problem, (such as algebraically, graphically, and numerically), then your work should
show these 3 ways also. That is , each video is a model for the work I want to see on your paper.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Evaluate the expression graphically.
1) f(2)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
A) -3 B) -4 C) -2 D) -1
2) f(-1)
x-5 -4 -3 -2 -1 1 2 3 4 5
y10987654321
-1-2-3-4-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y10987654321
-1-2-3-4-5
A) -3 B) -4 C) -2 D) -5
1
3) f(0)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
A) -3 B) -2 C) -1 D) -4
Evaluate f(x) at the given value of x.
4) f(x) = 2x3 + 3x2 - x + 26 for x = -2
A) 26 B) 24 C) 14 D) 12
Solve the problem.
5) A(x) = -0.015x3 + 1.05x gives the alcohol level in an average person's blood x hrs after drinking 8 oz of
100-proof whiskey. If the level exceeds 1.5 units, a person is legally drunk. Would a person be drunk after 7
hours?
A) Yes B) No
6) The following polynomial approximates the shark population in a particular area.
R(x) = -0.021x5 + 3.785x4 + 300, where x is the number of years from 1985. Use a graphing calculator to describe
the shark population from the years 1985 to 2010.
A) The population remains stable.
B) The population increases.
C) The population decreases.
7) The polynomial function I(t) = -0.1t2 + 1.6t represents the yearly income (or loss) from a real estate investment,
where t is time in years. After how many years does income begin to decline?
A) 10.7 yr B) 8 yr C) 16 yr D) 7 yr
Solve the equation.
8) 6x4 - 3x3 = 0
A) 0, - 1
2B) 0,
1
3C) 0,
1
2D) 0, -
1
3
Factor completely.
9) x2 - x - 20
A) (x + 1)(x - 9) B) (x + 5)(x - 4) C) (x + 4)(x - 5) D) (x - 5)(x + 5)
10) x2 - 6x - 40
A) (x - 4)(x - 10) B) (x - 4)(x + 1) C) (x - 4)(x + 10) D) (x + 4)(x - 10)
2
Solve the equation.
11) 16t3 - 49t = 0
A)7
4B) -
7
4,
7
4, 0 C) 0 D) ±
7
4
12) 3x3 + 26x2 = -48x
A) 6, 8
3B) -6, -
8
3C) -6, -
8
3, 0 D) 0, 6,
8
3
13) x3 + 14x2 + 59x = -70 (solve by graphing and 2nd calc intersect: y1 = x3 + 14x2 + 59x y2 = -70 )
A) -1, -2, -5 B) 7, 2, 5 C) 0, 7, 2, 5 D) -7, -2, -5
Solve the problem.
14) Ariel, a marine biologist, models a population P of crabs, t days after being left to reproduce, with the function
P(t) = -0.00006t3 + 0.016t2 + 7t + 1200 (solve by graphing and 2nd calc zero)
Assuming that this model continues to be accurate, when will this population become extinct? (Round to the
nearest day.)
A) 1512 days B) 911 days C) 707 days D) 547 days
ESSAY
15) An open top rectangular box is to formed by taking an 8.5" by 11" piece of paper, cutting out an x inch by x inch
square out of each corner, then folding the sides up to form the box. Go tohttp://calculusapplets.com/boxproblem.html to see an example and move the slider on the applet
to see the effect of different size cutouts on the resulting shape and volume of the box.
a) Draw a picture of the sheet of paper with the cutouts illustrated. And draw a 3-d picture of the resulting box.
b) Write the volume of the box, V, as a function of the cutout side, x.
c) What is the domain of V? That is, what is the feasible range of values that x can be?
d) Graph V(x) on its domain and find the maximum volume of the box and the cutout side, x, that produces this
maximum. (Use 2nd calc max to find the maximum.)
16) An open top cylindrical can is to be made such that its surface area (around the side and on the bottom) is 100
cm2 .
a) Write the volume, V, of the can as a function of the radius, r, of the can. Leave the symbol, π, in the equation -
do not approximate.
b) Graph V(r) on its domain (on the domain of feasible values of r).
c) Find the radius that produces the greatest volume for the can.
d) What is the height of the max volume can?
FROM THE BOOK
17) Do section 1.3 #13, 15; section 1.6 #7 http://youtu.be/Bg0eJX1DKc4 , 9, 11; Chapter 1 Review #33, 39
3
Answer KeyTestname: POLYNOMIAL FUNCTIONS REVIEW WORKSHEET
1) A
2) B
3) B
4) B
5) A
6) B
7) B
8) C
9) C
10) D
11) B
12) C
13) D
14) D
15) discuss the answer in class
16) max volume = 108.59 cm3 achieved when the radius is 3.26 cm, the height of the max volume can is h = 3.26 cm
17) see answers in the book
4
Rational, Power, and Root Functions Review Worksheet
Show all work on your paper as described in class. Video links are included throughout for instruction on how to do the
various types of problems. Important: Work the problems to match everything that was shown in the videos. For example:
Suppose a video shows 3 ways to do a problem, (such as algebraically, graphically, and numerically), then your work should
show these 3 ways also. That is , each video is a model for the work I want to see on your paper.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
What are the equations of the vertical and horizontal asymptotes of the graph of the given equation?
1) y = 8
x
A) Vertical: y = 0; horizontal: x = 0 B) Vertical: x = 0; horizontal: y = 0
C) Vertical: y = 0; horizontal: x = 8 D) Vertical: x = 0; horizontal: y = 8
2) y = 8
x + 9
A) Vertical: x = 0; horizontal: y = 8 B) Vertical: y = 9; horizontal: x = 8
C) Vertical: x = 8; horizontal: y = 9 D) Vertical: x = 0; horizontal: y = 9
3) y = 1
(x - 8)2 + 9 http://youtu.be/6J6UtGKE7Zs
A) Vertical: x = 8; horizontal: y = 9 B) Vertical: x = 8; horizontal: y = 0
C) Vertical: x = 0; horizontal: y = 8 D) Vertical: x = 9; horizontal: y = 8
Graph f(x) by a) finding asymptotes, x intercepts, and y intercept, b) on the graphing calculator to verify
4) f(x) = x + 2
x + 1
x-4 -3 -2 -1 1 2 3 4
y4
3
2
1
-1
-2
-3
-4
x-4 -3 -2 -1 1 2 3 4
y4
3
2
1
-1
-2
-3
-4
A)
x-4 -3 -2 -1 1 2 3 4
y4
3
2
1
-1
-2
-3
-4
x-4 -3 -2 -1 1 2 3 4
y4
3
2
1
-1
-2
-3
-4
B)
x-4 -3 -2 -1 1 2 3 4
y4
3
2
1
-1
-2
-3
-4
x-4 -3 -2 -1 1 2 3 4
y4
3
2
1
-1
-2
-3
-4
1
C)
x-4 -3 -2 -1 1 2 3 4
y4
3
2
1
-1
-2
-3
-4
x-4 -3 -2 -1 1 2 3 4
y4
3
2
1
-1
-2
-3
-4
D)
x-4 -3 -2 -1 1 2 3 4
y4
3
2
1
-1
-2
-3
-4
x-4 -3 -2 -1 1 2 3 4
y4
3
2
1
-1
-2
-3
-4
5) f(x) = x - 3
x - 4
x-8 -4 4 8
y8
4
-4
-8
x-8 -4 4 8
y8
4
-4
-8
A)
x-8 -4 4 8
y8
4
-4
-8
x-8 -4 4 8
y8
4
-4
-8
B)
x-8 -4 4 8
y8
4
-4
-8
x-8 -4 4 8
y8
4
-4
-8
2
C)
x-8 -4 4 8
y8
4
-4
-8
x-8 -4 4 8
y8
4
-4
-8
D)
x-8 -4 4 8
y8
4
-4
-8
x-8 -4 4 8
y8
4
-4
-8
6) f(x) = -2x - 3
x + 2
x-8 -4 4 8
y8
4
-4
-8
x-8 -4 4 8
y8
4
-4
-8
A)
x-8 -4 4 8
y8
4
-4
-8
x-8 -4 4 8
y8
4
-4
-8
B)
x-8 -4 4 8
y8
4
-4
-8
x-8 -4 4 8
y8
4
-4
-8
3
C)
x-8 -4 4 8
y8
4
-4
-8
x-8 -4 4 8
y8
4
-4
-8
D)
x-8 -4 4 8
y8
4
-4
-8
x-8 -4 4 8
y8
4
-4
-8
Provide an appropriate response.
7) Explain the behavior of the graph of f(x) as it approaches its vertical asymptote.
f(x) = 1
(x - 9)2
A) Approaches -∞ from the left and ∞ from the right
B) Approaches ∞ from the left and -∞ from the right
C) Approaches -∞ from the left and the right
D) Approaches ∞ from the left and the right
8) Explain the behavior of the graph of f(x) as it approaches its vertical asymptote.
f(x) = 3
x - 9
A) Approaches ∞ from the left and -∞ from the right
B) Approaches -∞ from the left and ∞ from the right
C) Approaches ∞ from the left and the right
D) Approaches -∞ from the left and the right
9) Explain the behavior of the graph of f(x) as it approaches its vertical asymptote.
f(x) = -1
x - 5
A) Approaches -∞ from the left and the right
B) Approaches -∞ from the left and ∞ from the right
C) Approaches ∞ from the left and the right
D) Approaches ∞ from the left and -∞ from the right
10) Explain the behavior of the graph of f(x) as it approaches its vertical asymptote.
f(x) = -2
(x - 3)2
A) Approaches -∞ from the left and the right
B) Approaches ∞ from the left and -∞ from the right
C) Approaches ∞ from the left and the right
D) Approaches -∞ from the left and ∞ from the right
4
11) Suppose a friend tells you that the graph of f(x) = x2 - 16
x - 4 has a vertical asymptote with equation x = 4. Is this
correct? If not, describe the behavior of the graph at x = 4.
A) This is incorrect. The graph has an oblique asymptote at x = 4.
B) This is correct.
C) This is incorrect. The graph is actually the graph of y = x - 4 with a "hole" at (-4, -8).
D) This is incorrect. The graph is actually the graph of y = x + 4 with a "hole" at (4, 8).
Solve the problem.
12) At an altitude of h feet above the surface of the earth, the approximate distance in miles that a person can see is
given by
d = 1.2247h1/2.
How far can a person see if he or she is 740 feet above the earth's surface? Round your answer to the nearest
tenth of a mile, if necessary.
A) 39 miles B) 33.3 miles C) 34.4 miles D) 35.7 miles
13) The formula
T = .07D1.5
can be used to approximate the duration of a storm, where T is the time in hours and D is the diameter of the
storm in miles. A storm that is 16.4 miles in diameter is heading toward a city. How long can the residents of the
city expect the storm to last? Round your answer to the nearest hundredth of an hour.
A) 5.88 hr B) 4.65 hr C) 4.08 hr D) 6.18 hr
14) In a manufacturing operation, the cost, c, is related to the manufacturing time, t, by the equation
c = t.
Find the exact value of c when t = 500. Do not concern yourself with units.
A) 500 B) 50 C) 10 5 D) 22
15) A manufacturer's cost is given by
C = 2003
n + 200,
where C is the cost, in dollars, and n is the number of parts produced. Find the cost when 343 parts are
produced.
A) $1600 B) $29 C) $900 D) $3904
16) The distance in miles that can be seen from above the surface of the ocean is given by
d = 1.4 h
where h is the height in feet above the surface of the water. How many feet above the water would a pirate have
to climb to see 17 mi? Round your answer to the nearest foot.
A) 74 ft B) 6 ft C) 570 ft D) 147 ft
17) To model the actual speed s (in miles per hour) in an accident which left a skid mark of l feet, police use the
formula
s = S l
L,
where S is the test-car speed (in miles per hour) and L is the test-skid length (in feet). Find the speed S = 54
mph, l = 150 ft, and L = 100 ft. Round your answer to the nearest unit.
A) 81 mph B) 104 mph C) 66 mph D) 44 mph
5
18) The time T in seconds required for a pendulum of length L feet to make one swing is given by
T = 2π L
32.
How long is a pendulum if it makes one swing in 3.00 sec? Round your answer to the nearest tenth of a foot.
A) 15.3 ft B) 8.0 ft C) 14.6 ft D) 7.3 ft
19) The radius r of a cone of height h (in inches) and volume V (in cubic inches) is given by
r = 3V
πh.
If the height of a cone is 8.9 in. and its volume is 198 in.3, find its radius to the nearest hundredth of an inch.
A) 1.54 in. B) 4.61 in. C) 2.6 in. D) 0.87 in.
20) Fred is in a row boat that is 3 miles from the shore of a lake. He wants to get to his house, which is 17 miles
down the shore, as shown below. He will row to shore and then jog the remaining distance along the shore. He
can row at 3 miles per hour and can jog at 6 miles per hour. At about what point along the shore should he
beach the boat and jog the rest of the way if he wants to get home as soon as possible? Round your answer to the
nearest tenth of a mile, if necessary. http://youtu.be/SI1fTmJ5v8k
68%
3 miles
17 miles
A) 15.3 miles from home B) 10.3 miles from home
C) 14.3 miles from home D) 16.3 miles from home
21) Two vertical poles of lengths 32 feet and 87 feet are situated on level ground 50 feet apart, as shown in the figure
below. A piece of wire is to be strung from the top of the 32-ft pole, to a stake in the ground, to the top of the
87-ft pole. At what distance from the 32-foot pole should the stake be located to minimize the amount of wire
used? Round your answer to the nearest tenth of a foot.
32 ft 87 ft
50 ft
A) 14.5 ft B) 10.1 ft C) 17.6 ft D) 13.7 ft
22) The battleship USS Tennessee is 170 miles due south of the destroyer USS Alaska and is sailing north at 40 mph.
If the USS Alaska is sailing east at 25 mph, how long will it be before the distance between the ships is at a
minimum? Express your answer in hours and minutes, rounded to the nearest minute.
A) 41 minutes B) 2 hours 5 minutes C) 3 hours 4 minutes D) 1 hour 26 minutes
6
23) The battleship USS Tennessee is 170 miles due south of the destroyer USS Alaska and is sailing north at 40 mph.
If the USS Alaska is sailing east at 25 mph, how far apart will the ships be when that distance is at a minimum?
Round your answer to the nearest tenth of a mile.
A) 17.8 miles B) 42.5 miles C) 90.1 miles D) 84.3 miles
24) The phone company needs to install a line from point A on shore to an island that is 8 miles away from point B,
as shown in the drawing below. The cost associated with installing underwater cable is $18,600 per mile, while
the cost associated with installing cable over land is $9000 per mile. For the project to be completed at minimum
cost, how much of the cable should be installed over land? In other words, at what distance from point A should
the company stop laying cable over land? Round your answer to the nearest tenth of a mile, if necessary.
8 miles
10 miles
A) 5.6 miles B) 6.7 miles C) 7.1 miles D) 4.8 miles
25) The phone company needs to install a line from point A on shore to an island that is 3 miles away from point B,
as shown in the drawing. The cost associated with underwater cable is $18,600 per mile, while the cost
associated with cable used over land is $9000 per mile. To the nearest hundred dollars, what is the minimum
cost for which this project can be completed?
3 miles
10 miles
A) $137,600 B) $139,600 C) $138,800 D) $135,400
FROM THE BOOK
26) Do section 1.6 #13, 15, 19, 21 http://youtu.be/wOt-IWgtVro
7
Answer KeyTestname: RATIONAL, POWER, AND ROOT FUNCTIONS WORKSHEET
1) B
2) D
3) A
4) B
5) D
6) C
7) D
8) B
9) D
10) A
11) D
12) B
13) B
14) C
15) A
16) D
17) C
18) D
19) B
20) A
21) D
22) C
23) C
24) A
25) C
26) see answers in the book
8
Exponential and Logarithmic Functions Review Worksheet
Show all work on your paper as described in class. Video links are included throughout for instruction on how to do the
various types of problems. Important: Work the problems to match everything that was shown in the videos. For example:
Suppose a video shows 3 ways to do a problem, (such as algebraically, graphically, and numerically), then your work should
show these 3 ways also. That is , each video is a model for the work I want to see on your paper.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Graph the function.
1) f(x) = 5x
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
A)
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
B)
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
C)
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
D)
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
1
2) f(x) = 1
5
x
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
A)
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
B)
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
C)
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
D)
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
Solve the problem.
3) The amount of particulate matter left in solution during a filtering process decreases by the equation
P(n) = 800(0.5)0.4n, where n is the number of filtering steps. Find the amounts left for n = 0 and n = 5. (Round to
the nearest whole number.)
A) 1600 ; 200 B) 800 ; 200 C) 800 ; 25 D) 800 ; 3200
4) The number of bacteria growing in an incubation culture increases with time according to B(x) = 1500(2)x,
where x is time in days. Find the number of bacteria when x = 0 and x = 3 .
A) 1500 ; 9000 B) 3000 ; 12,000 C) 1500 ; 12,000 D) 1500 ; 6000
2
5) A computer is purchased for $3500. Its value each year is about 75% of the value the preceding year. Its value, in
dollars, after t years is given by the exponential function V(t) = 3500(0.75)t. Find the value of the computer after
7 years. Round to the nearest cent.
A) $18,375.00 B) $350.40 C) $262.80 D) $467.19
6) The half-life of a certain radioactive substance is 7 years. Suppose that at time t = 0 , there are 22 g of the
substance. Then after t years, the number of grams of the substance remaining will be N(t) = 221
2
t/14. How
many grams of the substance will remain after 28 years? Round to the nearest hundredth when necessary.
A) 5.5 g B) 2.75 g C) 0.69 g D) 1.38 g
7) The space in a landfill decreases with time as given by the function F(t) = 260 - 30 log (20t + 5) acres where t is
measured in years. How much space is left when t = 6 ?
A) 197 acres B) 180 acres C) 110 acres D) 320 acres
Solve the equation.
8) 15 x = 22 (Round to the nearest hundredth.) http://youtu.be/0jLz4PByEGgA) 0.88 B) 1.38 C) 1.14 D) 0.17
9) 14 x - 2 = 26 (Round to the nearest hundredth.)
A) 3.35 B) 3.86 C) 3.23 D) 2.81
Solve the problem.
10) Sound levels in decibels can be computed by f(x) = 10log(x/So), where x is the intensity of the sound in watts per
square meter and So = 1.00 x 10-12 watt/m2. A certain noise produces 6.81 x 10-5 watt/m2 of power. What is
the decibel level of this noise? (Round to the nearest decibel)
A) 180 decibels B) 78 decibels C) 68 decibels D) 8 decibels
Solve the equation.
11) 6 3 x = 5 x + 1 (Round to the nearest thousandth.)
A) 1.898 B) 0.898 C) 0.427 D) 8.827
12) 3(1 + 2x) = 27 http://youtu.be/3Igo_ar7dVIA) 9 B) 3 C) 1 D) -1
13) 3x = 1
81
A)1
27B) -4 C)
1
4D) 4
Solve the problem.
14) How long will it take for the population of a certain country to triple if its annual growth rate is 4.4 %? (Round
to the nearest year.) http://youtu.be/RnfHYe5pRdAA) 11 yr B) 1 yr C) 68 yr D) 26 yr
3
15) How long will it take for the population of a certain country to double if its annual growth rate is 7.4 %? (Round
to the nearest year.)
A) 27 yr B) 4 yr C) 9 yr D) 1 yr
16) There are currently 79 million cars in a certain country, decreasing by 1.7 % annually. How many years will it
take for this country to have 57 million cars? (Round to the nearest year.)
A) 4 yr B) 182 yr C) 19 yr D) 13 yr
17) An economist predicts that the buying power B(x) of a dollar x years from now will be given by the formula
B(x) = 0.21x. How much will today's dollar be worth in 6 years? Round the answer to the nearest cent.
A) $1.46 B) $1.26 C) $0.39 D) $0.00
18) Suppose that y = 2 - log (100 - x)
0.40 can be used to calculate the number of years y for x percent of a population of
444 web-footed sparrows to die. Approximate the percentage (to the nearest whole per cent) of web-footed
sparrows that died after 4 years.
A) 97% B) 100% C) 95% D) 99%
19) Suppose f(x) = 34.2 + 1.4log (x + 1) models salinity of ocean water to depths of 1000 meters at a certain latitude. x
is the depth in meters and f(x) is in grams of salt per kilogram of seawater. Approximate the salinity (to the
nearest hundredth) when the depth is 710 meters.
A) 98.93 grams of salt per kilogram of seawater B) 96.13 grams of salt per kilogram of seawater
C) 38.19 grams of salt per kilogram of seawater D) 30.21 grams of salt per kilogram of seawater
20) Suppose f(x) = 30.6 + 1.4log (x + 1) models salinity of ocean water to depths of 1000 meters at a certain latitude. x
is the depth in meters and f(x) is in grams of salt per kilogram of seawater. Approximate the depth (to the
nearest tenth of a meter) where the salinity equals 37.
A) 37,276.9 meters B) -0.8 meters C) -1.0 meters D) 37,274.9 meters
FROM THE BOOK
21) Do section1.2 #15, 17, http://youtu.be/sf-mGSj5gCk ,27, 29 http://youtu.be/1swEiISxIuQ ;section 1.4 #47, 53 http://youtu.be/WVCzQXSaCD0
4
Answer KeyTestname: EXPONENTIAL AND LOGARITHMIC FUNCTIONS REVIEW WORKSHEET
1) D
2) A
3) B
4) C
5) D
6) A
7) A
8) C
9) C
10) B
11) C
12) C
13) B
14) D
15) C
16) C
17) D
18) A
19) C
20) D
21) see answers in the book
5
Trigonometric Functions Review Worksheet
Show all work on your paper as described in class. Video links are included throughout for instruction on how to do the
various types of problems. Important: Work the problems to match everything that was shown in the videos. For example:
Suppose a video shows 3 ways to do a problem, (such as algebraically, graphically, and numerically), then your work should
show these 3 ways also. That is , each video is a model for the work I want to see on your paper.
ESSAY.
1) Watch and take notes on this Radian Angles video: http://youtu.be/TArUro2zXcg
2) Take notes on these 3 Unit Circle Trig Explanation videos and know for a test:http://youtu.be/FrXpbS6pBlM , http://youtu.be/_9SA2VK4O2Y , http://youtu.be/7pci1NP0Bok
3) Go to http://www.youtube.com/watch?v=YfcIaUF2JqM After watching the 3 videos in the problem above, watch this video and copy down the unit circle with all the
angles and the corresponding points on the unit circle shown.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the exact value by drawing the unit circle, the angle, and the terminal point coordinates. Verify with your calculator.
4) sin 2π
A) 0 B)2
2C) 1 D) undefined
Find the exact value. Verify with your calculator.
5) cos 0
A) 0 B) 1 C)2
2D) undefined
6) tan 0
A) 1 B) 0 C)2
2D) undefined
7) tan π
A) 0 B) -1 C) 1 D) undefined
8) cos π
A) 0 B) -1 C) 1 D) undefined
9) sin (38π) http://youtu.be/KSeOYmnDdXwA) -1 B) 0 C) 1 D) undefined
10) sin (- π
2)
A) 1 B) 0 C) -1 D) undefined
11) cos (-π)
A) -1 B) 0 C) 1 D) undefined
1
12) cos π
4
A) -2
2B) 2 C)
3
2D)
2
2
13) cos 45°
A)2
2B) 2 C)
1
2D)
3
2
The figure shows an angle θ in standard position with its terminal side intersecting the unit circle. Evaluate the indicated
circular function value of θ.
14) Find cos θ.
- 5
13,
12
13
http://youtu.be/YXplSLI08Rg
A) - 12
13B) -
5
13C)
12
13D) -
5
12
2
15) Find tan θ.
7
25, -
24
25
A) - 25
7B) -
7
24C)
25
24D) -
24
7
16) Find tan θ.
- 5
13,
12
13
A) - 12
5B) -
13
12C)
12
5D) -
5
12
3
17) Find sin θ.
- 5
13,
12
13
A)12
13B) -
12
13C)
5
12D) -
5
13
Find the exact circular function value.
18) tan -3π
4
A) 3 B) -1 C)3
3D) 1
19) tan 7π
6
A)3
2B) 3 C)
3
3D) - 3
20) cos 2π
A) 1 B)1
2C) 0 D) -1
21) sin 4π
3
A)3
2B) -
1
2C) -
3
2D) -1
22) sin 3π
4
A) - 2
2B) -
1
2C)
2
2D)
1
2
4
23) cos -2π
3 http://youtu.be/Lb1FXS-riE4
A) - 3
2B) -
1
2C) undefined D)
3
2
Find the exact value. Do not use a calculator.
24) sin 405° http://youtu.be/vw7jiLIBeuQ
A) - 1
2B) -
2
2C)
2
2D)
1
2
25) cos 8π
3
A)3
2B) -
1
2C) -
3
2D)
1
2
Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use a calculator.
26) tan 9π
4
A) 1 B)3
3C) -1 D) 3
27) sin 22π
3
A)3
2B) -
3
2C) -1 D) -
1
2
28) cos 20π
3
A)1
2B) -
1
2C)
3
2D) -
3
2
29) tan 720°
A)3
3B) 0 C) 1 D) undefined
30) tan 390°
A)3
2B) 3 C) - 3 D)
3
3
31) sin 495°
A) - 1
2B) -
2
2C)
1
2D)
2
2
Graph the sinusoidal function.
5
32) y = 3 sin (πx) http://youtu.be/1CBtrE-qoaQ
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
A)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
B)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
C)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
D)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
6
33) y = -3 cos (πx)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
A)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
B)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
C)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
D)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
7
34) y = 3 sin (2x)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
A)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
B)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
C)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
D)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
8
35) y = 3 cos (πx)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
A)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
B)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
C)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
D)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
9
36) y = -4 sin (1
4x)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
A)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
B)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
C)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
D)
x-π π 2π 3π
y6
4
2
-2
-4
-6
x-π π 2π 3π
y6
4
2
-2
-4
-6
Solve the problem.
37) A surveyor is measuring the distance across a small lake. He has set up his transit on one side of the lake 150
feet from a piling that is directly across from a pier on the other side of the lake. From his transit, the angle
between the piling and the pier is 55°. What is the distance between the piling and the pier to the nearest foot?
http://youtu.be/gcyA7IobPAUA) 123 ft B) 86 ft C) 214 ft D) 105 ft
10
38) A radio transmission tower is 220 feet tall. How long should a guy wire be if it is to be attached 5 feet from the
top and is to make an angle of 25° with the ground? Give your answer to the nearest tenth of a foot.
A) 508.7 ft B) 242.7 ft C) 237.2 ft D) 520.6 ft
39) A building 180 feet tall casts a 90 foot long shadow. If a person looks down from the top of the building, what is
the measure of the angle between the end of the shadow and the vertical side of the building (to the nearest
degree)? (Assume the person's eyes are level with the top of the building.)
A) 63° B) 30° C) 27° D) 60°
40) A tree casts a shadow of 26 meters when the angle of elevation of the sun is 24°. Find the height of the tree to the
nearest meter.
A) 13 m B) 12 m C) 10 m D) 11 m
41) A twenty-five foot ladder just reaches the top of a house and forms an angle of 41.5° with the wall of the house.
How tall is the house? Round your answer to the nearest 0.1 foot.
A) 18.7 ft B) 18.6 ft C) 19 ft D) 18.8 ft
FROM THE BOOK
42) Do section 1.5 #19, 21 http://youtu.be/d6ziBWJ8rx0 , 23-27 odd
11
Answer KeyTestname: TRIGONOMETRIC FUNCTIONS REVIEW WORKSHEET
1)
2) see the video
3) see the video
4) A
5) B
6) B
7) A
8) B
9) B
10) C
11) A
12) D
13) A
14) B
15) D
16) A
17) A
18) D
19) C
20) A
21) C
22) C
23) B
24) C
25) B
26) A
27) B
28) B
29) B
30) D
31) D
32) B
33) C
34) D
35) C
36) D
37) C
38) A
39) C
40) B
41) A
42) see answers in the book
12
Limits and Continuity Worksheet
Show all work on your paper as described in class. Video links are included throughout for instruction on how to do the
various types of problems. Important: Work the problems to match everything that was shown in the videos. For example:
Suppose a video shows 3 ways to do a problem, (such as algebraically, graphically, and numerically), then your work should
show these 3 ways also. That is , each video is a model for the work I want to see on your paper.
ESSAY
1) Go to http://youtu.be/5vSUrN-nqwE and watch and take notes on "The Idea of a Limit" .
FROM THE BOOK
2) Read and take notes on section 1.8 Limits: The idea of a limit; Definition of a limit; Properties of Limits; One
and two sided limits; Limits at infinity; Continuity
3) Do section 1.8 #1, 2
ESSAY
4) Go to http://calculusapplets.com/tablelimits.html scroll down and hit "launch presentation" to
make the applet larger.
a) From the drop down menu arrow in the upper right corner choose 1. A nice line. Move the slider (in the lower
left corner) so the x values go toward c =1 from the left and write what number the y values approach. This is
limx → 1-
.5x Then move the slider so the x values go toward c =1 from the right and write what number the y
values approach. This is limx → 1+
.5x Are the left and right limits equal? So, what is the value of limx →1
.5x ?
b) From the drop down menu choose 2. A line with a displaced point. Move the slider the same way you did in
(a) and write down the left limit and the right limit.
c) From the drop down menu choose 5. A jump discontinuity. Move the slider the same way you did in (a) and
write down the left limit and the right limit. So, what is the value of limx →1
f(x) ?
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
5) What conditions, when present, are sufficient to conclude that a function f(x) has a limit as x approaches some
value of a?
A) f(a) exists, the limit of f(x) as x→a from the left exists, and the limit of f(x) as x→a from the right exists.
B) The limit of f(x) as x→a from the left exists, the limit of f(x) as x→a from the right exists, and these two
limits are the same.
C) Either the limit of f(x) as x→a from the left exists or the limit of f(x) as x→a from the right exists
D) The limit of f(x) as x→a from the left exists, the limit of f(x) as x→a from the right exists, and at least one of
these limits is the same as f(a).
1
Use the graph to evaluate the limit.
6) limx→-1
f(x)
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
1
-1
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
1
-1
A) -1 B)1
2C) ∞ D) -
1
2
7) limx→0
f(x)
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A) 1 B) does not exist C) -1 D) 0
2
8) limx→0
f(x)
x-2 -1 1 2 3 4 5
y12
10
8
6
4
2
-2
-4
x-2 -1 1 2 3 4 5
y12
10
8
6
4
2
-2
-4
A) 0 B) 6 C) does not exist D) -1
9) limx→0
f(x)
x-4 -3 -2 -1 1 2 3 4
y4
3
2
1
-1
-2
-3
-4
x-4 -3 -2 -1 1 2 3 4
y4
3
2
1
-1
-2
-3
-4
http://youtu.be/5Im6jcaoiAwA) 1 B) ∞ C) does not exist D) -1
3
10) limx→0
f(x)
x-4 -3 -2 -1 1 2 3 4
y4
3
2
1
-1
-2
-3
-4
x-4 -3 -2 -1 1 2 3 4
y4
3
2
1
-1
-2
-3
-4
A) does not exist B) ∞ C) -1 D) 1
11) limx→0
f(x)
x-4 -3 -2 -1 1 2 3 4
y
4
3
2
1
-1
-2
-3
-4
x-4 -3 -2 -1 1 2 3 4
y
4
3
2
1
-1
-2
-3
-4
A) -2 B) does not exist C) 0 D) 2
4
12) limx→0
f(x)
x-4 -3 -2 -1 1 2 3 4
y
4
3
2
1
-1
-2
-3
-4
x-4 -3 -2 -1 1 2 3 4
y
4
3
2
1
-1
-2
-3
-4
A) 0 B) -2 C) 1 D) does not exist
13) limx→0
f(x)
x-4 -3 -2 -1 1 2 3 4
y
4
3
2
1
-1
-2
-3
-4
x-4 -3 -2 -1 1 2 3 4
y
4
3
2
1
-1
-2
-3
-4
A) 2 B) -2 C) -1 D) does not exist
5
14) Find limx→(-1)-
f(x) and limx→(-1)+
f(x)
x-4 -2 2 4
y
2
-2
-4
-6
x-4 -2 2 4
y
2
-2
-4
-6
A) -5; -2 B) -7; -2 C) -7; -5 D) -2; -7
15) limx→0
f(x)
x-4 -3 -2 -1 1 2 3 4
y4
3
2
1
-1
-2
-3
-4
x-4 -3 -2 -1 1 2 3 4
y4
3
2
1
-1
-2
-3
-4
A) 0 B) does not exist C) -3 D) 3
6
Use the table of values of f to estimate the limit.
16) Let f(x) = x2 + 8x - 2, find limx→2
f(x). http://youtu.be/YHiZPKZ_gfM
x 1.9 1.99 1.999 2.001 2.01 2.1
f(x)
A)
x 1.9 1.99 1.999 2.001 2.01 2.1
f(x) 16.692 17.592 17.689 17.710 17.808 18.789 ; limit = 17.70
B)
x 1.9 1.99 1.999 2.001 2.01 2.1
f(x) 5.043 5.364 5.396 5.404 5.436 5.763 ; limit = 5.40
C)
x 1.9 1.99 1.999 2.001 2.01 2.1
f(x) 16.810 17.880 17.988 18.012 18.120 19.210 ; limit = 18.0
D)
x 1.9 1.99 1.999 2.001 2.01 2.1
f(x) 5.043 5.364 5.396 5.404 5.436 5.763 ; limit = ∞
17) Let f(x) = x - 4
x - 2, find lim
x→4f(x).
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x)
A)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x) 5.07736 5.09775 5.09978 5.10022 5.10225 5.12236 ; limit = 5.10
B)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745 ; limit = 1.20
C)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745 ; limit = ∞
D)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x) 3.97484 3.99750 3.99975 4.00025 4.00250 4.02485 ; limit = 4.0
7
18) Let f(x) = x2 - 5, find limx→0
f(x).
x -0.1 -0.01 -0.001 0.001 0.01 0.1
f(x)
A)
x -0.1 -0.01 -0.001 0.001 0.01 0.1
f(x) -2.9910 -2.9999 -3.0000 -3.0000 -2.9999 -2.9910 ; limit = -3.0
B)
x -0.1 -0.01 -0.001 0.001 0.01 0.1
f(x) -4.9900 -4.9999 -5.0000 -5.0000 -4.9999 -4.9900 ; limit = -5.0
C)
x -0.1 -0.01 -0.001 0.001 0.01 0.1
f(x) -1.4970 -1.4999 -1.5000 -1.5000 -1.4999 -1.4970 ; limit = -15.0
D)
x -0.1 -0.01 -0.001 0.001 0.01 0.1
f(x) -1.4970 -1.4999 -1.5000 -1.5000 -1.4999 -1.4970 ; limit = ∞
19) Let f(x) = x - 4
x2 - 5x + 4, find lim
x→4f(x).
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x)
A)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x) 0.2448 0.2344 0.2334 0.2332 0.2322 0.2226 ; limit = 0.2333
B)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x) -0.3448 -0.3344 -0.3334 -0.3332 -0.3322 -0.3226 ; limit = -0.3333
C)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x) 0.4448 0.4344 0.4334 0.4332 0.4322 0.4226 ; limit = 0.4333
D)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x) 0.3448 0.3344 0.3334 0.3332 0.3322 0.3226 ; limit = 0.3333
8
20) Let f(x) = x2 + 2x - 15
x2 - 2x - 3, find lim
x→3f(x).
x 2.9 2.99 2.999 3.001 3.01 3.1
f(x)
A)
x 2.9 2.99 2.999 3.001 3.01 3.1
f(x) 2.0256 2.0025 2.0003 1.9998 1.9975 1.9756 ; limit = 2
B)
x 2.9 2.99 2.999 3.001 3.01 3.1
f(x) 1.9256 1.9025 1.9003 1.8998 1.8975 1.8756 ; limit = 1.9
C)
x 2.9 2.99 2.999 3.001 3.01 3.1
f(x) 2.1256 2.1025 2.1003 2.0998 2.0975 2.0756 ; limit = 2.1
D)
x 2.9 2.99 2.999 3.001 3.01 3.1
f(x) -0.9048 -0.9900 -0.9990 -1.0010 -1.0101 -1.1053 ; limit = -1
21) Let f(x) = sin(5x)
x, find lim
x→0f(x).
x -0.1 -0.01 -0.001 0.001 0.01 0.1
f(x) 4.99791693 4.99791693
A) limit does not exist B) limit = 0 C) limit = 4.5 D) limit = 5
22) Let f(θ) = cos (5θ)
θ, find lim
θ→0f(θ).
x -0.1 -0.01 -0.001 0.001 0.01 0.1
f(θ) -8.7758256 8.7758256
A) limit = 5 B) limit does not exist C) limit = 0 D) limit = 8.7758256
Find the limit.
23) limx→18
2
A) 18 B) 3 2 C) 2 D) 2
24) limx→-9
(6x - 10)
A) -64 B) 44 C) 64 D) -44
Give an appropriate answer.
25) Let limx → -3
f(x) = 1 and limx → -3
g(x) = -10. Find limx → -3
[f(x) - g(x)]. http://youtu.be/E9fF0kbgShg
A) -9 B) -3 C) 11 D) 1
9
26) Let limx → 7
f(x) = 4 and limx → 7
g(x) = 5. Find limx → 7
[f(x) · g(x)].
A) 9 B) 5 C) 7 D) 20
27) Let limx → -8
f(x) = -7 and limx → -8
g(x) = -4. Find limx → -8
f(x)
g(x).
A)4
7B)
7
4C) -8 D) -3
Find the limit.
28) limx→2
(x3 + 5x2 - 7x + 1)
A) 15 B) does not exist C) 0 D) 29
29) limx→0
x3 - 6x + 8
x - 2
A) 4 B) 0 C) -4 D) Does not exist
30) limx→0
1 + x - 1
x http://youtu.be/tzUSjMBzBuk
A) 1/4 B) Does not exist C) 0 D) 1/2
Determine the limit by sketching an appropriate graph.
31) limx → 6-
f(x), where f(x) = -2x - 6 for x < 6
4x - 5 for x ≥ 6 http://youtu.be/MwTbTOgRSNg
For more info on graphing piecewise defined function in the calculator, seehttp://mathbits.com/mathbits/tisection/precalculus/piecewise.htm
A) -18 B) -4 C) -5 D) 19
32) limx → 6+
f(x), where f(x) = -4x - 3 for x < 6
5x - 2 for x ≥ 6
A) -27 B) 28 C) -2 D) -1
33) limx → 4+
f(x), where f(x) = x2 + 4 for x ≠ 4
0 for x = 4
A) 0 B) 16 C) 12 D) 20
34) lim
x → 5-f(x), where f(x) =
16 - x2 0 ≤ x < 4
4 4 ≤ x < 5
5 x = 5
A) 0 B) 4 C) Does not exist D) 5
35) lim
x → -7+f(x), where f(x) =
x -7 ≤ x < 0, or 0 < x ≤ 3
1 x = 0
0 x < -7 or x > 3
A) Does not exist B) 7 C) -7 D) -0
10
Find the limit, if it exists.
36) limx→0
x3 + 12x2 - 5x
5x
A) 5 B) 0 C) Does not exist D) -1
37) limx→1
x4 - 1
x - 1
A) 4 B) Does not exist C) 0 D) 2
38) limx → 10
x2 - 100
x - 10
A) 20 B) Does not exist C) 1 D) 10
39) limx → -9
x2 + 17x + 72
x + 9 http://youtu.be/G-sDRUmTbX0
A) 306 B) -1 C) 17 D) Does not exist
40) limx → 5
x2 + 3x - 40
x - 5
A) 13 B) Does not exist C) 0 D) 3
41) limx → 2
x2 + 2x - 8
x2 - 4
A) - 1
2B)
3
2C) Does not exist D) 0
42) limx → 3
x2 - 9
x2 - 7x + 12
A) - 6 B) 0 C) - 3 D) Does not exist
43) limh → 0
(x + h)3 - x3
h
A) 3x2 + 3xh + h2 B) 3x2 C) Does not exist D) 0
44) limx → 9
9 - x
9 - x http://youtu.be/1sF82HMZz4o
A) 0 B) -1 C) Does not exist D) 1
11
Compute the values of f(x) and use them to determine the indicated limit.
45) If f(x) = x4 - 1
x - 1, find lim
x → 1 f(x).
x 0.9 0.99 0.999 1.001 1.01 1.1
f(x)
A)
x 0.9 0.99 0.999 1.001 1.01 1.1
f(x) 1.032 1.182 1.198 1.201 1.218 1.392 ; limit = ∞
B)
x 0.9 0.99 0.999 1.001 1.01 1.1
f(x) 1.032 1.182 1.198 1.201 1.218 1.392 ; limit = 1.210
C)
x 0.9 0.99 0.999 1.001 1.01 1.1
f(x) 4.595 5.046 5.095 5.105 5.154 5.677 ; limit = 5.10
D)
x 0.9 0.99 0.999 1.001 1.01 1.1
f(x) 3.439 3.940 3.994 4.006 4.060 4.641 ; limit = 4.0
46) If f(x) = x3 - 6x + 8
x - 2, find lim
x → 0 f(x).
x -0.1 -0.01 -0.001 0.001 0.01 0.1
f(x)
A)
x -0.1 -0.01 -0.001 0.001 0.01 0.1
f(x) -1.22843 -1.20298 -1.20030 -1.19970 -1.19699 -1.16858 ; limit = ∞
B)
x -0.1 -0.01 -0.001 0.001 0.01 0.1
f(x) -4.09476 -4.00995 -4.00100 -3.99900 -3.98995 -3.89526 ; limit = -4.0
C)
x -0.1 -0.01 -0.001 0.001 0.01 0.1
f(x) -2.18529 -2.10895 -2.10090 -2.99910 -2.09096 -2.00574 ; limit = -2.10
D)
x -0.1 -0.01 -0.001 0.001 0.01 0.1
f(x) -1.22843 -1.20298 -1.20030 -1.19970 -1.19699 -1.16858 ; limit = -1.20
12
47) If f(x) = x - 4
x - 2, find lim
x → 4 f(x). http://youtu.be/q0OgTtfx0uk
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x)
A)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x) 3.97484 3.99750 3.99975 4.00025 4.00250 4.02485 ; limit = 4.0
B)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745 ; limit = 1.20
C)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745 ; limit = ∞
D)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x) 5.07736 5.09775 5.09978 5.10022 5.10225 5.12236 ; limit = 5.10
48) If f(x) = x - 2, find limx → 4
f(x).
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x)
A)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x) 3.9000 2.9000 1.9000 2.0000 3.0000 4.0000 ; limit = ∞
B)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x) 3.9000 2.9000 1.9000 2.0000 3.0000 4.0000 ; limit = 1.95
C)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x) 1.47736 1.49775 1.49977 1.50022 1.50225 1.52236 ; limit = 1.50
D)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x) -0.02516 -0.00250 -0.00025 0.00025 0.00250 0.02485 ; limit = 0
13
For the function f whose graph is given, determine the limit.
49) Find limx→-1-
f(x) and limx→-1+
f(x).
x-6 -4 -2 2 4 6
y
4
2
-2
-4
-6
-8
-10
x-6 -4 -2 2 4 6
y
4
2
-2
-4
-6
-8
-10
A) -5; -2 B) -7; -5 C) -2; -7 D) -7; -2
50) Find limx→2-
f(x) and limx→2+
f(x).
x-4 -3 -2 -1 1 2 3 4
y
8
6
4
2
-2
-4
-6
-8
x-4 -3 -2 -1 1 2 3 4
y
8
6
4
2
-2
-4
-6
-8
A) does not exist; does not exist B) 1; 1
C) -4; 3 D) 3; -4
14
51) Find limx→2+
f(x).
x-2 -1 1 2 3 4 5 6 7
f(x)
8
7
6
5
4
3
2
1
-1x-2 -1 1 2 3 4 5 6 7
f(x)
8
7
6
5
4
3
2
1
-1
A) 5 B) 1.3 C) -1 D) 4
52) Find limx→1-
f(x).
x-5 -4 -3 -2 -1 1 2 3 4 5
f(x)5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
f(x)5
4
3
2
1
-1
-2
-3
-4
-5
A) 2 B)1
2C) -1 D) does not exist
15
53) Find limx→1+
f(x).
x-5 -4 -3 -2 -1 1 2 3 4 5
f(x)5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
f(x)5
4
3
2
1
-1
-2
-3
-4
-5
A) 31
2B) does not exist C) 3 D) 4
54) Find limx→0
f(x).
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
A) 1 B) does not exist C) -1 D) 0
16
55) Find limx→0
f(x).
x-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8
y87654321
-1-2-3-4-5-6-7-8
x-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8
y87654321
-1-2-3-4-5-6-7-8
A) 0 B) does not exist C) 2 D) -2
56) Find limx→-1
f(x).
x-4 -2 2 4
y
4
2
-2
-4
A
x-4 -2 2 4
y
4
2
-2
-4
A
A) -1 B) - 2
3C) does not exist D)
2
3
17
57) Find limx→∞
f(x).
x-5 -4 -3 -2 -1 1 2 3 4 5
f(x)5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
f(x)5
4
3
2
1
-1
-2
-3
-4
-5
A) -2 B) ∞ C) 0 D) does not exist
Note: The previous problem brings up an issue: We can say limx→a
f(x) = ∞ indicating that as
x approaches the value a, the f(x) values get larger without bound. But the limit in this caseis still technically "does not exist" since infinity is not a number. So think of it as that infinityis just a special way the limit is failing to exist and writing infinity gives us more information(as opposed to failing because the limit from the left does not equal the limit from the right).
Find the limit.
58) limx→-2
1
x + 2 http://youtu.be/nTfty6Q4wNw
A) Does not exist B) -∞ C) 1/2 D) ∞
59) limx → 7+
1
(x - 7)2
A) -∞ B) 0 C) ∞ D) -1
60) limx → -4-
5
x2 - 16
A) ∞ B) -1 C) 0 D) -∞
61) limx→(π/2)+
tan x http://youtu.be/VyGOMJ-O0l4
A) 0 B) -∞ C) 1 D) ∞
62) limx→(-π/2)-
sec x
A) -∞ B) 1 C) ∞ D) 0
18
63) limx → 1-
x2 - 4x + 3
x3 - x
A) - 1 B) 0 C) ∞ D) -∞
64) limx → 4+
x2 - 6x + 8
x3 - 4x
A) 0 B) Does not exist C) ∞ D) -∞
65) limx → 3+
2
x2 - 9
A) 0 B) -∞ C) 1 D) ∞
66) limx→∞
7
x - 1 http://youtu.be/110FiQsvSfI
A) -1 B) 1 C) -8 D) 6
67) limx→-∞
5
5 - (9/x2)
A) - 5
4B) -∞ C) 1 D) 5
68) limx→-∞
-5 + (5/x)
7 - (1/x2)
A)5
7B) -
5
7C) ∞ D) -∞
69) limx→∞
x2 - 7x + 9
x3 - 6x2 + 14
A)9
14B) 0 C) 1 D) ∞
70) limx→-∞
-4x2 - 3x + 6
-18x2 - 4x + 9
A)2
9B) ∞ C) 1 D)
2
3
71) limx→-∞
cos 4x
x
A) 1 B) 0 C) 4 D) -∞
19
72) limx→ - ∞
2x3 + 4x2
x - 6x2
A) -∞ B) ∞ C) - 2
3D) 2
73) limx→∞
9x3 - 5x2 + 3x
-x3 - 2x + 6
A) ∞ B)3
2C) -9 D) 9
74) limx→∞
2x + 1
15x - 7
A)2
15B) 0 C) ∞ D) -
1
7
ESSAY
75) Definition of Continuity: Go tohttp://www.youtube.com/watch?v=hlorAjS0xWE&feature=topics and watch and take notes
on everything, and know for a test.
76) In the book, Continuity is discussed in section 1.8 - Read and take notes
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find all points where the function is discontinuous.
77)
A) x = 4 B) None C) x = 2 D) x = 4, x = 2
78)
A) x = -2, x = 1 B) x = 1 C) None D) x = -2
20
79)
A) x = 0, x = 2 B) x = 2 C) x = -2, x = 0, x = 2 D) x = -2, x = 0
80)
A) x = -2, x = 6 B) x = 6 C) None D) x = -2
81)
A) None B) x = 1, x = 4, x = 5 C) x = 4 D) x = 1, x = 5
82)
A) x = 1 B) x = 0, x = 1 C) x = 0 D) None
83)
A) None B) x = 3 C) x = 0 D) x = 0, x = 3
84)
A) None B) x = -2 C) x = 2 D) x = -2, x = 2
21
85)
A) x = -2, x = 0, x = 2 B) None C) x = 0 D) x = -2, x = 2
Provide an appropriate response.
86) Is f continuous at f(1), that is, at x = 1?
f(x) =
-x2 + 1,
4x,
-2,
-4x + 8
4,
-1 ≤ x < 0
0 < x < 1
x = 1
1 < x < 3
3 < x < 5
t-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
d6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
(1, -2)
t-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
d6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
(1, -2)
A) Yes B) No
87) Is f continuous at f(3), that is, at x = 3?
f(x) =
-x2 + 1,
3x,
-4,
-3x + 6
2,
-1 ≤ x < 0
0 < x < 1
x = 1
1 < x < 3
3 < x < 5
t-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
d6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
(1, -4)
t-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
d6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
(1, -4)
A) No B) Yes
22
88) Is f continuous at x = 0?
f(x) =
x3,
-3x,
5,
0,
-2 < x ≤ 0
0 ≤ x < 2
2 < x ≤ 4
x = 2
t-5 -4 -3 -2 -1 1 2 3 4 5
d10
8
6
4
2
-2
-4
-6
-8
-10
(2, 0)
t-5 -4 -3 -2 -1 1 2 3 4 5
d10
8
6
4
2
-2
-4
-6
-8
-10
(2, 0)
A) Yes B) No
89) Is f continuous at x = 4?
f(x) =
x3,
-2x,
6,
0,
-2 < x ≤ 0
0 ≤ x < 2
2 < x ≤ 4
x = 2
t-5 -4 -3 -2 -1 1 2 3 4 5
d10
8
6
4
2
-2
-4
-6
-8
-10
(2, 0)
t-5 -4 -3 -2 -1 1 2 3 4 5
d10
8
6
4
2
-2
-4
-6
-8
-10
(2, 0)
A) Yes B) No
90) Is f continuous on (-2, 4]? Note: A function is continuous on an interval if it is continuous atevery point in the interval
t-5 -4 -3 -2 -1 1 2 3 4 5
d10
8
6
4
2
-2
-4
-6
-8
-10
(2, 0)
t-5 -4 -3 -2 -1 1 2 3 4 5
d10
8
6
4
2
-2
-4
-6
-8
-10
(2, 0)
A) Yes B) No
23
ESSAY
91) Go to http://www.youtube.com/watch?v=Lgr-1ZKPnR4 Watch and take notes on the
Intermediate Value theorem. Use the Intermediate Value Theorem to prove that 6x3 + 5x2 + 4x + 7 = 0 has a
solution between -2 and -1.
92) Use the Intermediate Value Theorem to prove that 2x4 + 10x3 - 6x - 6 = 0 has a solution between -5 and -4.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find numbers a and b, or k, so that f is continuous at every point.
93)
f(x) =
-7,
ax + b,
21,
x < -4
-4 ≤ x ≤ 3
x > 3
http://youtu.be/6d0WlFdk2QoA) a = 4, b = 33 B) a = -7, b = 21 C) a = 4, b = 9 D) Impossible
94)
f(x) =
x2,
ax + b,
x + 6,
x < -5
-5 ≤ x ≤ -2
x > -2
A) a = 7, b = -10 B) a = -7, b = 10 C) a = -7, b = -10 D) Impossible
95)
f(x) =
3x + 8,
kx + 4,
if x < -1
if x ≥ -1
A) k = - 1 B) k = - 4 C) k = 7 D) k = 4
96)
f(x) =
x2,
x + k,
if x ≤ 4
if x > 4
A) k = 20 B) k = 12 C) k = -4 D) Impossible
97)
f(x) =
x2,
kx,
if x ≤ 2
if x > 2
A) k = 4 B) k = 2 C) k = 1
2D) Impossible
ESSAY
98) Formal Definition of a Limit (also called the "epsilon - delta definition"): Go to
http://www.youtube.com/watch?v=NAAhJ5ucYJg . Watch and take notes on the definition and
the proof of the problem shown in the video. In the book, the formal definition of the limit is in section 1.8
- Read!
24
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
99) Select the correct statement for the definition of the limit: limx→x0
f(x) = L
means that __________________
A) if given any number ε > 0, there exists a number δ > 0, such that for all x,
0 < x - x0 < ε implies f(x) - L < δ.
B) if given any number ε > 0, there exists a number δ > 0, such that for all x,
0 < x - x0 < δ implies f(x) - L < ε.
C) if given any number ε > 0, there exists a number δ > 0, such that for all x,
0 < x - x0 < ε implies f(x) - L > δ.
D) if given a number ε > 0, there exists a number δ > 0, such that for all x,
0 < x - x0 < δ implies f(x) - L > ε.
100) Identify the incorrect statements about limits.
I. The number L is the limit of f(x) as x approaches x0 if f(x) gets closer to L as x approaches x0.
II. The number L is the limit of f(x) as x approaches x0 if, for any ε > 0, there corresponds a δ > 0 such that
f(x) - L < ε whenever 0 < x - x0 < δ.
III. The number L is the limit of f(x) as x approaches x0 if, given any ε > 0, there exists a value of x for which
f(x) - L < ε.
A) I and II B) II and III C) I and III D) I, II, and III
Use the graph to find a δ > 0 such that for all x, 0 < x - x0 < δ ⇒ f(x) - L < ε. Write proofs also as shown in this video
for all linear function cases: http://youtu.be/oY-I0BD1Xg8
101)
x
y
0 x
y
0
y = x + 3
4.2
4
3.8
0.8 1 1.2
NOT TO SCALE
f(x) = x + 3
x0 = 1
L = 4
ε = 0.2
A) 3 B) 0.4 C) 0.2 D) 0.1
25
102)
x
y
0 x
y
0
y = 5x - 1
9.2
9
8.8
2
1.96 2.04
NOT TO SCALE
f(x) = 5x - 1
x0 = 2
L = 9
ε = 0.2
http://youtu.be/oY-I0BD1Xg8A) 0.08 B) 0.4 C) 7 D) 0.04
103)
f(x) = -5x - 1
x0 = -1
L = 4
ε = 0.2
x
y
0 x
y
0
y = -5x - 1
4.2
4
3.8
-1
-1.04 -0.96
NOT TO SCALE
A) 0.04 B) -0.04 C) 7 D) 0.4
26
104)
f(x) = -x + 2
x0 = -2
L = 4
ε = 0.2
x
y
0 x
y
0
y = -x + 2
4.2
4
3.8
-2.2 -2 -1.8
NOT TO SCALE
A) 0.4 B) -0.2 C) 6 D) 0.2
105)
x
y
0 x
y
0
y = 4
3x + 2
3.5
3.3
3.1
0.8 1 1.1
NOT TO SCALE
f(x) = 4
3x + 2
x0 = 1
L = 3.3
ε = 0.2
A) 0.3 B) 2.3 C) -0.3 D) 0.1
27
106)
f(x) = - 3
2x + 1
x0 = -2
L = 4
ε = 0.2
x
y
0 x
y
0
y = - 3
2x + 1
4.2
4
3.8
-2.1 -2 -1.9
NOT TO SCALE
A) -0.2 B) 0.1 C) 6 D) 0.2
107)
x
y
0 x
y
0
y = x
1.66
1.41
1.16
1.3575 2 2.7675
NOT TO SCALE
f(x) = x
x0 = 2
L = 2
ε = 1
4
A) 1.41 B) 0.6425 C) -0.59 D) 0.7675
28
108)
x
y
0 x
y
0
y = x - 2
1.25
1
0.75
2.5625 3 3.5625
NOT TO SCALE
f(x) = x - 2
x0 = 3
L = 1
ε = 1
4
A) 2 B) 0.5625 C) 0.4375 D) 1
109)
x
y
0 x
y
0
y = 2x2
3
2
1
1
0.71 1.22
NOT TO SCALE
f(x) = 2x2
x0 = 1
L = 2
ε = 1
A) 0.22 B) 0.51 C) 1 D) 0.29
29
110)
x
y
0 x
y
0
y = x2 - 1
9
8
7
3
2.83 3.16
NOT TO SCALE
f(x) = x2 - 1
x0 = 3
L = 8
ε = 1
A) 0.17 B) 5 C) 0.16 D) 0.33
ESSAY
111) Watch and take notes on this video which shows what happens when you have the wrong limit andyou try to prove it correct vs. when you have the correct limit: http://youtu.be/5iFwvdkc3OU
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
A function f(x), a point x0, the limit of f(x) as x approaches x0, and a positive number ε is given. Find a number δ > 0 such
that for all x, 0 < x - x0 < δ ⇒ f(x) - L < ε. (see video for f(x) = 3x2 below for the method to use to do these problems)
112) f(x) = 6x + 7, L = 19, x0 = 2, and ε = 0.01
A) 0.001667 B) 0.005 C) 0.008333 D) 0.003333
A function f(x), a point x0, the limit of f(x) as x approaches x0, and a positive number ε is given. Find a number δ > 0 such
that for all x, 0 < x - x0 < δ ⇒ f(x) - L < ε.
113) f(x) = 6x - 9, L = -3, x0 = 1, and ε = 0.01
A) 0.001667 B) 0.01 C) 0.003333 D) 0.000833
Find the limit L for the given function f, the point c, and the positive number ε. Then find a number δ > 0 such that, for all
x, 0 < |x - c|< δ ⇒ |f(x) - L| < ε.
114) f(x) = x2 -7x -18
x + 2, c = -2, ε = 0.02
A) L = 0; δ = 0.02 B) L = -11; δ = 0.02 C) L = -7; δ = 0.03 D) L = -18; δ = 0.03
A function f(x), a point c, the limit of f(x) as x approaches c, and a positive number ε is given. Find a number δ > 0 such
that for all x, 0 < x - c < δ ⇒ f(x) - L < ε.
115) f(x) = 3x2, L =12, c = 2, and ε = 0.1 http://youtu.be/Emw2prCwxEUA) δ = 0.00832 B) δ = 2.00832 C) δ = 1.99165 D) δ = 0.00835
116) f(x) = -8x + 6, L = -18, c = 3, and ε = 0.01
A) δ = 0.00125 B) δ = 0.0025 C) δ = 0.005 D) δ = -0.003333
30
117) f(x) = 9x - 10, L = 17, c = 3, and ε = 0.01
A) δ = 0.002222 B) δ = 0.000556 C) δ = 0.001111 D) δ = 0.003333
A function f(x), a point x0, the limit of f(x) as x approaches x0, and a positive number ε is given. Find a number δ > 0 such
that for all x, 0 < x - x0 < δ ⇒ f(x) - L < ε.
118) f(x) = -2x - 6, L = -12, x0 = 3, and ε = 0.01
A) 0.005 B) 0.01 C) 0.0025 D) -0.003333
119) f(x) = 3x2, L =48, x0 = 4, and ε = 0.1
A) 3.99583 B) 4.00416 C) 0.00417 D) 0.00416
ESSAY
Prove the limit statement
120) limx→1
(5x - 4) = 1 Review this video seen earlier for how to write a proof of a limit statement:
http://www.youtube.com/watch?v=NAAhJ5ucYJg
121) limx→6
2x2 - 7x- 30
x - 6 = 17
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use your calculator to plot the function near the point x0 being approached. From your plot guess the value of the
limit.
122) limx→25
x - 5
x - 25 http://youtu.be/D6l-uv_Q56I
A)1
5B) 0 C)
1
10D) 5
123) limx→0
81 + x - 81 - x
x
A) 9 B) 0 C)1
9D)
1
18
124) limx→0
25 - x - 5
x
A) - 1
10B) 10 C)
1
10D) 5
125) limx→ -1
x2 - 1
x2 + 3 - 2
A)1
4B) 2 C) 1 D) 4
31
ESSAY
Solve the problem BY HAND with ALGEBRA
126) Evaluate lim
x → 5 x2 - 25
x - 5. http://youtu.be/beP0b18Fktg
Solve the problem.
127) Evaluate lim
x → 4
x - 4
x - 2. http://youtu.be/beP0b18Fktg (same as previous video)
128) Evaluate lim
x → 9
x - 3
x - 9.
129) Evaluate lim
x → 3
2x - 6
x2 - 4x + 3.
130) Evaluate lim
x → 3
x2 - x - 6
x - 3.
131) Find the trigonometric limit: lim
θ→0
θ2
cosθ. Use calculator methods
132) Find the trigonometric limit: lim
θ→0
tanθ
θ. Use calculator methods
Note: The following problems pertain to sums, differences, products, and quotients oflimits.
133) Go tohttp://tutorial.math.lamar.edu/Classes/CalcI/LimitProofs.aspx#Extras_Limit_LimitProp Write down the nine properties of limits listed there. The proofs of each property are also shown. Know the
proofs for properties 7, 1, and 2 for a test.
http://tutorial.math.lamar.edu/Classes/CalcI/LimitProofs.aspx#Extras_Limit_LimitProp
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
134) Provide a short sentence that summarizes the general limit principle given by the formal notation
limx→a
[f(x) ± g(x)] = limx→a
f(x) ± limx→a
g(x) = L ± M, given that limx→a
f(x) = L and limx→a
g(x) = M.
A) The limit of a sum or a difference is the sum or the difference of the limits.
B) The limit of a sum or a difference is the sum or the difference of the functions.
C) The sum or the difference of two functions is continuous.
D) The sum or the difference of two functions is the sum of two limits.
32
135) The statement "the limit of a constant times a function is the constant times the limit" follows from a
combination of two fundamental limit principles. What are they?
A) The limit of a product is the product of the limits, and a constant is continuous.
B) The limit of a constant is the constant, and the limit of a product is the product of the limits.
C) The limit of a function is a constant times a limit, and the limit of a constant is the constant.
D) The limit of a product is the product of the limits, and the limit of a quotient is the quotient of the limits.
Give an appropriate answer.
136) Let limx → 2
f(x) = -5 and limx → 2
g(x) = -8. Find limx → 2
[f(x) - g(x)].
A) -5 B) -13 C) 3 D) 2
137) Let limx → -9
f(x) = -1 and limx → -9
g(x) = 10. Find limx → -9
[f(x) · g(x)].
A) -9 B) 9 C) 10 D) -10
138) Let limx → -6
f(x) = 9 and limx → -6
g(x) = 8. Find limx → -6
f(x)
g(x).
A)9
8B) 1 C) -6 D)
8
9
33
Answer KeyTestname: LIMITS AND CONTINUITY WORKSHEET
1)
2)
3) see answers in the book. For #2 discuss in class.
4) see the applet
5) B
6) B
7) D
8) A
9) C
10) A
11) A
12) B
13) B
14) D
15) B
16) C
17) D
18) B
19) D
20) A
21) D
22) B
23) C
24) A
25) C
26) D
27) B
28) A
29) C
30) D
31) A
32) B
33) D
34) B
35) C
36) D
37) A
38) A
39) B
40) A
41) B
42) A
43) B
44) C
45) D
46) B
47) A
48) D
49) C
50) D
34
Answer KeyTestname: LIMITS AND CONTINUITY WORKSHEET
51) D
52) A
53) C
54) B
55) B
56) D
57) B
58) A
59) C
60) A
61) B
62) C
63) A
64) A
65) D
66) A
67) C
68) B
69) B
70) A
71) B
72) B
73) C
74) A
75)
76)
77) A
78) B
79) C
80) B
81) A
82) D
83) B
84) D
85) C
86) B
87) A
88) A
89) A
90) B
91) Let f(x) = 6x3 + 5x2 + 4x + 7 and let y0 = 0. f(-2) = -29 and f(-1) = 2. Since f is continuous on [-2, -1] and since y0 = 0 is
between f(-2) and f(-1), by the Intermediate Value Theorem, there exists a c in the interval (-2 , -1) with the property
that f(c) = 0. Such a c is a solution to the equation 6x3 + 5x2 + 4x + 7 = 0.
92) Let f(x) = 2x4 + 10x3 - 6x - 6 and let y0 = 0. f(-5) = 24 and f(-4) = -110. Since f is continuous on [-5, -4] and since y0 =
0 is between f(-5) and f(-4), by the Intermediate Value Theorem, there exists a c in the interval (-5, -4) with the
property that f(c) = 0. Such a c is a solution to the equation 2x4 + 10x3 - 6x - 6 = 0.
93) C
94) C
95) A
35
Answer KeyTestname: LIMITS AND CONTINUITY WORKSHEET
96) B
97) B
98)
99) B
100) C
101) C
102) D
103) A
104) D
105) D
106) B
107) B
108) C
109) A
110) C
111)
112) A
113) A
114) B
115) A
116) A
117) C
118) A
119) D
120)
Let ε > 0 be given. Choose δ = ε/5. Then 0 < x - 1 < δ implies that
(5x - 4) - 1 = 5x - 5
= 5(x - 1)
= 5 x - 1 < 5δ = ε
Thus, 0 < x - 1 < δ implies that (5x - 4) - 1 < ε
121) Let ε > 0 be given. Choose δ = ε/2. Then 0 < x - 6 < δ implies that
2x2 - 7x- 30
x - 6 - 17 =
(x - 6)(2x + 5)
x - 6 - 17
= (2x + 5) - 17 for x ≠ 6
= 2x - 12
= 2(x - 6)
= 2 x - 6 < 2δ = ε
Thus, 0 < x - 6 < δ implies that 2x2 - 7x- 30
x - 6 - 17 < ε
122) C
123) C
124) A
125) D
126) 10
127) 4
128)1
6
129) 1
130) 5
36
Answer KeyTestname: LIMITS AND CONTINUITY WORKSHEET
131) 0
132) 1
133)
134) A
135) B
136) C
137) D
138) A
37
Derivatives Worksheet 1 - Understanding the Derivative
Show all work on your paper as described in class. Video links are included throughout for instruction on how to do
the various types of problems. Important: Work the problems to match everything that was shown in the videos. For
example: Suppose a video shows 3 ways to do a problem, (such as algebraically, graphically, and numerically), then
your work should show these 3 ways also. That is , each video is a model for the work I want to see on your paper.
ESSAY.
1) A car is traveling on a straight track at a constant velocity moving away from the starting line. At time t =
0 hours the car is at point A, 5 miles from the starting line, and at time t = 2 hours the car is at point B, 25
miles away from the starting line. http://youtu.be/LKn61xIo4lsa) Express the distance of the car from the starting line, d(t), as a function of time, t.
b) Compute the average velocity of the car on its journey from point A to point B using a difference
quotient.
c) Graph d(t) and illustrate the average velocity calculation with auxiliary lines on the graph.
2) A car is traveling on a straight track at a constant velocity moving away from the starting line. At time t =
0 hours the car is at point A, 10 miles away from the starting line, and at time t = 6 hours the car is at point
B, 100 miles away from the starting line.
a) Express the distance of the car from the starting line, d(t), as a function of time, t.
b) Compute the average velocity of the car on its journey from point A to point B using a difference
quotient.
c) Graph d(t) and illustrate the average velocity calculation with auxiliary lines on the graph.
3) A toy car is traveling on a straight track at a constant velocity moving away from the starting line. At time
t = 3 seconds the car is at point A, 7 ft. from the starting line, and at time t = 6 seconds the car is at point B,
13 ft. away from the starting line.
a) Express the distance of the car from the starting line, d(t), as a function of time, t.
b) Compute the average velocity of the car on its journey from point A to point B using a difference
quotient.
c) Graph d(t) and illustrate the average velocity calculation with auxiliary lines on the graph.
4) A person is walking on a straight track such that his/her position, d(t), in feet from the starting line as a
function of time, t seconds, is given by d(t) = t2 . http://youtu.be/1z2q_dT7oQ8a) Compute the average velocity of the person over the time interval from t = 1 sec. to t = 3 sec. using a
difference quotient.
b) Graph d(t) and illustrate the average velocity calculation with auxiliary lines on the graph.
c) What constant velocity could the person have walked over the same time interval to arrive at the same
position at the end of the time interval?
1
5) A person is walking on a straight track such that his/her position, d(t), in feet from the starting line as a
function of time, t seconds, is given by d(t) = -t2 + 4t .
a) Where is the person at t = 0 sec.?
b) Where is the person at t = 4 sec.?
c) Describe the journey of the person on the track and include information about the maximum distance
from the starting line the person reaches.
d) Compute the average velocity of the person over the time interval from t = 0 sec. to t = 4 sec. using a
difference quotient.
e) Graph d(t) and illustrate the average velocity calculation with auxiliary lines on the graph.
f) What constant velocity could the person have walked over the same time interval to arrive at the same
position at the end of the time interval? Describe the journey of the person in that case.
FROM THE BOOK
6) Do section 2.1 # 1
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds.
NOTE: "Displacement" means the change in position (end position - start position), not the total distance
traversed while going forward or back on the way from the start to the end.
7) s = 2t2 + 4t + 2, 0 ≤ t ≤ 2
Find the body's displacement and average velocity for the given time interval.
http://youtu.be/WX5WNbR1WFcA) 20 m, 10 m/sec B) 16 m, 16 m/sec C) 16 m, 8 m/sec D) 12 m, 12 m/sec
8) s = 4t - t2, 0 ≤ t ≤ 4
Find the body's displacement and average velocity for the given time interval.
A) 32 m, -4 m/sec B) 32 m, 8 m/sec C) 0 m, 0 m/sec D) -32 m, -8 m/sec
9) s = - t3 + 7t2 - 7t, 0 ≤ t ≤ 7
Find the body's displacement and average velocity for the given time interval.
A) 637 m, 91 m/sec B) -49 m, -14 m/sec C) 49 m, 7 m/sec D) -49 m, -7 m/sec
ESSAY.
10) A ball is thrown upward. The height of the ball above the ground is given in the table below:
t (sec) 0 1 2 3 4 5 6
h (ft.) 6 90 142 162 150 106 30
You want to estimate the exact (instantaneous) velocity of the ball at t = 2 seconds. Use a 1 second time
interval to compute the average velocity of the ball over the time interval from t = 2 to 3 seconds, and then
use that as the estimate of the instantaneous) velocity at t = 2 seconds. http://youtu.be/Gke1L4mVARI
2
11) A ball is thrown upward. The height of the ball above the ground is given in the table below:
t (sec) 0 1 2 3 4 5 6
h (ft.) 6 90 142 162 150 106 30
You want to estimate the exact (instantaneous) velocity of the ball at t = 4 seconds. Use a 1 second time
interval to compute the average velocity of the ball over the time interval from t = 4 to 5 seconds, and then
use that as the estimate of the instantaneous velocity at t = 4 seconds.
12) A ball is thrown upward. The height of the ball above the ground is given in the table below:
t (sec) 0 1 2 3 4 5 6
h (ft.) 6 90 142 162 150 106 30
You are tired of estimation, so this time you want to find the instantaneous velocity of the ball at t = 1
seconds. Try using a 0 second time interval to compute the average velocity of the ball over the time
interval from t = 4 to 4 seconds (not a misprint!), and then use that as the answer to the instantaneous
velocity at t = 4 seconds. What is the trouble? !!
13) Go to http://en.wikipedia.org/wiki/History_of_calculus , read, and write a brief paragraph
on what Isaac Newton and Gottried Leibniz had to do with Calculus, including dates. Also, from the
Newton section, include info about Newton's idea about "x = x + o". (After your study of derivatives in
this class you should go back and read this wikipedia page again - you'll be amazed that you can actually
understand it!!)
14) Walking Man: A man is walking on a straight path with a position function d(t) where d is his postion on
the path in feet and t is time in seconds. The following video shows this motion:
Go to: http://youtu.be/rWOv5Eb7ojw This video is called "Walking Man" . Hit Full Screen on the
video.
a) Graph d(t) on the interval of 0 to 5 seconds on your own paper. ( d(t) is given on the video.)
b) Play the video and observe the motion of the man and the graph below.
c) Describe how the position of the man changes over time.
d) Describe how the velocity and acceleration change over time.
e) What is the relationship between the graphic of the man walking on the path and the graph below?
f) Playback the walk and stop it at 4 seconds. Write down the position and velocity values showing.
g) Compute d(t) for t = 4 sec. using the given d(t) function. Verify that your calculated d(4) value is the
same as the position indicated by the video.
h) Mark the cooresponding point on your d(t) graph.
15) Refering to the preceding Walking Man problem, now you want to use the d(t) function to estimate the
instantaneous velocity of the man at t = 3 sec. and see if your calculations match up with the velocity value
indicated on the video.
a) Using the d(t) function, calculate the average velocity of the man on a very small time interval from t =
3 to t = 3.01 sec. (that's a .01 second time interval).
b) Using the d(t) function, find the average velocity of the man on a very small time interval from t = 3 to t
= 3.0001 sec. (that's a .0001 second time interval !).
c) What does the video show as the instantaneous velocity at t = 3 seconds? Are you getting there with
your average velocity calculations? How could you get even better results?
3
16) Refering to the Walking Man problem, now you want to use d(t) to estimate the instantaneous velocity of
the man at t = 2 sec. and see if your calculations match up with the velocity from the video . Run the video,
stop at t = 2 sec. and see what velocity the video shows .
a) Using the d(t) function, find the average velocity of the man on a very small time interval from t = 2 to t
= 2.001 sec. (that's a .001 second time interval).
b) Using the d(t) function, find the average velocity of the man on a very small time interval from t = 2 to t
= 2.00001 sec. (that's a .00001 second time interval !).
c) What does the video show as the instantaneous velocity at t = 2 seconds? Are you getting there with
your average velocity calculations? How could you get even better results?
17) Refering to the Walking Man problem, and using your knowledge of limits:
a) Use numerical methods (calculator) to find limh→0
d(3+h)-d(3)
h , that is , find the limit of the average
velocity from t = 3 to 3+h seconds as h goes to 0 sec. This will be the true instantaneous velocity at t = 3
seconds.
b) Run the video and stop it at 3 seconds and see if velocity value shown verifies with your answer .
18) Refering to the Walking Man problem, and using your knowledge of limits:
Use algebraic methods to find limh→0
d(3+h)-d(3)
h , that is , find the limit of the average velocity from t = 3
to 3+h seconds as h goes to 0 sec. This will be the true instantaneous velocity at t = 3 seconds.
http://youtu.be/N0_kHV0kd0Q
Note: limh→0
f(3+h)-f(3)
h , for a function f(x) in general, is called the Derivative of f(x) at the point x = 3.
And the general Point Definition of the Derivative of f(x) at the point x = a is limh→0
f(a+h)-f(a)
h . It gives
the instantaneous rate of change of the function f(x) at the point x = a. If the function models distance as a
function of time, as in the Walking Man problem, then this derivative is the instantaneous velocity at time t =
a units.
19) Refering to the Walking Man problem, and using your knowledge of limits:
Use algebraic methods to find limh→0
d(t+h)-d(t)
h , that is , find the limit of the average velocity from t to
t+h seconds as h goes to 0 sec. This will yield an expression that will give the instantaneous velocity at
ANY time t seconds we choose!! http://youtu.be/K_sYh2kjaZQNote: This answer is the expression for the instantaneous velocity function, v(t) .
4
20) Based on your answer to the previous problem, find the following velocity values and then run the
Walking Man video to see if your calculations check out.
a) v(0), (the velocity at t = 0 sec.)
b) v(1)
c) v(2.5)
d) v(4)
e) v(5)
20f) Take notes and know for a test: The general Limit Definition of the Derivative of f at x
is limh→0
f(x+h)-f(x)
h . This limit produces a function that gives the instantaneous rate of
change of the function f at any point x (any point for which the limit exists). If the functionmodels distance as a function of time, as in the Walking Man problem, then this derivative isthe instantaneous velocity function, v(t) and gives us the velocity of an object at any time twe choose.
NOTE: The more consise notation for the derivative of f at x, or the derivative of f(x), is f'(x)and is read "f prime of x". Thus we have:
f '(x) = the derivative of f(x) = rate of change of f at x = limh→0
f(x+h)-f(x)
h . See defintions in
the book sections 2.1 to 2.3 Know these defintions for a test!
Solve the problem.
21) Find f'(x) using the limit definition of the derivative given f(x) = x2 + x + 1 .
http://youtu.be/m1JtDxOpCmk
22) Find g'(z) using the limit definition of the derivative given g(z) = z2 - 5z + 7, .
23) Find f'(x) using the limit definition of the derivative given f(x) = 3x2 -4x + 1 .
24) Find f'(x) using the limit definition of the derivative given f(x) = x3 + 2x
25) Use the limit definition of the derivative to find f'(x) given f(x) = 3x + 2 .
26) Use the limit definition of the derivative to find g'(x) given g(x) = 4x3 - 1 .
27) A ball thrown vertically upward at time t = 0 (seconds) has height y(t) = 96t - 16t2 (ft) at time t.
a) Graph y(t)
b) Find the velocity function, v(t) , using the limit definition of the derivative.
c) What is the velocity of the ball at t = 2 sec.?
d) What is the velocity of the ball at t = 4 sec. ? What does the negative sign mean?
e) When is the velocity function equal to 0? What physical events are occuring at each of the times when
the velocity is 0?
http://youtu.be/nNvPsEqUDhI
5
28) A ball thrown vertically upward at time t = 0 (seconds) has height h(t) = 64t - 16t2 (ft) at time t. The
velocity function is v(t) = 64 - 32t
a) Graph h(t)
b) Use 2nd calc max to find the time, tmax , when the ball reaches its maximum height, and illustrate the
point ( tmax , hmax) on your graph.
c) Using the velocity function, v(t) , evaluate v( tmax) .
d) Do you think the ball indeed comes to a stop instantaneously before coming back down? Why?
29) Walking Man 2: A man is walking on a straight path with a position function d(t) where d is his postion
on the path in feet and t is time in seconds. The following video shows this motion:
Go to: http://youtu.be/dAJE_cbXQWg This video is called "Walking Man 2" (it has a different d(t)
function than Walking Man did . Hit Full Screen on the video.
a) Graph d(t) on the interval of 0 to 5 seconds on your own paper. ( d(t) is given on the video.)
b) Play the video and observe the motion of the man and the graph below it.
c) Describe how the position of the man changes over time.
d) Describe how the velocity and acceleration change over time, particularly, when are they positive and
when are they negative and why?
e) What is the relationship between the graphic of the man walking on the path and the graph below?
f) Playback the walk and stop it at 2 seconds. Write down the velocity value showing. What special is
happening at t = 2 seconds? at 4 seconds? Illustrate these points on your graph.
g) What special is happening at t = 3 seconds? Illustrate that point on your graph.
h) Mark the cooresponding point on your d(t) graph.
NOTE: We know the Derivative of a function f(x) is the (instantaneous) rate of change of f at x. There is
another interpretation: The Derivative of f(x) gives the slope of the tangent line to the graphof f(x) at the point x. The following problems help you understand this concept.
30) A ball thrown vertically upward at time t = 0 (seconds) has height f(t) = 96t - 16t2 (ft) at time t.
a) Go to http://youtu.be/BpZnYjZxWG4 . Hit Full Screen on the video. For best quality select 720p.
Watch the video several times and observe the graph and what the graph is showing.
b) Below is a picture of the video at the beginning, when the time increment is h = 2 sec. Use the
information on the picture to find the equation of the secant line shown and the equation of the tangent
line.
c) Watch the video live and pause the video when h = 1 sec. and find the equation of the secant line shown.
d) Watch the video live and pause the video when h = .2 sec. and find the equation of the secant line
shown.
e) Describe the relationship between the average rate of change of the function, the secant line, the
derivative of f(t), and the tangent line.
6
NOTE: The slope of the tangent line to the graph of a function f(x) at a point x0 is the
derivative of f(x) at x0 , f'(x0 ) .
7
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the slope of the line tangent to the graph at the given point. Use the limit definition of the derivative, or
numerical methods (small h = .001), to find the slope.
31) y = 8x + 7, x = -2
A) m = -16 B) m = -8 C) m = 16 D) m = 8
32) y = x2 + 6x - 8, x = -2 http://youtu.be/k2e0CU8fcTwA) m = -4 B) m = 2 C) m = -6 D) m = -8
Find an equation for the tangent line to the curve at the given point. AND graph the curve and the tangent line on
your TI and copy to your paper. It better look exactly tangent or something is wrong!
33) y = x2 - 2, (-3, 7) Hint: To save yourself some time, use y' = 2x
A) y = -3x - 11 B) y = -6x - 11 C) y = -6x - 22 D) y = -6x - 20
34) h(x) = t3 - 16t + 3, (4, 3) Hint: To save yourself some time, use h'(x) = 3t2 - 16
A) y = 3 B) y = 35t - 125 C) y = 32t - 125 D) y = 32t + 3
ESSAY.
35) For the graph of f(x) below, estimate f'(x) at the given x values by using the grids and the fact that f'(x) =
slope of the tangent line at the point x. http://youtu.be/jj3hwpe85Z8
x-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11
y5
4
3
2
1
-1
-2
-3
-4
-5
x-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11
y5
4
3
2
1
-1
-2
-3
-4
-5
a) x = -2
b) x = -1
c) x = -0.4
d) x = 0
e) x = 1
f) x = 2
g) x = 3.5
h) x = 5
8
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Estimate the slope of the curve at the indicated point.
36) There are no grids for these so you just pick the only value that is possible among the choices.
A) 1 B) Undefined C) 0 D) -1
37)
A) 1 B) 0 C) -1 D) Undefined
38)
A) Undefined B) 1 C) -1 D) 0
39)
A) 1 B) Undefined C) 0 D) -1
40)
A) Undefined B) 0 C) 1 D) -1
9
41)
A) -2 B)1
2C) 2 D) -
1
2
42)
A) - 1
2B) -2 C) 2 D)
1
2
43)
A)1
20B) -2 C) -
1
20D) 2
FROM THE BOOK
44) Do section 2.1 #3, 5, 7, 13, 23; section 2.2 #1, 3, 5, 7, 9, 11, 15, 19; For 25 and 27 estimate the derivative with
numerical methods (small h = .001); 39, 43, 45, 47
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
10
45) The graph of y = f(x) in the accompanying figure is made of line segments joined end to end. Graph the
derivative of f. http://youtu.be/Rb4-iTE3ays
x
y
(-5, 0)
(-3, 2)
(0, -1)
(3, 5) (6, 5)
x
y
(-5, 0)
(-3, 2)
(0, -1)
(3, 5) (6, 5)
x
y
x
y
A)
x-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
B)
x-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
C)
x-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
D)
x-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
11
46) Use the following information to graph the function f over the closed interval [-5, 6].
i) The graph of f is made of closed line segments joined end to end.
ii) The graph starts at the point (-5, 1).
iii) The derivative of f is the step function in the figure shown here.
So the step function shown is the derivative of f(x), you draw the graph of the original f(x)
x-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
y
x
y
A)
x
y
(-5, 1)
(-3, 5)
(0, 2)
(3, 5)
(6, -1)
x
y
(-5, 1)
(-3, 5)
(0, 2)
(3, 5)
(6, -1)
B)
x
y
(-5, 1)
(-3, 6)
(0, 2)
(3, 5)
(6, 0) x
y
(-5, 1)
(-3, 6)
(0, 2)
(3, 5)
(6, 0)
C)
x
y
(-5, 1)
(-3, 5)
(0, 1)
(3, 5)
(6, -1)
x
y
(-5, 1)
(-3, 5)
(0, 1)
(3, 5)
(6, -1)
D)
x
y
(-5, 1)
(-3, 5)
(0, 2)
(3, 6)
(6, 0) x
y
(-5, 1)
(-3, 5)
(0, 2)
(3, 6)
(6, 0)
12
47) Match the graph of the function to the graph of its derivative. http://youtu.be/TnYLrTMqG2Y
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
A)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
B)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
C)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
D)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
13
48) Match the graph of the function with the graph of its derivative.
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
A)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
B)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
C)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
D)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
14
49) Match the graph of the function to the graph of its derivative.
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
A)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
B)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
C)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
D)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
15
The graph of a function is given. Choose the answer that represents the graph of its derivative.
50)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
A)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
B)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
C)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
D)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
16
51)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
A)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
B)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
C)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
D)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
17
52)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
A)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
B)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
C)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
D)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
18
53)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
A)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
B)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
C)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
D)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
19
54)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
A)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
B)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
C)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
D)
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
x-15 -10 -5 5 10 15
y15
10
5
-5
-10
-15
FROM THE BOOK
55) Do section 2.3 #1-11 odd, 17 - 35 odd, 41 http://youtu.be/QDHwSUCYwO0
ESSAY.
56) Read and take notes on "An Alternate Notation for the Derivative" in section 2.4 in the book. It
explains that dy
dx is another notation for f '(x), the derivative.
FROM THE BOOK
57) Do section 2.4 #1-5 odd, 11, 15, 21
20
ESSAY.
58) Go to http://youtu.be/fE5EXbNxLEQ and take notes on the "2nd Derivative" video and
pay attention to the relationship between f" and the concavity of f and inflection points.
FROM THE BOOK
59) Read section 2.5 and take notes
60) Do section 2.5 #1-13 odd, 17- 21 odd, 27, 29. Video for 11: http://youtu.be/zZXW8T8EKksVideo for 21: http://youtu.be/V_GkVXyLXD4
21
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
61) Select an appropriate graph of a twice-differentiable function y = f(x) that passes through the points
(- 2,1) , - 6
3,5
9, (0,0),
6
3,5
9 and ( 2,1), and whose first two derivatives have the following sign patterns.
y′ : + - + -
- 2 0 2
y′′ :
- + -
- 6
3
6
3
http://youtu.be/bJWayQoZFZEA)
x-3 -2 -1 1 2 3
y16
12
8
4
-4
-8
-12
-16
x-3 -2 -1 1 2 3
y16
12
8
4
-4
-8
-12
-16
B)
x-3 -2 -1 1 2 3
y2
1.5
1
0.5
-0.5
-1
-1.5
-2
x-3 -2 -1 1 2 3
y2
1.5
1
0.5
-0.5
-1
-1.5
-2
C)
x-3 -2 -1 1 2 3
y16
12
8
4
-4
-8
-12
-16
x-3 -2 -1 1 2 3
y16
12
8
4
-4
-8
-12
-16
D)
x-3 -2 -1 1 2 3
y4
3
2
1
-1
-2
-3
-4
x-3 -2 -1 1 2 3
y4
3
2
1
-1
-2
-3
-4
ESSAY.
62) Sketch a continuous curve y = f(x) with the following properties:
f(2) = 3; f′′(x) > 0 for x > 4; and f′′(x) < 0 for x < 4 .
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
63) The graph below shows the first derivative of a function y = f(x). Select a possible graph of f that passes
through the point P. http://youtu.be/dIbjDzq3WWU
f′
22
x
y
P
x
y
P
A)
x
y
x
y
B)
x
y
x
y
C)
x
y
x
y
D)
x
y
x
y
23
64) The graph below shows the first derivative of a function y = f(x). Select a possible graph f that passes
through the point P.
f′
x
y
P
x
y
P
A)
x
y
x
y
B)
x
y
x
y
C)
x
y
x
y
D)
x
y
x
y
24
65) The graph below shows the first derivative of a function y = f(x). Select a possible graph f that passes
through the point P.
f′
x
y
P
x
y
P
A)
x
y
x
y
B)
x
y
x
y
C)
x
y
x
y
D)
x
y
x
y
25
66) Using the following properties of a twice-differentiable function y = f(x), select a possible graph of f.
x y Derivatives
x < 2 y′ > 0,y′′ < 0
-2 11 y′ = 0,y′′ < 0
-2 < x < 0 y′ < 0,y′′ < 0
0 -5 y′ < 0,y′′ = 0
0 < x < 2 y′ < 0,y′′ > 0
2 -21 y′ = 0,y′′ > 0
x > 2 y′ > 0,y′′ > 0
A)
x-4 -3 -2 -1 1 2 3 4
y24
16
8
-8
-16
-24
x-4 -3 -2 -1 1 2 3 4
y24
16
8
-8
-16
-24
B)
x-4 -3 -2 -1 1 2 3 4
y24
16
8
-8
-16
-24
x-4 -3 -2 -1 1 2 3 4
y24
16
8
-8
-16
-24
C)
x-4 -3 -2 -1 1 2 3 4
y24
16
8
-8
-16
-24
x-4 -3 -2 -1 1 2 3 4
y24
16
8
-8
-16
-24
D)
x-4 -3 -2 -1 1 2 3 4
y24
16
8
-8
-16
-24
x-4 -3 -2 -1 1 2 3 4
y24
16
8
-8
-16
-24
FROM THE BOOK
67) Read section 2.6 Differentiability and take notes. Know the proof of "A differentiablefunction is continuous" for a test.
68) Do section 2.6 #1-9 odd. Video for #5: http://youtu.be/csRj7wNmXzQ
26
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Given the graph of f, find any values of x at which f ′ is not defined.
69)
A) x = 0 B) x = 1 C) x = -1 D) x = 2
70)
A) x = -3, 0, 3 B) x = -2, 2 C) x = -3, 3 D) x = -2, 0, 2
71)
A) x = 2 B) x = -2, 0, 2 C) x = 0 D) x = -2, 2
72)
A) x = 1 B) x = 2 C) x = 0 D) x = 0, 1, 2
73)
A) x = 2 B) x = 1, 2, 3
C) x = 1, 3 D) Defined for all values of x
27
74)
A) x = -1, 0, 1 B) x = 0
C) x = -1, 1 D) Defined for all values of x
75)
A) x = 0 B) x = -2, 2
C) x = -2, 0, 2 D) Defined for all values of x
76)
A) x = 3 B) x = 0, 3
C) x = 0 D) Defined for all values of x
NOTE: Now we get some rules that give us short-cuts to differentiation ! We learn how to find the
derivative of a constant, a constant multiple, sum and difference, powers, and putting it all together,
polynomials. (Other functions come later.) Using these short-cuts without understanding what the
derivative really is is just useless rote memorization - That's why we spent all this time on understanding the
derivative. So now that you understand derivatives you won't have to use the limit definition of the
derivative to find derivative functions from now on! Well, if you run into a function displayed as data (as is
frequently the case in the real world), then you can always fall back on your understanding to solve the
problem. This way you have every situation covered.
FROM THE BOOK
77) Go to http://youtu.be/QVNz5LVNKjA and take notes on the this video about the
derivative of a constant, a constant multiple, sum and difference, and powers. Know fora test. Then read section 3.1 and take notes from the book. AND know (for a test) theproofs of Theorem 3.1, 3.2, and the Power rule (also explained in the video). I want youto know why these short-cut rules work!!
28
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the derivative using the short-cut rules.
78) y = 14 - 14x2 http://youtu.be/nl0NGmZHd5gA) 14 - 28x B) 14 - 14x C) -28x D) -28
Find the derivative.
79) y = 5 - 4x3
A) -12x B) -8x2 C) 5 - 12x2 D) -12x2
80) y = 2x4 - 8x3 + 9 http://youtu.be/3cbiT9J_-3IA) 8x3 - 24x2 - 7 B) 4x3 + 3x2 C) 8x3 - 24x2 D) 4x3 + 3x2 - 7
81) s = 3t2 + 4t + 1
A) 3t2 + 4 B) 6t2 + 4 C) 3t + 4 D) 6t + 4
82) y = 7x-2 + 14x3 + 10x
A) -14x-3 + 42x2 + 10 B) -14x-1 + 42x2
C) -14x-3 + 42x2 D) -14x-1 + 42x2 + 10
83) y = 5x2 + 12x + 4x-3
A) 5x + 4x-4 B) 10x + 12 + 12x-4 C) 10x + 12 - 12x-4 D) 10x - 12x-4
84) w = z-5 - 1
z http://youtu.be/3cbiT9J_-3I
A) -5z-6 - 1
z2B) z-6 +
1
z2C) 5z-6 -
1
z2D) -5z-6 +
1
z2
85) r = 4
s3 -
5
s
A)12
s4 -
5
s2B)
4
s4 -
5
s2C) -
12
s2 +
5
s2D) -
12
s4 +
5
s2
86) y = 1
7x2 +
1
5x
A) - 2
7x3 -
1
5x2B) -
1
7x3 +
1
5x2C) -
2
7x -
1
5x2D)
2
7x3 +
1
5x2
Find the second derivative. Graph the derivative and graph the 2nd derivative. Illustrate that the 2nd derivative is
giving you the slope values of the derivative function.
87) y = 6x2 + 3x - 4 http://youtu.be/zXkPGvHEJKQA) 12 B) 6 C) 0 D) 12x + 3
29
Find the second derivative.
88) y = 7x4 - 4x2 + 8
A) 28x2 - 8x B) 28x2 - 8 C) 84x2 - 8x D) 84x2 - 8
89) s = 2t3
3 + 2
A) 4t + 2 B) 2t2 C) 4t D) 2t
90) y = 5x2 + 11x + 5x-3
A) 10 + 60x-1 B) 10 - 60x-5 C) 10 + 60x-5 D) 10x + 11 - 15x-4
91) r = 3
s3 -
5
s
A)3
s5 -
5
s3B)
36
s5 -
10
s3C) -
9
s4 +
5
s2D)
36
s5 +
10
s3
FROM THE BOOK
92) Do section 3.1 #1-47 odd ; video of selected problems: http://youtu.be/cctPUbcBGNk
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
93) The position of a body moving on a coordinate line is given by s = t2 - 6t + 7, with s in meters and t in
seconds. When, if ever, during the interval 0 ≤ t ≤ 6 does the body change direction?
http://youtu.be/E0TsxC0htLwA) t = 6 sec B) t = 12 sec
C) t = 3 sec D) no change in direction
****Videos will have been added after this point to the online
version of these notes by the time you get here!****
94) At time t, the position of a body moving along the s-axis is s = t3 - 15t2 + 72t m. Find the body's
acceleration each time the velocity is zero.
A) a(6) = 6 m/sec2, a(4) = -6 m/sec2 B) a(12) = 72 m/sec2, a(8) = 12 m/sec2
C) a(6) = 0 m/sec2, a(4) = 0 m/sec2 D) a(6) = -6 m/sec2, a(4) = 6 m/sec2
95) At time t ≥ 0, the velocity of a body moving along the s-axis is v = t2 - 9t + 8. When is the body moving
backward?
A) 1 < t < 8 B) 0 ≤ t < 1 C) t > 8 D) 0 ≤ t < 8
30
96) At time t ≥ 0, the velocity of a body moving along the s-axis is v = t2 - 11t + 10. When is the body's
velocity increasing?
A) t < 5.5 B) t < 10 C) t > 5.5 D) t > 10
97) A ball dropped from the top of a building has a height of s = 256 - 16t2 meters after t seconds. How long
does it take the ball to reach the ground? What is the ball's velocity at the moment of impact?
A) 16 sec, -512 m/sec B) 4 sec, 128 m/sec C) 8 sec, -64 m/sec D) 4 sec, -128 m/sec
98) A rock is thrown vertically upward from the surface of an airless planet. It reaches a height of
s = 120t - 2t2 meters in t seconds. How high does the rock go? How long does it take the rock to reach its
highest point?
A) 3570 m, 30 sec B) 3600 m, 60 sec C) 1800 m, 30 sec D) 7080 m, 60 sec
99) The area A = πr2 of a circular oil spill changes with the radius. At what rate does the area change with
respect to the radius when r = 9 ft?
A) 18 ft2/ft B) 18π ft2/ft C) 81π ft2/ft D) 9π ft2/ft
100) The driver of a car traveling at 60 ft/sec suddenly applies the brakes. The position of the car is s = 60t - 3t2,
t seconds after the driver applies the brakes. How far does the car go before coming to a stop?
A) 600 ft B) 10 ft C) 1200 ft D) 300 ft
101) The size of a population of mice after t months is P = 100(1 + 0.2t + 0.02t2). Find the growth rate at t = 13
months.
A) 172 mice/month B) 72 mice/month C) 144 mice/month D) 36 mice/month
102) The number of gallons of water in a swimming pool t minutes after the pool has started to drain is
Q(t) = 50(20 - t)2. How fast is the water running out at the end of 11 minutes?
A) 900 gal/min B) 450 gal/min C) 2025 gal/min D) 4050 gal/min
FROM THE BOOK
103) Do section 3.1 # 55 (graph the function and the tangent line together - see that your tangent line is the
perfect fit to the curve), 57, 65, 67, 69, 71
NOTE: Now we learn how to find the derivative of exponential functions, f(x) = ax
104) Read section 3.2 "Derivatives of Exponential Functions and the Number e" ; "A formula
for the Dervative of ax ", and take notes.
105) Do section 3.2 # 1-25 odd, 39, 41, 43, 45
NOTE: Now we learn short-cut rules for finding the derivative of the product and quotient
of two functions.
31
106) Go to http://math.ucsd.edu/~wgarner/math20a/prodrule.htm and take notes on the
derivation of the Product Rule (know all the steps for a test!) Note: The way the rule looksin the book is a bit different but it is equivalent. We'll use the rule as stated on this site.
107) Go to http://math.ucsd.edu/~wgarner/math20a/quotrule.htm and take notes on the
derivation of the Quotient Rule (know all the steps for a test!) Note: The way the rulelooks in the book is a bit different but it is equivalent. We'll use the rule as stated on thissite.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find y ′.
108) y = (2x - 3)(2x + 1)
A) 8x - 8 B) 8x - 2 C) 4x - 4 D) 8x - 4
109) y = (3x - 3)(5x3 - x2 + 1)
A) 15x3 + 18x2 - 54x + 3 B) 60x3 - 18x2 + 54x + 3
C) 45x3 + 54x2 - 18x + 3 D) 60x3 - 54x2 + 6x + 3
110) y = (x2 - 3x + 2)(2x3 - x2 + 5)
A) 10x4 - 28x3 + 21x2 + 6x - 15 B) 2x4 - 24x3 + 21x2 + 6x - 15
C) 10x4 - 24x3 + 21x2 + 6x - 15 D) 2x4 - 28x3 + 21x2 + 6x - 15
111) y = 1
x2 + 7 x2 -
1
x2 + 7
A)4
x5 + 14x B) -
4
x5 - 14x C) -
1
x5 + 14x D)
4
x3 + 14x
Find the derivative of the function.
112) y = x2 - 3x + 2
x7 - 2
A) y ′ = -5x8 + 18x7 - 14x6 - 3x + 6
(x7 - 2)2B) y ′ =
-5x8 + 19x7 - 14x6 - 4x + 6
(x7 - 2)2
C) y ′ = -5x8 + 18x7 - 13x6 - 4x + 6
(x7 - 2)2D) y ′ =
-5x8 + 18x7 - 14x6 - 4x + 6
(x7 - 2)2
113) y = x3
x - 1
A) y ′ = -2x3 + 3x2
(x - 1)2B) y ′ =
2x3 + 3x2
(x - 1)2C) y ′ =
2x3 - 3x2
(x - 1)2D) y ′ =
-2x3 - 3x2
(x - 1)2
32
114) g(x) = x2 + 5
x2 + 6x
A) g ′(x) = 2x3 - 5x2 - 30x
x2(x + 6)2B) g ′(x) =
4x3 + 18x2 + 10x + 30
x2(x + 6)2
C) g ′(x) = 6x2 - 10x - 30
x2(x + 6)2D) g ′(x) =
x4 + 6x3 + 5x2 + 30x
x2(x + 6)2
115) y = x2 + 8x + 3
x
A) y ′ = 3x2 + 8x - 3
xB) y ′ =
3x2 + 8x - 3
2x3/2C) y ′ =
2x + 8
xD) y ′ =
2x + 8
2x3/2
116) y = x2 + 2x - 2
x2 - 2x + 2
A) y ′ = 4x2 + 8x
(x2 - 2x + 2)2B) y ′ =
4x2 - 8x
(x2 - 2x + 2)2C) y ′ =
-4x2 + 8x
(x2 - 2x + 2)2D) y ′ =
-4x2 - 8x
(x2 - 2x + 2)2
117) r = θ - 4
θ + 4
A) r ′ = 4
θ(θ + 4)2B) r ′ =
4
θ + 4
C) r ′ = 8
(θ + 4) θ2 - 16D) r ′ = -
4
θ(θ + 4)2
118) z = 3x2ex
A)dz
dx = 6xex + 3x2ex B)
dz
dx = 3xex + 6x2ex C)
dz
dx = 3xex + 3x2ex D)
dz
dx = 6xex - 3x2ex
FROM THE BOOK
119) Do section 3.3 #1-33 odd, 41, 45, 47
NOTE: Now we learn the Chain Rule for finding the derivative of the Composition of twofunctions. First a review of Composition.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the requested composition or operation.
120) f(x) = 3x + 13, g(x) = 4x - 1
Find (f ∘ g)(x).
A) 12x + 16 B) 12x + 51 C) 12x + 10 D) 12x + 12
33
121) f(x) = x + 7, g(x) = 8x - 11
Find (f ∘ g)(x).
A) 8 x + 7 - 11 B) 2 2x + 1 C) 8 x - 4 D) 2 2x - 1
122) f(x) = 4x2 + 2x + 8, g(x) = 2x - 5
Find (g ∘ f)(x).
A) 4x2 + 4x + 11 B) 8x2 + 4x + 21 C) 8x2 + 4x + 11 D) 4x2 + 2x + 3
123) f(x) = x - 3
7, g(x) = 7x + 3
Find (g ∘ f)(x).
A) x B) 7x + 18 C) x - 3
7D) x + 6
Perform the requested composition or operation.
124) Find (f ∘ g)(8) when f(x) = -4x - 5 and g(x) = -4x2 + 5x + 3.
A) -34 B) -5658 C) -49 D) 847
125) Find (g ∘ f)(4) when f(x) = 9x - 6 and g(x) = -8x2 - 2x + 9.
A) -1149 B) -7251 C) -291 D) -285
Use the graphs to evaluate the expression.
126) (g ∘ f)(-3)
y = f(x) y = g(x)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
A) 1.5 B) 0.5 C) -2.5 D) -1
34
127) (f ∘ g)(-2)
y = f(x) y = g(x)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
A) -2 B) -1.5 C) 0 D) 1
128) (f ∘ g)(0)
y = f(x) y = g(x)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
(1, -1)
(2, 2)
(0, -2)
(-1, -1)
(-2, 2)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
(1, -1)
(2, 2)
(0, -2)
(-1, -1)
(-2, 2)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
(1, 2)
(2, 4)
(-1, -2)
(-2, -4)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
(1, 2)
(2, 4)
(-1, -2)
(-2, -4)
A) -1 B) 1 C) -2 D) -3
35
129) (g ∘ f)(-1)
y = f(x) y = g(x)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
(1, -1)
(2, 2)
(0, -2)
(-1, -1)
(-2, 2)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
(1, -1)
(2, 2)
(0, -2)
(-1, -1)
(-2, 2)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
(1, 2)
(2, 4)
(-1, -2)
(-2, -4)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
(1, 2)
(2, 4)
(-1, -2)
(-2, -4)
A) -2 B) -3 C) -1 D) -4
130) g(f(4))
y = f(x) y = g(x)
x-5 5
y5
-5
x-5 5
y5
-5
x-5 5
y5
-5
x-5 5
y5
-5
A) -2 B) 8 C) 4 D) 6
Use the tables to evaluate the expression if possible.
131) Find (g ∘ f)(11).
x 7 9 11
f(x) 49 81 121
x 81 121 64 49 100
g(x) 67 107 50 35 86
A) 89 B) 38 C) 67 D) 107
132) Find (f ∘ g)(3).
x 11 7 3 5
f(x) 22 14 6 10
x 5 3 6 4
g(x) 9 5 11 7
A) 10 B) 3 C) 14 D) 5
36
133) Find (g ∘ f)(10).
x 10 13 11 22
f(x) 11 20 55 57
x 12 22 10 11
g(x) 23 19 22 21
A) 55 B) 21 C) 10 D) 19
134) Find (f ∘ f)(8).
x 8 11 9 7
f(x) 9 8 43 45
x 10 8 11 9
g(x) 19 15 21 17
A) 43 B) 8 C) 17 D) 21
135) Find (g ∘ g)(3).
x 3 6 4 8
f(x) 4 6 13 15
x 5 8 3 4
g(x) 9 5 8 7
A) 7 B) 13 C) 15 D) 5
ESSAY.
136) Go to http://math.ucsd.edu/~wgarner/math20a/chainrule.htm and take notes on the
derivation of the Chain Rule (know all the steps for a test!)
FROM THE BOOK
137) Read section 3.4 Chain Rule and take notes
138) Do section 3.4 #1-43 odd, 59, 79
NOTE: Now we'll learn about the derivatives of Trigonometric functions
37
ESSAY.
139) Consider the graph of f(x) = sin(x) below.
x-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11
y5
4
3
2
1
-1
-2
-3
-4
-5
x-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11
y5
4
3
2
1
-1
-2
-3
-4
-5
a) Make a table of estimated slope values of the graph of f(x) for x values incrementing by 0.5 on the
interval from x = 0 to 6.5
b) Graph the derivative of f(x) using your data from (a). What classic trig function does it look like?
140) Another way to estimate the graph of the derivative of f(x) = sin(x) (or any function) is to graph the
difference quotient with h = .001, or other small h value, in your TI as follows:
y1 = (f(x+.001)-f(x))/.001 . So in the case of f(x) = sin(x) that would be: y1= (sin(x+.001)-sin(x))/.001
Use this method to graph the derivative of f(x) = sin(x) and report which trig function results.
141) Here is a limit that is critical to the proof that the derivative of sin(x) is cos(x).
Consider limh→0
sin(h)
h
a) Find the limit by graphical methods, that is, graph y1= sin(x)/x (remember, your calculator must be in
radian mode, not degrees) on x: -1 to 1 ; y: 0 to 2 and copy to your paper. Use 2nd calc value and put in x
= .001 . What is your estimate for the limit using this "h" value of .001?
b) Evaluate using smaller h values. What do you conclude the limit is?
142) Here is another limit that is critical to the proof that the derivative of sin(x) is cos(x).
Consider limh→0
1-cos(h)
h
a) Find the limit by graphical methods, that is, graph y1= (1-cos(x))/x (remember, your calculator must
be in radian mode, not degrees) on x: -1 to 1 ; y: -1 to 1 and copy to your paper. Use 2nd calc value and
put in x = .001 . What is your estimate for the limit using this "h" value of .001?
b) Evaluate using smaller h values. What do you conclude the limit is?
143) Go to http://www-math.mit.edu/~djk/18_01/chapter05/proof02.html and copy down the proof that the
derivative of sin(x) is cos(x), and know the proof for a test.
38
FROM THE BOOK
144) Find the informal justification of d
dx(sin(x)) = cos(x) in section 3.5, write it down with an illustration and
know for a test.
145) Find the informal justification of d
dx(cos(x)) = -sin(x) in section 3.5, write it down and know for a test.
ESSAY.
146) Estimate the graph of the derivative of f(x) = cos(x) using the difference quotient with h = .001 in your TI
as follows: y1= (cos(x+.001)-cos(x))/.001
Use this method to graph the derivative of f(x) = cos(x) and report which trig function results.
FROM THE BOOK
147) Find the proof of d
dx(tan(x)) =
1
cos2 x in section 3.5, write it down and know for a test.
ESSAY.
148) Estimate the graph of the derivative of f(x) = tan(x) using the difference quotient with h = .001 in your TI
as follows: y1= (tan(x+.001)-tan(x))/.001 Then put in y2 = 1/cos(x)^2 . What do you notice?
FROM THE BOOK
149) Do section 3.5 #3 - 41 odd, 45, 47, 49
ESSAY.
Solve the problem.
150) A rocket is launched vertically and is tracked by a radar station located on the ground 6 mi from the
launch pad. Suppose that the elevation angle θ of the line of sight to the rocket is increasing at 5° per
second when θ = 60°. What is the velocity of the rocket at this instant?
151) Write an equation of the line that is tangent to the curve y = x sin x at the point P with x-coordinate π
2 .
FROM THE BOOK
152) In section 3.6, copy down and know the proofs of
a) d
dx(x
1
2 ) = 1
2x
-1
2 b) d
dx(ln(x)) =
1
x c)
d
dx(ax) = ln(a)ax
d) d
dx(arctan(x)) =
1
1+x2 e)
d
dx(arcsin(x)) =
1
1-x2
39
153) Do section 3.6 #1-33 odd, 43
154) Read section 3.7 Implicit Functions and take notes
ESSAY.
155) Go to http://www.flashandmath.com/mathlets/calc/implicit/implicit.html Enter in the
implicit equation x2 + y2 = 4 into the applet (you would type in x^2+y^2=4 according to the info on the
site on how to enter syntax).
a) Copy the graph to your paper.
b) Use implicit differentiation to find dy
dx
c) Use dy
dx to find the slope of the curve at the point ( 2 , 2 )
d) Find the equation of the tangent line to the curve at ( 2 , 2 ) and graph the line on your graph.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use implicit differentiation to find dy/dx.
156) 2xy - y2 = 1
A)y
x - yB)
y
y - xC)
x
y - xD)
x
x - y
157) x3 + 3x2y + y3 = 8
A) - x2 + 3xy
x2 + y2B) -
x2 + 2xy
x2 + y2C)
x2 + 2xy
x2 + y2D)
x2 + 3xy
x2 + y2
158) y x + 1 = 4
A)y
2(x + 1)B) -
y
2(x + 1)C) -
2y
x + 1D)
2y
x + 1
159) xy + x + y = x2y2
A)2xy2 + y + 1
-2x2y - x - 1B)
2xy2 - y
2x2y + xC)
2xy2 + y
2x2y - xD)
2xy2 - y - 1
-2x2y + x + 1
160) cos xy + x7 = y7
A)7x6 - x sin xy
7y6B)
7x6 + y sin xy
7y6 - x sin xyC)
7x6 - y sin xy
7y6 + x sin xyD)
7x6 + x sin xy
7y6
40
At the given point, find the slope of the curve or the line that is tangent to the curve, as requested. Then Go tohttp://www.flashandmath.com/mathlets/calc/implicit/implicit.html , enter in the implicit equation, copy
the graph to your paper and illustrate the slope or tangent line on your graph. (For a multiplication you have to type
for example: 9*x instead of 9x)
161) y6 + x3 = y2 + 9x, slope at (0, 1)
A) - 3
2B)
3
2C)
9
8D)
9
4
162) x6y6 = 64, slope at (2, 1)
A) -32 B) - 1
2C) 2 D) -
1
4
163) 5x2y - π cos y = 6π, slope at (1, π)
A) - π
2B) π C) 0 D) -2π
164) 6x2y - π cos y = 7π, tangent at (1, π)
A) y = - π
2x +
3π
2B) y = -2πx + π C) y = πx D) y = -2πx + 3π
165) y4 + x3 = y2 + 12x, tangent at (0, 1)
A) y = 6x + 1 B) y = - 3x - 1 C) y = - 2x D) y = 3x + 1
FROM THE BOOK
166) Do section 3.7 #1-17 odd, 19-25 ALL (for 19-25 go tohttp://www.flashandmath.com/mathlets/calc/implicit/implicit.html enter in the implicit
equation, copy the graph to your paper and illustrate the slope or tangent line requested on your
graph.)
Note: Local Linearity - Tangent Line Approximation. When you zoom in on a point
on the graph of f(x) where f(x) is differentiable, the graph appears very much straight like a line the closer
you zoom in - Local Linearity. This is because the requirement of the derivative existing at the point
prohibits that jagged edge look from occuring there, in other words, the graph is smooth at the point. Any
time you zoom in on something smoothly curved it starts looking more straight. Thus, using a tangent line at
the point to approximate values of the function near the point will be pretty good. That's what these next
problem focus on.
ESSAY.
Find the linearization L(x) (the tangent line) of f(x) at x = a AND evaluate the difference between L and f at a point
nearby. That is, find L(a + 0.1) - f(a + 0.1) to see how far off the linear approximation is from the actual function
value if we move 0.1 away to the right. Graph f(x) and L(x) to verify if your difference result makes sense.
167) f(x) = 4x2 - 4x + 4, a = -5
168) f(x) = 8x + 36, a = 0
41
169) f(x) = sin x, a = 0
170) f(x) = tan x, a = π
FROM THE BOOK
171) Do section 3.9 #1 - 7 odd, 11, 13, 19, 25
Note: Four Important Theorems: Mean Value Theorem; Rolle's Theorem; IncreasingFunction Theorem; Constant Function Theorem
172) In Section 3.10: Read and take notes about The Mean Value Theorem
ESSAY.
173) Go to http://math.ucsd.edu/~wgarner/math20a/mvt.htm and take notes on the proofs of the
Mean Value Theorem and Rolle's Theorem. Know the proofs for a test.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the value or values of c that satisfy the equation f(b) - f(a)
b - a = f′(c) in the conclusion of the Mean Value Theorem
for the function and interval.
174) f(x) = x2 + 3x + 4, [-3, 2]
A) -3, 2 B) 0, - 1
2C) -
1
2D) -
1
2,
1
2
175) f(x) = x + 112
x, [7, 16]
A) -4 7, 4 7 B) 4 7 C) 7, 16 D) 0, 4 7
176) f(x) = ln (x - 3), [4, 6] Round to the nearest thousandth.
A) ±4.820 B) 4.820 C) 5.820 D) 5.885
Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval.
177) f(x) = x1/3, -4,2
A) Yes B) No
178) g(x) = x3/4, 0,5
A) Yes B) No
179) s(t) = t(3 - t), -1,5
A) Yes B) No
42
ESSAY.
Answer the question.
180) A trucker handed in a ticket at a toll booth showing that in 2 hours he had covered 230 miles on a toll road
with speed limit 65 mph. The trucker was cited for speeding. Why?
181) A marathoner ran the 26.2 mile New York City Marathon in 2.7 hrs. Did the runner ever exceed a speed of
9 miles per hour?
Provide an appropriate response.
182) The function f(x) = -5x 0 ≤ x < 1
0 x = 1is zero at x = 0 and x = 1 and differentiable on (0, 1), but its derivative on
(0,1) is never zero. Does this example contradict Rolle's Theorem?
183) Decide if the statement is true or false. If false, explain.
The points (-1, -1) and (1, 1) lie on the graph of f(x) = 1
x. Therefore, the Mean Value Theorem says that
there exists some value x = c on (-1, 1) for which f′(x) = 1 - (-1)
1 - (-1) = 1.
FROM THE BOOK
184) In Section 3.10 read and take notes on The Increasing Function Theorem and The Constant Function
Theorem. Know the proofs for a test.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing.
185) f′(x) = (3 - x)(4 - x)
A) Decreasing on (-∞, 3); increasing on (4, ∞)
B) Decreasing on (3, 4); increasing on (-∞, 3) ∪ (4, ∞)
C) Decreasing on (-∞, -3) ∪ (-4, ∞); increasing on (-3, -4)
D) Decreasing on (-∞, 3) ∪ (4, ∞); increasing on (3, 4)
186) f′(x) = x1/3(x - 6)
A) Decreasing on (0, 6); increasing on (6, ∞)
B) Decreasing on (-∞, 0) ∪ (6, ∞); increasing on (0, 6)
C) Decreasing on (0, 6); increasing on (-∞, 0) ∪ (6, ∞)
D) Increasing on (0, ∞)
187) f′(x) = (x - 1) e-x
A) Increasing on (-∞, -1); decreasing on (-1, ∞) B) Decreasing on (-∞, 1); increasing on (1, ∞)
C) Decreasing on (-∞, -1); increasing on (-1, ∞) D) Increasing on (-∞, 1); decreasing on (1, ∞)
188) f′(x) = (x + 5)2 e-x
A) Decreasing on (-∞, -5); increasing on (-5, ∞)
B) Never decreasing; increasing on (-∞, -5) ∪ (-5, ∞)
C) Never decreasing; increasing on (-∞, 5) ∪ (5, ∞)
D) Never increasing; decreasing on (-∞, -5) ∪ (-5, ∞)
43
Answer KeyTestname: DERIVATIVES WORKSHEET 1 - UNDERSTANDING THE DERIVATIVE
1) d(t) = 10t+5, 10 mi/hr
2) d(t) = 15t+10, 15 mi/hr
3) d(t) = 2t+1, 2 ft/sec
4) 4 ft/sec, The person could walk at a 4 ft/sec velocity to arrive at the same end point at the same time.
5) a) at the start line, b) back at the start line, c) goes forward until the 2 sec. mark where he/she is 4 ft. from the
start, then turns around and goes back to the start line, d) 0 ft/sec. f) he/she could have walked 0 ft/sec and
ended up at the same place, that is, just stay at the start line and not move.
6) see answers in the book
7) C
8) C
9) D
10) 20 ft/sec (positive means the ball is going up)
11) -44 ft/sec (negative means the ball is going up) This shows the difference between velocity and speed. The ball
is going at a speed of 44 ft/sec in the downward direction. The velcity of the ball is -44 ft/sec)
12) (90-90)/(1-1) = 0/0 is undefined. That's the trouble , so we can quit and go home cuz there's no way to do it ...
OR IS THERE?!!
13) Paragraph essay
14) discuss in class
15) a) 6.01 ft/sec, b) 6.0001 ft/sec, c) discuss in class
16) a) 4.001 ft/sec, b) 4.00001 ft/sec, c) discuss in class
17) 6 ft/sec
18) 6 m/sec
19) 2t
20) a) 0, b) 2 c) 5 d) 8 e) 10
21) 2x + 1
22) a) 2z - 5
23) 6x-4
24) 3x2 + 2
25) f'(x) = 3
26) 12x2
27) b) v(t) = 96 - 32t, c) 32 ft/sec. , d) -32 ft/sec, the ball is going down.
28) b) tmax = 2 sec. , and hmax = 64 ft. , c) v(2) = 0 ft/sec , d) Yes, it does, because ... you answer why in class!
29) discuss in class
30) b) secant line: y = 32t + 48 , tangent line: y = 64t + 16 , c) y = 48t + 32 , d) y = 60.8t + 19.2 , e) check answer in class.
31) D
32) B
33) B
34) C
35) The following were obtained by plugging into the actual f '(x) function, so yours will be approximate: a) 5.6 b) 1.8
c) .096 d) -.8 e) -2.2 f) -2.4 g) -.45 h) 4.2
36) C
37) A
38) C
39) D
40) A
41) B
42) A
44
Answer KeyTestname: DERIVATIVES WORKSHEET 1 - UNDERSTANDING THE DERIVATIVE
43) D
44) see answers in the book
45) B
46) A
47) B
48) A
49) D
50) D
51) C
52) B
53) B
54) B
55) see answers in the book
56)
57) see answers in the book
58)
59) section 2.5
60) see answers in the book
61) B
62) Answers will vary. A general shape is indicated below:
63) A
64) C
65) B
66) D
67)
68) see answers in the book
69) A
70) B
71) C
72) C
73) D
45
Answer KeyTestname: DERIVATIVES WORKSHEET 1 - UNDERSTANDING THE DERIVATIVE
74) B
75) C
76) B
77)
78) C
79) D
80) C
81) D
82) A
83) C
84) D
85) D
86) A
87) A
88) D
89) C
90) C
91) B
92) see answers in the book
93) C
94) A
95) A
96) C
97) D
98) C
99) B
100) D
101) B
102) A
103) see answers in the book
104) Read and take notes
105) see answers in the book
106) http://math.ucsd.edu/~wgarner/math20a/prodrule.htm
107) http://math.ucsd.edu/~wgarner/math20a/quotrule.htm108) D
109) D
110) A
111) A
112) D
113) C
114) C
115) B
116) C
117) A
118) A
46
Answer KeyTestname: DERIVATIVES WORKSHEET 1 - UNDERSTANDING THE DERIVATIVE
119) see answers in the book. You can also get the answer to any derivative problem by going to
http://www.wolframalpha.com/ Type in: derivative of (type in your function) and click the"=" button.
120) C
121) D
122) C
123) A
124) D
125) B
126) B
127) C
128) C
129) A
130) C
131) D
132) A
133) B
134) A
135) D
136) http://math.ucsd.edu/~wgarner/math20a/chainrule.htm137) Read and take notes
138) see answers in the book. You can also see answers by going to http://www.wolframalpha.com/ Typein: derivative of (type in your function) and click the "=" button.
139) a) show table b) y = cos(x) .
140) y = cos(x)
141) a) .99999983 b) 1
142) a) .0005 b) 0
143) http://www-math.mit.edu/~djk/18_01/chapter05/proof02.html
144) see section 3.5
145) see section 3.5
146) y = -sin(x)
147) see section 3.5 Derivative of the Tangent Function
148) y = 1
cos2 x is the resulting graph, the estimated graph of the derivative of tan(x) and the graph of
1
cos2 x
coincide.
149) see answers in the book. You can also see more info at WolframAlpha: http://www.wolframalpha.com/Type in: derivative of (type in your function) and click the "=" button.
150)2π
3 mi/s ≈ 7540 mi/h
151) y = x
152) see section 3.6
153) see answers in the book. Also, you can check answers using the WolframAlpha site
154) read book and take notes
155) b) y' = -x
y c) slope = -1 d) y = -x + 2 2
156) B
47
Answer KeyTestname: DERIVATIVES WORKSHEET 1 - UNDERSTANDING THE DERIVATIVE
157) B
158) B
159) D
160) C
161) D
162) B
163) D
164) D
165) A
166) see answers in the book. Also you can check the graphs of the equations by entering in the equations into the
WolframAlpha site. (Type in like: graph y^6+x^3=y^2+9x ). You can even get the derivative answers by
typing in like: derivative y^6+x^3=y^2+9x
167) L(x) = -44x - 96 , L(a + 0.1) - f(a + 0.1) = -.04
168) L(x) = 2
3x + 6 , L(a + 0.1) - f(a + 0.1) = .00036631
169) L(x) = x , L(a + 0.1) - f(a + 0.1) = .00016658
170) L(x) = x - π , L(a + 0.1) - f(a + 0.1) = -.00033467
171) see answers in the book
172) Section 3.10 in the book
173) http://math.ucsd.edu/~wgarner/math20a/mvt.htm174) C
175) B
176) B
177) B
178) A
179) B
180) As the trucker's average speed was 115 mph, the Mean Value Theorem implies that the trucker must have been
going that speed at least once during the trip.
181) Yes, the Mean Value Theorem implies that the runner attained a speed of 9.7 mph, which was her average speed
throughout the marathon.
182) This example does not contradict Rolle's Theorem because the function f is not continuous on the closed interval
[0, 1]. In particular, f is not continuous at the right end point x = 1.
183) False. The function has a non-removable discontinuity at x = 0. The mean value theorem does not apply.
184) section 3.10
185) B
186) C
187) B
188) B
48
Derivatives Worksheet 2 - Using the Derivative
Show all work on your paper as described in class. Video links are included throughout for instruction on how to do
the various types of problems. Important: Work the problems to match everything that was shown in the videos. For
example: Suppose a video shows 3 ways to do a problem, (such as algebraically, graphically, and numerically), then
your work should show these 3 ways also. That is , each video is a model for the work I want to see on your paper.
More videos will have been added to the online version of this worksheet by the time you get here!
FROM THE BOOK
1) In Section 4.1 Read and take notes on: Local Maxima and Minima; How to detect a local maximum or
minimum (critical point and critical value); Local Extrema and Critical Points; The First-Derivative Test
for Local Maxima and Minima; The Second-Derivative Test for Local Maxima and Minima; Concavity and
Inflection Points
ESSAY.
2) Go to http://youtu.be/KqbsIJ3v4Y4 . "Increasing, Decreasing, and Tangent line slopes.avi" Use full
screen. Take notes on this video and answer the following:
a) What is this video showing? What does the blue segment represent?
b) (Fill in on your paper) When the blue line goes up from left to right then the sign of the slope is ________,
and the function is _____________. When the blue line goes down from left to right then the sign of the slope
is ________, and the function is _____________.
c) Stop the video at a = -1 . What is happening there?
3) Go to http://youtu.be/CWznFtlzbs4 "Derivative Increasing, Decreasing, and Concavity.avi" Use full
screen. Take notes on this video and answer the following:
a) What is this video showing?
b) Stop the video at a = 0. What is happening here?
4) Go to http://www.youtube.com/watch?v=aJuJOB6NTuc "Increasing/Decreasing , Local
Maximums/Minimums". Take notes on this video and answer the following:
a) Is the point A a local minimum? What is the derivative of the function there?
b) What kind of point is point F?
c) f '(6) is what value?
d) f '(7) is what value?
e) Exlain why point c is both a local min and a local max.
5) Go to http://www.youtube.com/watch?v=-W4d0qFzyQY "Finding Intervals of
Increase/Decrease Local Max/Mins" Take notes on this video and answer the following:
a) If you want to find all the local extrema (local maximums or minimums), what do you do with the first
derivative, and how do you tell if you have found a local maximum or a local minimum?
1
6) Go tohttp://www.youtube.com/user/EducatorVids?v=Z7QWpBU1ePU&feature=pyv&ad=8624604428
&kw=second%20derivatives%20test "First Derivative Test, Second Derivative Test" Take notes on this
video and answer the following:
a) What is the "first derivative test" and what is it used for. What is the "2nd derivative test" and what is it used
for?
7) Go to http://www.youtube.com/watch?v=c1N8zyVhWxM "Concavity, Inflection Points and
Second Derivatives" and take notes on this video.
8) Go to http://www.youtube.com/watch?v=wRBCvDy2jEY "Second Derivative Test" and take
notes on this video.
9) Go to http://www.youtube.com/watch?v=QtXCIxB6kW8 "Finding Local
Maximums/Minimums - Second Derivative Test" and take notes on this video.
NOTE: Now we do problems using the concepts and techniques in the videos. It's always a good idea to
verify answers also by going to http://www.wolframalpha.com and entering in like: local extrema of
(x-5)e^(-x) , or like: inflection points of (x-3)^2(x+2) You get a graph and the answers.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Using the derivative of f(x) given below, determine the critical points of f(x).
10) f′(x) = (x + 10)(x + 9)
A) -10, -9 B) -19 C) 0, -10, -9 D) 9, 10
11) f′(x) = (x - 3)2(x + 2)
A) -3, 0, 2 B) -3, -2, 3 C) -3, 2 D) -2, 3
12) f′(x) = (x - 5) e-x
A) 6 B) -6 C) -5 D) 5
Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing.
13) f′(x) = (3 - x)(4 - x)
A) Decreasing on (-∞, -3) ∪ (-4, ∞); increasing on (-3, -4)
B) Decreasing on (-∞, 3); increasing on (4, ∞)
C) Decreasing on (3, 4); increasing on (-∞, 3) ∪ (4, ∞)
D) Decreasing on (-∞, 3) ∪ (4, ∞); increasing on (3, 4)
14) f′(x) = x1/3(x - 6)
A) Decreasing on (0, 6); increasing on (6, ∞)
B) Increasing on (0, ∞)
C) Decreasing on (0, 6); increasing on (-∞, 0) ∪ (6, ∞)
D) Decreasing on (-∞, 0) ∪ (6, ∞); increasing on (0, 6)
2
15) f′(x) = (x - 1) e-x
A) Decreasing on (-∞, 1); increasing on (1, ∞) B) Increasing on (-∞, 1); decreasing on (1, ∞)
C) Decreasing on (-∞, -1); increasing on (-1, ∞) D) Increasing on (-∞, -1); decreasing on (-1, ∞)
16) f′(x) = (x + 5)2 e-x
A) Decreasing on (-∞, -5); increasing on (-5, ∞)
B) Never decreasing; increasing on (-∞, -5) ∪ (-5, ∞)
C) Never increasing; decreasing on (-∞, -5) ∪ (-5, ∞)
D) Never decreasing; increasing on (-∞, 5) ∪ (5, ∞)
Use the maximum/minimum finder on a graphing calculator (that's 2nd calc max or 2nd calc min on the TI) to
determine the approximate location of all local extrema.
17) f(x) = 0.1x3 -15x2 - 23x - 14
A) Approximate local maximum at -100.761; approximate local minimum at 0.761
B) Approximate local maximum at -0.761; approximate local minimum at 100.761
C) Approximate local minimum at -100.761; approximate local maximum at 0.761
D) Approximate local minimum at -0.761; approximate local maximum at 100.761
18) f(x) = 0.1x4 - x3- 15x2 + 59x + 6
A) Approximate local maximum at 1.652; approximate local minima at -6.73 and 12.445
B) Approximate local maximum at 1.646; approximate local minima at -6.693 and 12.611
C) Approximate local maximum at 1.815; approximate local minima at -6.837 and 12.498
D) Approximate local maximum at 1.735; approximate local minima at -6.777 and 12.542
19) f(x) = x4 - 3x3- 21x2 + 74x - 71
A) Approximate local maximum at 1.553; approximate local minima at -2.993 and 3.651
B) Approximate local maximum at 1.562; approximate local minima at -3.114 and 3.64
C) Approximate local maximum at 1.54; approximate local minima at -3.006 and 3.817
D) Approximate local maximum at 1.604; approximate local minima at -3.089 and 3.735
20) f(x) = x4 - 4x3- 53x2 - 86x + 62
A) Approximate local maximum at 0.975; approximate local minima at -3.194 and 7.145
B) Approximate local maximum at 0.86; approximate local minima at -3.248 and 7.229
C) Approximate local maximum at -0.944; approximate local minima at -3.192 and 7.136
D) Approximate local maximum at 0.861; approximate local minima at -3.153 and 7.236
21) f(x) = 0.1x5 + 5x4 - 8x3- 15x2 - 6x + 77
A) Approximate local maxima at -41.132 and -0.273; approximate local minima at -0.547 and 1.952
B) Approximate local maxima at -41.126 and -0.329; approximate local minima at -0.543 and 1.896
C) Approximate local maxima at -41.043 and -0.177; approximate local minima at -0.587 and 1.992
D) Approximate local maxima at -41.191 and -0.214; approximate local minima at -0.55 and 2.011
3
Find the open intervals on which the function is increasing and decreasing. Identify the function's local and
absolute extreme values, if any, saying where they occur.
22)
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
x-6 -4 -2 2 4 6
y6
4
2
-2
-4
-6
A) increasing on (-2, 2); decreasing on (-6, -2) and (2, 6);
absolute maximum at (2, 4); absolute minimum at (-2, -4)
B) increasing on (-2, 2); decreasing on (-6, -2) and (2, 6);
no absolute maximum; no absolute minimum
C) increasing on (-2, 2); decreasing on (0, 6);
absolute maximum at (2, 4); absolute minimum at (-2, -4)
D) increasing on (-2, 2); decreasing on (-6, 0);
absolute maximum at (2, 4); absolute minimum at (-2, -4)
23)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
A) increasing on (-3, 0); decreasing on (-5, -3) and (2, 5)
absolute maximum at (-5, 0); local maximum at (0, -1) and (2, -1);
absolute minimum at (5, -4)
B) increasing on (-3, 0); decreasing on [-5, -3) and (2, 5]
absolute maximum at (-5, 0); absolute minimum at (5, -4)
C) increasing on (-3, 0); decreasing on (-5, -3) and (2, 5)
absolute maximum at (-5, 0); local minimum at (-3, -4) and (5, -4)
D) increasing on (-3, 1); decreasing on (-5, -3) and (0, 5)
absolute maximum at (-5, 0); no absolute minimum
4
24)
x-5 -4 -3 -2 -1 1 2 3 4 5
y8
7
6
5
4
3
2
1
-1x-5 -4 -3 -2 -1 1 2 3 4 5
y8
7
6
5
4
3
2
1
-1
A) increasing on (-2, 0) and (2, 4); decreasing on (0, 2);
absolute maximum at (4, 6); absolute minimum at (-2, 0) and (2, 0)
B) increasing on (-2, 0) and (2, 4); decreasing on (0, 2);
absolute maximum at (4, 6); local maximum at (0, 2); absolute minimum at (-2, 0) and (2, 0)
C) increasing on (-2, 0) and (2, 4); decreasing on (0, 2);
absolute maximum at (4, 6) and(0,2); absolute minimum at (-2, 0) and (2, 0)
D) increasing on (2, 4); decreasing on (0, 2);
absolute maximum at (4, 6); local maximum at (0, 2); absolute minimum at (-2, 0) and (2, 0)
Find the largest open interval where the function is changing as requested.
25) Increasing y = 7x - 5
A) (-5, ∞) B) (-∞, 7) C) (-5, 7) D) (-∞, ∞)
26) Increasing f(x) = 1
4x2 -
1
2x
A) (-1, 1) B) (-∞, -1) C) (1, ∞) D) (-∞, ∞)
27) Increasing y = (x2 - 9)2
A) (-∞, 0) B) (3, ∞) C) (-3, 0) D) (-3, 3)
28) Increasing f(x) = x2 - 2x + 1
A) (-∞, 0) B) (0, ∞) C) (-∞, 1) D) (1, ∞)
29) Increasing f(x) = 1
x2 + 1
A) (1, ∞) B) (0, ∞) C) (-∞, 1) D) (-∞, 0)
30) Decreasing f(x) = 4 - x
A) (-∞, 4) B) (-4, ∞) C) (4, ∞) D) (-∞, -4)
31) Decreasing f(x) = ∣x - 8∣
A) (-8, ∞) B) (8, ∞) C) (-∞, -8) D) (-∞, 8)
5
32) Decreasing f(x) = x3 - 4x
A)2 3
3, ∞ B) -
2 3
3,
2 3
3C) -∞, -
2 3
3D) -∞, ∞
Identify the function's local and absolute (global) extreme values, if any, saying where they occur. They mean to do
it by hand using f'(x) tp find the critical points and then use the first and/or 2nd derivative test to classify the point
(to decide whether it is a max or a min or neither).
33) f(x) = -x3- 9x2 - 24x + 2
A) local maximum at x = 4; local minimum at x = 2
B) local maximum at x = -4; local minimum at x = -2
C) local maximum at x = -2; local minimum at x = -4
D) local maximum at x = 2; local minimum at x = 4
Identify the function's local and absolute extreme values, if any, saying where they occur.
34) f(r) = (r - 9) 3
A) local minimum: x = 0; local maximum: x = 9 B) local minimum: x = 0
C) no local extrema D) local minimum: x = 9
35) h(x) = x - 2
x2 + 3x + 6
A) local minimum at x = -2; no local maxima
B) no local extrema
C) local minimum at x = -5; local maximum at x = 6
D) local minimum at x = -2; local maximum at x = 6
36) f(x) = x2 + 2x + 2
A) absolute maximum: 1 at x = -1
B) absolute minimum: 1 at x = -1
C) no local extrema
D) relative minimum: 1 at x = -1; relative maximum: -1 at x = 1
Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the
extreme values, if any, are absolute.
37) f(x) = (x + 7)2, -∞ < x ≤ 0
A) local maximum: 49 at x = 0;
local and absolute absolute minimum: 0 at x = -7
B) no local extrema; no absolute extrema
C) local and absolute minimum: 0 at x = -7
D) local and absolute maximum: 49 at x = 0;
local and absolute minimum: 0 at x = -7
38) f(x) = x2 - 8x, -∞ < x ≤ 8
A) local minimum: 0 at x = 8; local and absolute maximum:-16 at x = 4
B) local minimum: -16 at x = 4; local and absolute maximum: 0 at x = 8
C) local and absolute minimum: -16 at x = 4; local and absolute maximum: 0 at x = 8
D) local and absolute minimum: -16 at x = 4; local maximum: 0 at x = 8
6
39) g(t) = t3
3 -
9
2t2 + 8t, 0 ≤ t < ∞
A) local minimum: 23
6 at x =1; local and absolute maximum: -
160
3 at x = 8
B) local and absolute minimum: - 160
3 at x = 8; local maximum:
23
6 at x =1
C) local minimum: 0 at x = 0; local and absolute minimum: - 160
3 at x = 8; local maximum:
23
6 at x =1
D) local minimum: 23
6 at x =1; local maximum: 0 at x = 0; absolute maximum: -
160
3 at x = 8
40) h(x) = x3 + 2x2 + 5x + 4, -∞ < x ≤ 0
A) local and absolute maximum: 4 at x = 0;
local and absolute minimum: 3 at x = -1
B) local and absolute minimum: 3 at x = -1;
C) local and absolute maximum: 4 at x = 0; D) no local extrema; no absolute extrema
41) f(x) = 4 - x2, -2 ≤ x < 2
A) local and absolute minimum: 0 at x = -2;
local and absolute maximum: 2 at x = 0
B) local and absolute minimum: 0 at x = -2 and x = 2;
local and absolute maximum: 2 at x = 0
C) no local extrema; no absolute extrema
D) local and absolute maximum: 0 at x = -2;
local and absolute minimum: 2 at x = 0
Find the extrema of the function on the given interval, and say where they occur.
42) sin 4x, 0 ≤ x ≤ π
2
A) local maxima: 1 at x = π
8 and 0 at x =
π
2;
local minimum: -1 at x = 3π
8
B) local maxima: 1 at x = π
4 and 0 at x =
π
2;
local minima: 0 at x = 0 and -1 at x = 3π
8
C) local maxima: 1 at x = π
8 and 0 at x =
π
2;
local minima: 0 at x = 0 and -1 at x = 3π
8
D) local maxima: 1 at x = π
8 and 0 at x =
π
4;
local minimum: 0 at x = 0
7
43) sin x + cos x, 0 ≤ x ≤ 2π
A) local maxima: 1 at x = 0 and - 2 at x = 7π
4;
local minima: 1 at x = 2π and 2 at x = π
4
B) local maxima: 1 at x = 0 and - 2 at x = 5π
4;
local minima: 1 at x = 2π and 2 at x = π
4
C) local maxima: 1 at x = 2π and 2 at x = π
4;
local minima: 1 at x = 0 and - 2 at x = 5π
4
D) local maxima: 1 at x = 2π and 2 at x = π
4;
local minima: 1 at x = 0 and - 2 at x = 7π
4
Provide an appropriate response.
44) Find the absolute maximum and minimum values of f(x) = 2x - ex on [0, 1].
A) Maximum = ln 4 - 2 at x = ln 2, minimum = 1 at x = 0
B) Maximum = ln 2 - 2 at x = ln 2, minimum = -1 at x = 0
C) Maximum = 0 at x = ln 2, minimum = 1 at x = 0
D) Maximum = ln 4 - 2 at x = ln 2, minimum = -1 at x = 0
Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave
up and concave down.
45)
x-10 -5 5 10
y10
5
-5
-10
x-10 -5 5 10
y10
5
-5
-10
A) Local minimum at x = 1; local maximum at x = -1; concave up on (0, ∞); concave down on (-∞, 0)
B) Local minimum at x = 1; local maximum at x = -1; concave down on (-∞, ∞)
C) Local minimum at x = 1; local maximum at x = -1; concave up on (-∞, ∞)
D) Local minimum at x = 1; local maximum at x = -1; concave down on (0, ∞); concave up on (-∞, 0)
8
46)
x-10 -5 5 10
y10
5
-5
-10
x-10 -5 5 10
y10
5
-5
-10
A) Local maximum at x = 1; local minimum at x =-1; concave up on (0, ∞); concave down on (-∞, 0)
B) Local minimum at x = 1; local maximum at x =-1; concave up on (0, ∞); concave down on (-∞, 0)
C) Local minimum at x = 1; local maximum at x =-1; concave down on (0, ∞); concave up on (-∞, 0)
D) Local maximum at x = 1; local minimum at x =-1; concave up on (-∞, ∞)
47)
x-10 -5 5 10
y10
5
-5
-10
x-10 -5 5 10
y10
5
-5
-10
A) Local minimum at x = 3; local maximum at x = -3 ; concave down on (0, ∞); concave up on (-∞, 0)
B) Local maximum at x = 3; local minimum at x = -3 ; concave up on (0, -3) and (3, ∞); concave down on
(-3, 3)
C) Local minimum at x = 3; local maximum at x = -3 ; concave up on (0, -3) and (3, ∞); concave down on
(-3, 3)
D) Local minimum at x = 3; local maximum at x = -3 ; concave up on (0, ∞); concave down on (-∞, 0)
9
48)
x-10 -5 5 10
y10
5
-5
-10
x-10 -5 5 10
y10
5
-5
-10
A) Local minimum at x = 3 ; local maximum at x = -3 ; concave up on (-∞, -3) and (3, ∞); concave down
on (-3, 3)
B) Local minimum at x = 3 ; local maximum at x = -3 ; concave up on (0, ∞); concave down on (-∞, 0)
C) Local minimum at x = 3 ; local maximum at x = -3 ; concave down on (0, ∞); concave up on (-∞, 0)
D) Local minimum at x = 3 ; local maximum at x = -3 ; concave down on (-∞, -3) and (3, ∞); concave up
on (-3, 3)
49)
x-10 10
y
10
-10
x-10 10
y
10
-10
A) Local minimum at x = 0; local maximum at x = 2; concave down on (0, ∞); concave up on (-∞, 0)
B) Local minimum at x = 0; local maximum at x = 2; concave up on (0, ∞); concave down on (-∞, 0)
C) Local minimum at x = 2; local maximum at x = 0; concave up on (0, ∞); concave down on (-∞, 0)
D) Local minimum at x = 2; local maximum at x = 0; concave down on (0, ∞); concave up on (-∞, 0)
Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points.
10
50) y = 8x
x2 + 16
x
y
x
y
A) absolute maximum: 0, 1
2
no inflection point
x-4 -2 2 4
y6
4
2
-2
-4
-6
x-4 -2 2 4
y6
4
2
-2
-4
-6
B) local minimum: -4, - 1
2
local maximum: 4, 1
2
inflection point: (0,0)
x-4 -2 2 4
y3
2
1
-1
-2
-3
x-4 -2 2 4
y3
2
1
-1
-2
-3
C) local minimum: (4, -1)
local maximum: (-4, 1)
inflection point: (0, 0)
x-4 -2 2 4
y6
4
2
-2
-4
-6
x-4 -2 2 4
y6
4
2
-2
-4
-6
D) local minimum: (-4, -1)
local maximum: (4, 1)
inflection points: (0, 0), (-4 3, -2 3),
(4 3, 2 3)
x-4 -2 2 4
y6
4
2
-2
-4
-6
x-4 -2 2 4
y6
4
2
-2
-4
-6
11
51) y = 2x3 - 9x2 + 12x
x
y
x
y
A) local minimum: (2, 4)
local maximum: (1, 5)
inflection point: 3
2,
9
2
x-8 -4 4 8
y
24
12
-12
-24
x-8 -4 4 8
y
24
12
-12
-24
B) local minimum: (0, 0)
local maximum: (0, 0)
inflection point: (0, 0)
x-8 -4 4 8
y
432
216
-216
-432
x-8 -4 4 8
y
432
216
-216
-432
C) no extrema
inflection point: (0, 0)
x-8 -4 4 8
y
72
36
-36
-72
x-8 -4 4 8
y
72
36
-36
-72
D) local minimum: (1, 10)
no inflection point
x-8 -4 4 8
y
24
12
-12
-24
x-8 -4 4 8
y
24
12
-12
-24
12
52) y = x1/3(x2 - 175)
x
y
x
y
A) local minimum: 5, -1503
5
local maximum: -5, 1503
5
inflection point: (0, 0)
x-20 -10 10 20
y400
300
200
100
-100
-200
-300
-400
x-20 -10 10 20
y400
300
200
100
-100
-200
-300
-400
B) local minimum: (0, 0)
no inflection points
x-20 -10 10 20
y400
300
200
100
-100
-200
-300
-400
x-20 -10 10 20
y400
300
200
100
-100
-200
-300
-400
C) local minimum: ± 75, - 75
2
local maximum: (0, 0)
inflection points: ±5, - 25
3
x-20 -10 10 20
y100
75
50
25
-25
-50
-75
-100
x-20 -10 10 20
y100
75
50
25
-25
-50
-75
-100
D) no extrema
inflection point: (0, 0)
x-20 -10 10 20
y600
450
300
150
-150
-300
-450
-600
x-20 -10 10 20
y600
450
300
150
-150
-300
-450
-600
13
53) y = x2
x2 + 2
x
y
x
y
A) local minimum: (0, 0)
no inflection points
x-3 -2 -1 1 2 3
y0.375
0.25
0.125
-0.125
-0.25
-0.375
x-3 -2 -1 1 2 3
y0.375
0.25
0.125
-0.125
-0.25
-0.375
B) local minimum: (0, 0)
inflection points: - 6
3,
1
4,
6
3,
1
4
x-3 -2 -1 1 2 3
y0.75
0.5
0.25
-0.25
-0.5
-0.75
x-3 -2 -1 1 2 3
y0.75
0.5
0.25
-0.25
-0.5
-0.75
C) local minimum: 0, - 1
2
no inflection points
x-3 -2 -1 1 2 3
y0.75
0.5
0.25
-0.25
-0.5
-0.75
x-3 -2 -1 1 2 3
y0.75
0.5
0.25
-0.25
-0.5
-0.75
D) local minimum: 0, 1
2
no inflection points
x-3 -2 -1 1 2 3
y1.5
1
0.5
-0.5
-1
-1.5
x-3 -2 -1 1 2 3
y1.5
1
0.5
-0.5
-1
-1.5
14
54) y = x + cos 2x, 0 ≤ x ≤ π
x1 2 3
y
4
3
2
1
-1
x1 2 3
y
4
3
2
1
-1
A) local minimum: 5π
12,
5π - 6 3
12
local maximum: π
12,
π + 6 3
12
inflection points: π
4,
π
4,
3π
4,
3π
4
x1 2 3
y
4
3
2
1
-1
x1 2 3
y
4
3
2
1
-1
B) no local extrema
inflection point: π
2,
π
2
x1 2 3
y
4
3
2
1
-1
x1 2 3
y
4
3
2
1
-1
15
C) local minimum: π
4, -1
local maximum: 3π
4, 3
inflection point: π
2, 1
x1 2 3
y
4
3
2
1
-1
x1 2 3
y
4
3
2
1
-1
D) local minimum: (1.444, -0.246)
local maximum: (0.126, 1.031)
inflection points: (0.785, 0.393), (2.356, 1.178)
x1 2 3
y
4
3
2
1
-1
x1 2 3
y
4
3
2
1
-1
55) y = x 17 - x2
x
y
x
y
16
A) local maximum: 34
3,
2 · 173/2 · 3
9
no inflection point.
x-16 -12 -8 -4 4 8 12 16
y32
24
16
8
-8
-16
-24
-32
x-16 -12 -8 -4 4 8 12 16
y32
24
16
8
-8
-16
-24
-32
B) local minimum: - 34
2, -
17
2
local maximum: 34
2,
17
2
inflection point: (0, 0)
x-4 -3 -2 -1 1 2 3 4
y8
6
4
2
-2
-4
-6
-8
x-4 -3 -2 -1 1 2 3 4
y8
6
4
2
-2
-4
-6
-8
C) local maximum: (0, 17)
no inflection points.
x-4 -3 -2 -1 1 2 3 4
y8
6
4
2
-2
-4
-6
-8
x-4 -3 -2 -1 1 2 3 4
y8
6
4
2
-2
-4
-6
-8
D) local minimum: - 51
3, -
2 · 173/2 · 3
9
local maximum: 51
3,
2 · 173/2 · 3
9
inflection point: (0, 0)
x-4 -3 -2 -1 1 2 3 4
y32
24
16
8
-8
-16
-24
-32
x-4 -3 -2 -1 1 2 3 4
y32
24
16
8
-8
-16
-24
-32
Sketch the graph and show all local extrema and inflection points.
17
56) y = -x4 + 2x2 - 7
x-10 -5 5 10
y
10
5
-5
-10
x-10 -5 5 10
y
10
5
-5
-10
A) Absolute maxima: (-1, -6), (1, -6)
Local minimum: (0, -7)
Inflection points: -1
3,
2
3,
1
3,
2
3
x-10 -5 5 10
y
10
5
-5
-10
x-10 -5 5 10
y
10
5
-5
-10
B) Absolute minima: (-1, 6), (1, 6)
Local maximum: (0, 7)
Inflection point: -1
3,
58
9,
1
3,
58
9
x-10 -5 5 10
y
10
5
-5
-10
x-10 -5 5 10
y
10
5
-5
-10
C) Absolute maxima: (-1, -6), (1, -6)
Inflection points: -1
3,
2
3,
1
3,
2
3
x-10 -5 5 10
y
10
5
-5
-10
x-10 -5 5 10
y
10
5
-5
-10
D) Absolute maxima: (-1, -6), (1, -6)
Local minimum: (0, -7)
No inflection points
x-10 -5 5 10
y
10
5
-5
-10
x-10 -5 5 10
y
10
5
-5
-10
18
57) y = x - sin x, 0 ≤ x ≤ 2π
xπ2
π 3π2
2π
y
6
4
2
xπ2
π 3π2
2π
y
6
4
2
A) Local minimum: (0, 0)
Local maximum: (2π, 2π)
No inflection points
xπ2
π 3π2
2π
y
6
4
2
xπ2
π 3π2
2π
y
6
4
2
B) Local minimum: (0, 0)
Local maximum: (2π, 2π)
Inflection point: (π, π)
xπ2
π 3π2
2π
y
6
4
2
xπ2
π 3π2
2π
y
6
4
2
C) Local minimum: (0, 0)
Local maximum: (2π, 2π)
No inflection points
xπ2
π 3π2
2π
y
6
4
2
xπ2
π 3π2
2π
y
6
4
2
D) Local minimum: (0, 0)
Local maximum: (2π, 2π)
Inflection point: (π, π)
xπ2
π 3π2
2π
y
6
4
2
xπ2
π 3π2
2π
y
6
4
2
19
58) y = ln (8 - x2)
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
x-5 -4 -3 -2 -1 1 2 3 4 5
y5
4
3
2
1
-1
-2
-3
-4
-5
A) Local minimum (0, ln 8)
No inflection point
x-5 -4 -3 -2 -1 1 2 3 4 5
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x-5 -4 -3 -2 -1 1 2 3 4 5
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
B) No extrema
No inflection point
x-5 -4 -3 -2 -1 1 2 3 4 5
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x-5 -4 -3 -2 -1 1 2 3 4 5
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
C) Local maximum (0, ln 8)
No inflection point
x-5 -4 -3 -2 -1 1 2 3 4 5
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x-5 -4 -3 -2 -1 1 2 3 4 5
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
D) Local minimum (0, -ln 8)
No inflection point
x-5 -4 -3 -2 -1 1 2 3 4 5
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x-5 -4 -3 -2 -1 1 2 3 4 5
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
20
59) y = ex - 3e-x - 4x
x-8 -6 -4 -2 2 4 6 8
y8
6
4
2
-2
-4
-6
-8
x-8 -6 -4 -2 2 4 6 8
y8
6
4
2
-2
-4
-6
-8
A) Local minimum (2, -1)
No inflection point
x-8 -6 -4 -2 2 4 6 8
y8
6
4
2
-2
-4
-6
-8
x-8 -6 -4 -2 2 4 6 8
y8
6
4
2
-2
-4
-6
-8
B) Local maximum (0, -2)
Local minimum (ln 3, 2 - 4 ln 3)
No inflection point
x-8 -6 -4 -2 2 4 6 8
y8
6
4
2
-2
-4
-6
-8
x-8 -6 -4 -2 2 4 6 8
y8
6
4
2
-2
-4
-6
-8
C) Local minimum 1
2 ln 3, - 2 ln 3
No inflection point
x-8 -6 -4 -2 2 4 6 8
y8
6
4
2
-2
-4
-6
-8
x-8 -6 -4 -2 2 4 6 8
y8
6
4
2
-2
-4
-6
-8
D) Local maximum (0, -2)
Local minimum (ln 3, 2 - 4 ln 3)
Inflection point 1
2 ln 3, - 2 ln 3
x-8 -6 -4 -2 2 4 6 8
y8
6
4
2
-2
-4
-6
-8
x-8 -6 -4 -2 2 4 6 8
y8
6
4
2
-2
-4
-6
-8
For the given expression y′, find y'' and sketch the general shape of the graph of y = f(x).
21
60) y' = x2
3 - 5
x
y
x
y
A)
x
y
x
y
B)
x
y
x
y
C)
x
y
x
y
D)
x
y
x
y
22
61) y′ = x2(2 - x)
x
y
x
y
A)
x
y
x
y
B)
x
y
x
y
C)
x
y
x
y
D)
x
y
x
y
23
62) y' = sin x, 0 ≤ x ≤ 2π
x
y
x
y
A)
x
y
x
y
B)
x
y
x
y
C)
x
y
x
y
D)
x
y
x
y
24
63) y′ = x-2/3(x - 6)
x
y
x
y
A)
x
y
x
y
B)
x
y
x
y
C)
x
y
x
y
D)
x
y
x
y
Solve the problem.
25
64) The graphs below show the first and second derivatives of a function y = f(x). Select a possible graph of f
that passes through the point P.
f′ f′′
x
y
P
x
y
P
x
y
P
x
y
P
A)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
B)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
C)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
D)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
26
65) The graphs below show the first and second derivatives of a function y = f(x). Select a possible graph f that
passes through the point P.
f′ f′′
x
y
P
x
y
P
x
y
P
x
y
P
A)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
B)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
C)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
D)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
27
66) The graphs below show the first and second derivatives of a function y = f(x). Select a possible graph f that
passes through the point P.
f′ f′′
x
y
P
x
y
P
x
y
P
x
y
P
A)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
B)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
C)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
D)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
28
67) The graphs below show the first and second derivatives of a function y = f(x). Select a possible graph f that
passes through the point P.
f′ f′′
x
y
P
x
y
P
x
y
P
x
y
P
A)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
B)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
C)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
D)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
29
68) The graphs below show the first and second derivatives of a function y = f(x). Select a possible graph f that
passes through the point P.
f′ f′′
x
y
P
x
y
P
x
y
P
x
y
P
A)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
B)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
C)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
D)
x
y
x
y
[NOTE: Graph vertical scales
may vary from graph to graph.]
Graph the rational function.
30
69) y = x - 4
x2 - 7x + 12
x-8 -4 4 8
y
8
4
-4
-8
x-8 -4 4 8
y
8
4
-4
-8
A)
x-8 -4 4 8
y
8
4
-4
-8
x-8 -4 4 8
y
8
4
-4
-8
B)
x-8 -4 4 8
y
8
4
-4
-8
x-8 -4 4 8
y
8
4
-4
-8
C)
x-8 -4 4 8
y
8
4
-4
-8
x-8 -4 4 8
y
8
4
-4
-8
D)
x-8 -4 4 8
y
8
4
-4
-8
x-8 -4 4 8
y
8
4
-4
-8
31
70) y = x2 + x - 56
x2 - x - 42
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
A)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
B)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
C)
x-10 -8 -6 -4 -2 2 4 6 8
y8
6
4
2
-2
-4
-6
-8
x-10 -8 -6 -4 -2 2 4 6 8
y8
6
4
2
-2
-4
-6
-8
D)
x-10 -8 -6 -4 -2 2 4 6 8
y8
6
4
2
-2
-4
-6
-8
x-10 -8 -6 -4 -2 2 4 6 8
y8
6
4
2
-2
-4
-6
-8
32
71) y = x2
x2 + 2
x-3 -2 -1 1 2 3
y
1.5
1
0.5
-0.5
x-3 -2 -1 1 2 3
y
1.5
1
0.5
-0.5
A)
x-3 -2 -1 1 2 3
y
1.5
1
0.5
-0.5
x-3 -2 -1 1 2 3
y
1.5
1
0.5
-0.5
B)
x-3 -2 -1 1 2 3
y
1.5
1
0.5
-0.5
x-3 -2 -1 1 2 3
y
1.5
1
0.5
-0.5
C)
x-3 -2 -1 1 2 3
y
1.5
1
0.5
-0.5
x-3 -2 -1 1 2 3
y
1.5
1
0.5
-0.5
D)
x-3 -2 -1 1 2 3
y
1.5
1
0.5
-0.5
x-3 -2 -1 1 2 3
y
1.5
1
0.5
-0.5
33
72) y = x - 1
x2 - 1
-6 -4 -2 2 4 6
4
2
-2
-4
-6 -4 -2 2 4 6
4
2
-2
-4
A)
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
B)
-6 -4 -2 2 4 6
4
2
-2
-4
-6 -4 -2 2 4 6
4
2
-2
-4
C)
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
D)
-6 -4 -2 2 4 6
4
2
-2
-4
-6 -4 -2 2 4 6
4
2
-2
-4
34
73) y = 18x
x2 + 9
x-4 -2 2 4
y6
4
2
-2
-4
-6
x-4 -2 2 4
y6
4
2
-2
-4
-6
A)
x-4 -2 2 4
y6
4
2
-2
-4
-6
x-4 -2 2 4
y6
4
2
-2
-4
-6
B)
x-4 -2 2 4
y3
2
1
-1
-2
-3
x-4 -2 2 4
y3
2
1
-1
-2
-3
C)
x-4 -2 2 4
y6
4
2
-2
-4
-6
x-4 -2 2 4
y6
4
2
-2
-4
-6
D)
x-4 -2 2 4
y6
4
2
-2
-4
-6
x-4 -2 2 4
y6
4
2
-2
-4
-6
35
74) y = 4
x2
x-8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
A)
x-8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
B)
x-8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
C)
x-8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
D)
x-8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
36
75) y = x
x2 - 25
x-10 10
y
5
-5
x-10 10
y
5
-5
A)
x-10 10
y
5
-5
x-10 10
y
5
-5
B)
x-10 10
y
10
-10
x-10 10
y
10
-10
C)
x-10 10
y
5
-5
x-10 10
y
5
-5
D)
x-10 10
y
5
-5
x-10 10
y
5
-5
37
Solve the problem.
76) Using the following properties of a twice-differentiable function y = f(x), select a possible graph of f.
x y Derivatives
x < 2 y′ > 0,y′′ < 0
-2 11 y′ = 0,y′′ < 0
-2 < x < 0 y′ < 0,y′′ < 0
0 -5 y′ < 0,y′′ = 0
0 < x < 2 y′ < 0,y′′ > 0
2 -21 y′ = 0,y′′ > 0
x > 2 y′ > 0,y′′ > 0
A)
x-4 -3 -2 -1 1 2 3 4
y24
16
8
-8
-16
-24
x-4 -3 -2 -1 1 2 3 4
y24
16
8
-8
-16
-24
B)
x-4 -3 -2 -1 1 2 3 4
y24
16
8
-8
-16
-24
x-4 -3 -2 -1 1 2 3 4
y24
16
8
-8
-16
-24
C)
x-4 -3 -2 -1 1 2 3 4
y24
16
8
-8
-16
-24
x-4 -3 -2 -1 1 2 3 4
y24
16
8
-8
-16
-24
D)
x-4 -3 -2 -1 1 2 3 4
y24
16
8
-8
-16
-24
x-4 -3 -2 -1 1 2 3 4
y24
16
8
-8
-16
-24
38
77) Select an appropriate graph of a twice-differentiable function y = f(x) that passes through the points
(- 2, 1), - 6
3,
5
9, (0, 0),
6
3,
5
9 and ( 2, 1), and whose first two derivatives have the following sign
patterns.
y′ : + - + -
- 2 0 2
y′′ :
+ - +
- 6
3
6
3
A)
x-3 -2 -1 1 2 3
y2
1.5
1
0.5
-0.5
-1
-1.5
-2
x-3 -2 -1 1 2 3
y2
1.5
1
0.5
-0.5
-1
-1.5
-2
B)
x-3 -2 -1 1 2 3
y16
12
8
4
-4
-8
-12
-16
x-3 -2 -1 1 2 3
y16
12
8
4
-4
-8
-12
-16
C)
x-3 -2 -1 1 2 3
y4
3
2
1
-1
-2
-3
-4
x-3 -2 -1 1 2 3
y4
3
2
1
-1
-2
-3
-4
D)
x-3 -2 -1 1 2 3
y16
12
8
4
-4
-8
-12
-16
x-3 -2 -1 1 2 3
y16
12
8
4
-4
-8
-12
-16
ESSAY.
Provide an appropriate response.
78) If f(x) is a differentiable function and f′ (c) = 0 at an interior point c of f's domain, and if f′′(x) > 0 for all x in
the domain, must f have a local minimum at x = c? Explain.
79) Sketch a smooth curve through the origin with the following properties:
f′(x) > 0 for x < 0; f′ (x) < 0 for x > 0; f′′(x) approaches 0 as x approaches -∞; and
f′′(x) approaches 0 as x approaches ∞.
39
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Identify the function's local and absolute extreme values, if any, saying where they occur.
80) f(x) = x3- 3x2 + 3x - 1
A) local maximum at x = 1
B) local minimum at x = 1
C) local maximum at x = 1; local minimum at x = -1
D) no local extrema
81) f(x) = x3 + 6.5x2 + 12x + 3
A) local maximum at x = -3; local minimum at x = - 4
3
B) local maximum at x = 4
3; local minimum at x = 3
C) local maximum at x = 4; local minimum at x = 1
D) local maximum at x = - 1; local minimum at x = -4
ESSAY.
Provide an appropriate response.
82) The accompanying figure shows a portion of the graph of a function that is twice-differentiable at all x
except at x = p. At each of the labeled points, classify y′ and y′′ as positive, negative, or zero.
40
83) The graph below shows the position s = f(t) of a body moving back and forth on a coordinate line.
(a) When is the body moving away from the origin? Toward the origin? At approximately what times is
the (b) velocity equal to zero? (c) Acceleration equal to zero? (d) When is the acceleration positive?
Negative?
84) For x > 0, sketch a curve y = f(x) that has f(1) = 0 and f′(x) = - 1
x. Can anything be said about the concavity
of such a curve? Give reasons for your answer.
85) Sketch a continuous curve y = f(x) with the following properties:
f(2) = 3; f′′(x) > 0 for x > 4; and f′′(x) < 0 for x < 4 .
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the largest open interval where the function is changing as requested.
86) Decreasing f(x) = - x + 3
A) (-∞, 3) B) (-3, ∞) C) (-∞, -3) D) (3, ∞)
Provide an appropriate response.
87) Suppose the derivative of the function y = f(x) is y' = (x - 2)2(x + 7). At what points, if any, does the graph
of f have a local minimum or local maximum?
A) local maximum at x = -7 B) local minimum at x = -7
C) local minimum at x = 2 D) no local minimum or local maximum
88) Suppose that the second derivative of the function y = f(x) is y'' = (x - 4)(x + 8). For what x-values does the
graph of f have an inflection point?
A) 4, -8 B) -4, 8 C) 4, 8 D) -4, -8
FROM THE BOOK
89) Do section 4.1 # 1-21 odd, 28-31 ALL, 35, 37, 41, 45
Curve Sketching
Solve the problem.
90) Find the intervals on which the function f(x) = 4x5 - 5x4 is increasing and decreasing. Sketch the graph of
y = f(x), and identify any local maxima and minima. Any global extrema should also be identified.
41
91) Find the interval on which the function f(x) = 5x2 + 10x - 7 is increasing and decreasing. Sketch the graph
of y = f(x), and identify any local maxima and minima. Any global extrema should also be identified.
92) Find the interval on which the function f(x) = (x - 2)2(x + 3)2 is increasing and decreasing. Sketch the graph
of y = f(x), and identify any local maxima and minima. Any global extrema should also be identified.
93) Find the intervals on which the function f(x) = x2/3(10 - x) is increasing and decreasing. Sketch the graph
of y = f(x), and identify any local maxima and minima. Any global extrema should also be identified.
94) Find the intervals on which the function f(x) = 3x5 - 20x3 is increasing and decreasing. Sketch the graph of
y = f(x), and identify any local maxima and minima. Any global extrema should also be identified.
95) Find the intervals on which the function f(x) = 3
x is increasing and decreasing. Sketch the graph of of
y = f(x), and identify any local maxima and minima. Any global extrema should also be identified.
96) Find the intervals on which the function f(x) = x3 - 27x is increasing and decreasing. Sketch the graph of
y = f(x), and identify any local maxima and minima. Any global extrema should also be identified.
Optimization
97) Find the maximum possible area of a rectangle with perimeter 180 meters.
98) A rectangular corral is to be constructed with an internal divider parallel to two opposite sides, and 1200
meters of fencing will be used to make the corral (including the divider). What is the maximum possible
area that such a corral can have?
99) What is the greatest amount by which a number in the interval [0, 1] can exceed its cube?
100) A farmer has 600 yards of fencing with which to build a rectangular corral. He will use a long straight wall
to form one side of the corral (to save fencing). What is the maximum possible area that can be enclosed in
this way?
101) Two nonnegative numbers have sum 1. What is the maximum possible value of the sum of their squares?
102) Find the shape of the right circular cylinder of maximal surface area (including the top and the bottom)
inscribed in a sphere of radius R.
103) Two cubes have total volume 250 cm3. What is the maximum possible surface area they can have? The
minimum?
104) A rectangle has each diagonal of length 5. What is the maximum possible perimeter of such a rectangle?
105) The product of two positive integers is 1600. What is the minimum possible value of their sum?
42
106) A rectangle has its base on the x-axis and its upper two vertices on the graph of y = 12 - x2. What is the
maximum possible area of such a rectangle?
107) The sum of two nonnegative numbers x and y is 8. Find the maximum possible value of the expression
x2 + y3.
108) A mass of clay of volume 432 in.3 is formed into two cubes. What is the minimum possible total surface
area of the two cubes? What is the maximum?
109) A rancher has 1700 meters of fencing with which to build two widely separated corrals; one is to be
square, and the other is to be twice as long as it is wide. What is the maximum possible total area that the
rancher can thereby enclose?
110) Find the maximum possible volume of a right circular cylinder inscribed in a sphere of radius R.
111) A farmer has 480 meters of fencing. He wishes to enclose a rectangular plot of land and to divide the plot
into three equal rectangles with two parallel lengths of fence down the middle. What dimensions will
maximize the enclosed area? Be sure to verify that you have found the maximum enclosed area.
112) The sum of two nonnegative numbers is 10. Find the minimum possible value of the sum of their cubes.
113) A rectangle has a line of fixed length L reaching from one vertex to the midpoint of one of the far sides.
What is the maximum possible area of such a rectangle?
114) Write an equation of the straight line through the point (7, 1) that cuts off the least area from the first
quadrant. Be sure to verify that your area is minimal.
115) A circle is dropped into the graph of the parabola y = x2. How small can the radius of the circle be and yet
allow the circle to touch the parabola at two different points?
116) Find the point on the graph of y = x that is closest to the point (3, 0). Be sure to verify that it is indeed
closest.
117) The sum of two nonnegative numbers is 48. What is the smallest possible value of the sum of their
squares?
118) A rectangular box is to have a base three times as long as it is wide. The total surface area of the box is to
be 20 ft2. Find the maximum possible volume of such a box.
119) A right circular cone has a slant height of 10 ft. Find the maximum possible volume of such a cone. Verify
that your answer is maximal.
120) A rectangular poster is to contain 8250 cm2 of print in the shape of a smaller rectangle; the margins at top
and bottom must each be 22 cm, and those at the sides must each be 15 cm. What are the dimensions of
such a poster having the least possible total area?
43
121) A wastebasket is to have as its base an equilateral triangle, its sides are to be vertical, and its volume is to
be 8 ft3. What is the minimum possible surface area of such a wastebasket?
122) A railroad will operate a special a special excursion train if at least 200 people subscribe. The fare will be
$8 per person if 200 people subscribe, but will decrease 1 cent for each additional person who subscribes.
What number of passengers will bring the railroad maximum revenue?
123) An aquarium has a square base made of slate costing 8¢/in.2 and four glass sides costing 3¢/in.2. The
volume of the aquarium is to be 36,000 in.3. Find the dimensions of the least expensive such aquarium.
124) A poster is to contain 96 in.2 of print, and each copy must have 3-in. margins at top and bottom and 2-in.
margins on each side. What are the dimensions of such a poster having the least possible area?
125) A wall 8 m high stands 4 m away from a building. What is the length of the shortest ladder that will lean
over the wall and touch the building? Use as independent variable the angle that the ladder makes with
the ground.
126) A rectangular sheet of thin metal is 5 m wide and 8 m long. Four small equal squares are cut from its
corners and the projections of the resulting cross-shaped piece of metal are bent upward and welded to
make an open-topped box with a rectangular base. What is the maximum possible volume of such a box?
127) You plan to build a playing field in the shape of a rectangle with a semicircular region at each end, so that
races can be held around the perimeter of the field. If you want the total perimeter of the field to be 1000
m, what dimensions should your field have to maximize the area of the rectangular portion of the field?
44
128) You are planning to close off a corner of the first quadrant with a line segment 15 units long running from
(x, 0) to (0, y). Show that the area of the triangle enclosed by the segment is largest when x = y.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
129) A private shipping company will accept a box for domestic shipment only if the sum of its length and girth
(distance around) does not exceed 90 in. What dimensions will give a box with a square end the largest
possible volume?
A) 15 in. × 15 in. × 30 in. B) 15 in. × 15 in. × 75 in.
C) 15 in. × 30 in. × 30 in. D) 30 in. × 30 in. × 30 in.
45
130) A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas
the semicircle is of tinted glass that transmits only one-fifth as much light per unit area as clear glass does.
The total perimeter is fixed. Find the proportions of the window that will admit the most light. Neglect the
thickness of the frame.
A)width
height =
20
10 + πB)
width
height =
5
10 + 4πC)
width
height =
20
10 + 4πD)
width
height =
20
5 + 4π
131) A trough is to be made with an end of the dimensions shown. The length of the trough is to be 24 feet long.
Only the angle θ can be varied. What value of θ will maximize the trough's volume?
θ θ
A) 54° B) 6° C) 30° D) 32°
46
132) A rectangular sheet of perimeter 36 cm and dimensions x cm by y cm is to be rolled into a cylinder as
shown in part (a) of the figure. What values of x and y give the largest volume?
A) x = 14 cm; y = 4 cm B) x = 12 cm; y = 6 cm
C) x = 11 cm; y = 7 cm D) x = 13 cm; y = 5 cm
133) The 9 ft wall shown here stands 30 feet from the building. Find the length of the shortest straight beam
that will reach to the side of the building from the ground outside the wall.
9' wall
30'
A) 53.3 ft B) 51.3 ft C) 39 ft D) 52.3 ft
47
134) The strength S of a rectangular wooden beam is proportional to its width times the square of its depth.
Find the dimensions of the strongest beam that can be cut from a 12-in.-diameter cylindrical log. (Round
answers to the nearest tenth.)
12"
A) w = 5.9 in.; d = 10.8 in. B) w = 7.9 in.; d = 10.8 in.
C) w = 6.9 in.; d = 9.8 in. D) w = 7.9 in.; d = 8.8 in.
135) The stiffness of a rectangular beam is proportional to its width times the cube of its depth. Find the
dimensions of the stiffest beam than can be cut from a 14-in.-diameter cylindrical log. (Round answers to
the nearest tenth.)
14"
A) w = 8.0 in.; d = 13.1 in. B) w = 6.0 in.; d = 13.1 in.
C) w = 8.0 in.; d = 11.1 in. D) w = 7.0 in.; d = 12.1 in.
136) A small frictionless cart, attached to the wall by a spring, is pulled 10 cm back from its rest position and
released at time t = 0 to roll back and forth for 4 sec. Its position at time t is
s = 1 - 10 cos πt. What is the cart's maximum speed? When is the cart moving that fast? What is the
magnitude of of the acceleration then?
A) 10π ≈ 31.42 cm/sec; t = 0.5 sec, 1.5 sec, 2.5 sec, 3.5 sec; acceleration is 0 cm/sec2
B) π ≈ 3.14 cm/sec; t = 0.5 sec, 1.5 sec, 2.5 sec, 3.5 sec; acceleration is 0 cm/sec2
C) 10π ≈ 31.42 cm/sec; t = 0.5 sec, 2.5 sec; acceleration is 1 cm/sec2
D) 10π ≈ 31.42 cm/sec; t = 0 sec, 1 sec, 2 sec, 3 sec; acceleration is 0 cm/sec2
137) At noon, ship A was 12 nautical miles due north of ship B. Ship A was sailing south at 12 knots (nautical
miles per hour; a nautical mile is 2000 yards) and continued to do so all day. Ship B was sailing east at
7 knots and continued to do so all day. The visibility was 5 nautical miles. Did the ships ever sight each
other?
A) Yes. They were within 4 nautical miles of each other.
B) No. The closest they ever got to each other was 7.0 nautical miles.
C) No. The closest they ever got to each other was 6.0 nautical miles.
D) Yes. They were within 3 nautical miles of each other.
48
138) If the price charged for a candy bar is p(x) cents, then x thousand candy bars will be sold in a certain city,
where p(x) = 106 - x
28. How many candy bars must be sold to maximize revenue?
A) 2968 thousand candy bars B) 2968 candy bars
C) 1484 candy bars D) 1484 thousand candy bars
139) Find the number of units that must be produced and sold in order to yield the maximum profit, given the
following equations for revenue and cost:
R(x) = 60x - 0.5x2
C(x) = 9x + 4.
A) 52 units B) 69 units C) 55 units D) 51 units
140) Suppose c(x) = x3 - 18x2 + 10,000x is the cost of manufacturing x items. Find a production level that will
minimize the average cost of making x items.
A) 9 items B) 11 items C) 8 items D) 10 items
141) Determine the dimensions of the rectangle of largest area that can be inscribed in a semicircle of radius 3.
A) h = 3 2, w = 3 2
2, B) h = 3 2, w = 2 C) h =
3 2
2, w = 3 2 D) h = 2, w = 3 2
ESSAY.
142) A storage building is to be shaped like a box with a square base (its floor). The floor costs $3/m2, the walls
cost $7/m2, and the flat roof costs $5/m2. The volume of the building is to be 12,544 m3. What is the shape
of the least expensive such building?
143) You need a cardboard container in the shape of a right circular cylinder of volume 54π in.2. What radius r
and height h would minimize its total surface area (including top and bottom)?
144) What is the maximum possible volume of a right circular cylinder with total surface area 600π in.2
(including the top and the bottom)?
145) Find the minimum possible value of the sum of a real number and its square.
146) What point on the parabola y = x2 is closest to the point (3, 0)?
147) A rectangular box with square base and no top is to have a volume of exactly 1000 cm3. What is the
minimum possible surface area of such a box?
148) Find the coordinates of the point or points on the curve 2y2 = 5x + 5 which is (are) closest to the origin (0,
0).
149) The sum of the squares of two nonnegative real numbers x and y is 18. What is the minimum possible
value of x + y?
49
FROM THE BOOK
150) Do section 4.2 # 3 - 11 odd, 13-21 odd, 25, 29, 33
151) Do section 4.4 #1-9 odd, 17-37 odd, 47, 49
Rates and Related Rates
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
152) A company knows that the unit cost C and the unit revenue R from the production and sale of x units are
related by C = R2
134,000 + 8437. Find the rate of change of unit revenue when the unit cost is changing by
$11/unit and the unit revenue is $4000.
A) $184.25/unit B) $843.70/unit C) $513.98/unit D) $220.00/unit
153) Water is falling on a surface, wetting a circular area that is expanding at a rate of 6 mm2/s. How fast is the
radius of the wetted area expanding when the radius is 170 mm? (Round your answer to four decimal
places.)
A) 0.0353 mm/s B) 178.0234 mm/s C) 0.0112 mm/s D) 0.0056 mm/s
154) A wheel with radius 3 m rolls at 15 rad/s. How fast is a point on the rim of the wheel rising when the point
is π/3 radians above the horizontal (and rising)? (Round your answer to one decimal place.)
A) 90.0 m/s B) 45.0 m/s C) 22.5 m/s D) 11.3 m/s
155) Assume that the profit generated by a product is given by P(x) = 3 x, where x is the number of units sold.
If the profit keeps changing at a rate of $600 per month, then how fast are the sales changing when the
number of units sold is 1900? (Round your answer to the nearest dollar per month.)
A) $156,920/month B) $9/month C) $17,436/month D) $8718/month
156) A piece of land is shaped like a right triangle. Two people start at the right angle of the triangle at the same
time, and walk at the same speed along different legs of the triangle. If the area formed by the positions of
the two people and their starting point (the right angle) is changing at 2 m2/s, then how fast are the people
moving when they are 3 m from the right angle? (Round your answer to two decimal places.)
A) 1.33 m/s B) 0.67 m/s C) 0.33 m/s D) 4.48 m/s
Solve the problem. Round your answer, if appropriate.
157) Water is discharged from a pipeline at a velocity v (in ft/sec) given by v = 1634p(1/2), where p is the
pressure (in psi). If the water pressure is changing at a rate of 0.209 psi/sec, find the acceleration (dv/dt) of
the water when p = 34.0 psi.
A) 140 ft/sec2 B) 996 ft/sec2 C) 29.3 ft/sec2 D) 47.6 ft/sec2
158) One airplane is approaching an airport from the north at 149 km/hr. A second airplane approaches from
the east at 246 km/hr. Find the rate at which the distance between the planes changes when the
southbound plane is 27 km away from the airport and the westbound plane is 17 km from the airport.
A) -257 km/hr B) -514 km/hr C) -128 km/hr D) -385 km/hr
50
159) Water is being drained from a container which has the shape of an inverted right circular cone. The
container has a radius of 6.00 inches at the top and a height of 10.0 inches. At the instant when the water in
the container is 9.00 inches deep, the surface level is falling at a rate of 1.2 in./sec. Find the rate at which
water is being drained from the container.
A) 110 in.3/s B) 159 in.3/s C) 129 in.3s D) 105 in.3/s
160) A man 6 ft tall walks at a rate of 5 ft/sec away from a lamppost that is 18 ft high. At what rate is the length
of his shadow changing when he is 45 ft away from the lamppost? (Do not round your answer)
A)5
4 ft/sec B)
75
2 ft/sec C)
5
2 ft/sec D)
5
8 ft/sec
161) The volume of a sphere is increasing at a rate of 6 cm3/sec. Find the rate of change of its surface area when
its volume is 32π
3 cm3. (Do not round your answer.)
A) 4 cm2/sec B) 12π cm2/sec C)8
3 cm2/sec D) 6 cm2/sec
162) The volume of a rectangular box with a square base remains constant at 500 cm3 as the area of the base
increases at a rate of 3 cm2/sec. Find the rate at which the height of the box is decreasing when each side of
the base is 18 cm long. (Do not round your answer.)
A)125
81 cm/sec B)
125
486 cm/sec C)
125
8748 cm/sec D)
1
108 cm/sec
163) The radius of a right circular cylinder is increasing at the rate of 4 in./sec, while the height is decreasing at
the rate of 9 in./sec. At what rate is the volume of the cylinder changing when the radius is 6 in. and the
height is 12 in.?
A) -36π in.3/sec B) 252π in.3/sec C) -36 in.3/sec D) -42 in.3/sec
FROM THE BOOK
164) Do section 4.6 #1-9 odd, 15, 17, 21 - 31 odd, 35, 37, 43, 45
51
Answer KeyTestname: DERIVATIVES WORKSHEET 2 - USING THE DERIVATIVE
1) see section 4.1
2) Watch the video - pause it in various places and see what is happening.
3) Watch the video
4) Watch the video.
5) Watch the video
6) Watch the video
7) Watch the video
8) Watch the video
9) Watch the video
10) A
11) D
12) A
13) C
14) C
15) A
16) B
17) B
18) D
19) D
20) C
21) A
22) A
23) B
24) B
25) D
26) C
27) B
28) D
29) D
30) A
31) D
32) B
33) C
34) C
35) D
36) B
37) A
38) D
39) C
40) C
41) A
42) C
43) C
44) D
45) A
46) B
47) D
48) B
52
Answer KeyTestname: DERIVATIVES WORKSHEET 2 - USING THE DERIVATIVE
49) C
50) D
51) A
52) A
53) B
54) A
55) B
56) A
57) B
58) C
59) D
60) D
61) B
62) A
63) A
64) D
65) C
66) D
67) D
68) A
69) C
70) B
71) B
72) B
73) D
74) B
75) A
76) C
77) A
78) Yes. The point x = c is either a local maximum, a local minimum, or an inflection point. But, since f ′′(x) > 0 for all
x in the domain, there are no inflection points and the curve is everywhere concave up and thus cannot have a
local maximum. Hence, there is a local minimum at x = c.
79) Answers will vary. A general shape is indicated below:
80) D
81) A
53
Answer KeyTestname: DERIVATIVES WORKSHEET 2 - USING THE DERIVATIVE
82) a: both y′ and y′′ are undefined.
b: y′ =0 and y′′ > 0
c: y′ > 0 and y′′ = 0
d: y′ = 0 and y′′ = 0
e: y′ > 0 and y′′ = 0
f: y′ = 0 and y′′ < 0
g: y′ < 0 and y′′ = 0
83) (a). Moving towards to origin on (1, 2) and (5.7, 7); moving away from the origin on (0, 1), (2, 5.7), and (7, 10).
(b). Velocity is zero at the extrema. These occur at t ≈ 1 sec and t ≈ 5.7 sec.
(c). Acceleration is zero at the inflection points. These occur at t ≈2.3 sec, t ≈ 4 sec, t ≈ 5.1 sec, t ≈ 7 sec, and t ≈ 8.5
sec.
(d). Acceleration is positive where f(t) is concave up and negative where it is concave down. Acceleration is
positive on (2.3, 4), (5.1, 7), and (8.5, 10). Acceleration is negative on (0, 2.3), (4, 5.1), and (7, 8.5).
84)
Since f′′(x) = 1
x2 > 0 for all x > 0, then the function is everywhere concave up.
85) Answers will vary. A general shape is indicated below:
54
Answer KeyTestname: DERIVATIVES WORKSHEET 2 - USING THE DERIVATIVE
86) B
87) B
88) A
89) see answers in the book. You can also verify answers by graphing the function on your TI and do 2nd calc max
or min. Or you can go to WolframAlpha.com and enter in like: local extrema of x^3-x , or you can enter in like:
inflection points of x^3-x , or like: absolute maximum of y=-x^2+3 Try it!
90) local maximum at (0, 0)
local minimum at (1, -1)
increasing on (-∞, 0) ∪ (1, ∞)
decreasing on (0, 1)
x-4 -2 2 4
y
10
5
-5
-10
x-4 -2 2 4
y
10
5
-5
-10
91) global minimum at (-1, -12)
increasing on (-1, ∞)
decreasing on (-∞, -1)
x-4 -2 2 4
y
10
5
-5
-10
x-4 -2 2 4
y
10
5
-5
-10
55
Answer KeyTestname: DERIVATIVES WORKSHEET 2 - USING THE DERIVATIVE
92) global minima at (-3, 0) and (2, 0)
local maximum at (- 1
2,
625
16)
increasing on (-3, - 1
2) ∪ (2, ∞)
decreasing on (-∞, -3) ∪ (- 1
2, 2)
x-4 -2 2 4
y40
30
20
10
-10
x-4 -2 2 4
y40
30
20
10
-10
93) local minimum at (0, 0) (and a cusp)
local maximum at (4, 123
2)
increasing on (0, 4)
decreasing on (-∞, 0) ∪ (4, ∞)
x-10 -8 -6 -4 -2 2 4 6 8
y2018161412108642
-2-4-6-8
-10-12-14-16-18-20
x-10 -8 -6 -4 -2 2 4 6 8
y2018161412108642
-2-4-6-8
-10-12-14-16-18-20
56
Answer KeyTestname: DERIVATIVES WORKSHEET 2 - USING THE DERIVATIVE
94) local minimum at (2, -64)
local maximum at (-2, 64)
increasing on (-∞, -2) ∪ (2, ∞)
decreasing on (-2, 2) (but with a horizontal tangent at (0, 0))
x-4 -2 2 4
y100908070605040302010
-10-20-30-40-50-60-70-80-90
-100
x-4 -2 2 4
y100908070605040302010
-10-20-30-40-50-60-70-80-90
-100
95) no extrema
decreasing on (-∞, 0) ∪ (0, ∞)
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
57
Answer KeyTestname: DERIVATIVES WORKSHEET 2 - USING THE DERIVATIVE
96) local maximum at (-3, 54)
local minimum at (3, -54)
increasing on (-∞, -3) ∪ (3, ∞)
decreasing on (-3, 3)
x-10 -5 5 10
y200
100
-100
-200
x-10 -5 5 10
y200
100
-100
-200
97) A = 2025 m2
98) 60,000 m2
99)2 3
9
100) 45,000 yd2
101) 1
102) r = 1
2 +
1
2 5R, h = 2R
5 - 5
10
1/2
(Must satisfy r2 + 1
2h
2 = R2)
103) min = 150 3
4 cm2, max = 300 cm2
104) max = 10 2
105) S = 80
106) 32
107) max = 512
108) max = 432 in.2, min = 216 3
4 in.2
109) 180,625 m2 (using all the fence for the square corral)
110)4 3
9πR3
111) A = 7,200, width = 60, length = 120
112) S = 250
113) L2
114) x + 7y = 14
58
Answer KeyTestname: DERIVATIVES WORKSHEET 2 - USING THE DERIVATIVE
115) r > 1
2
116)5
2,
5
2
117) S = 1,152
118)5 10
3 ft3
119)2000π
9 3
120) 105 cm wide × 154 cm high
121) 12 3 ft2
122) 500
123) 30 in. × 30 in. × 40 in.
124) 12 in. × 18 in.
125) 4 1 + 22/3 3/2 m
126) 18 m3
127) The rectangular region should be 500
π m × 250 m, with the radius of each semicircular region being
250
π m.
128) If x , y represent the legs of the triangle, then x2 + y2 = 152.
Solving for y, y = 225 - x2
A(x) = xy = x 225 - x2
A'(x) = - x2
2 225 - x2 +
225 - x2
2
Solving A'(x) = 0, x = ± 15 2
2
Substitute and solve for y: (15 2
2)2
+ y2 = 225 ; y = 15 2
2 ∴ x = y.
129) A
130) C
131) C
132) B
133) D
134) C
135) D
136) A
137) C
138) D
139) D
140) A
141) C
142) 28 m × 28 m × 16 m
143) r = 3 in., h = 6 in.
144) 2000π in.3
59
Answer KeyTestname: DERIVATIVES WORKSHEET 2 - USING THE DERIVATIVE
145) - 1
4
146) (1, 1)
147) 3003
4 cm2
148) (-1, 0)
149) 18
150) see answers in the book
151) see answers in the book
152) A
153) D
154) C
155) C
156) B
157) C
158) A
159) A
160) C
161) D
162) C
163) B
164) See answers in the book
60
Integration Worksheet 1 - Understanding the Definite Integral
Show all work on your paper as described in class. Video links are included throughout for instruction on how to do
the various types of problems. Important: Work the problems to match everything that was shown in the videos. For
example: Suppose a video shows 3 ways to do a problem, (such as algebraically, graphically, and numerically), then
your work should show these 3 ways also. That is , each video is a model for the work I want to see on your paper.
More videos will have been added to the online version of this worksheet by the time you get here!
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Estimate the value of the quantity.
1) The table shows the velocity of a remote controlled race car moving along a dirt path for 8 seconds.
Estimate the distance traveled by the car using 8 sub-intervals of length 1 with left-end point values.
Time
(sec)
Velocity
(in./sec)
0
1
2
3
4
5
6
7
8
0
10
16
12
22
25
27
12
5
A) 114 in. B) 124 in. C) 248 in. D) 129 in.
2) The table shows the velocity of a remote controlled race car moving along a dirt path for 8 seconds.
Estimate the distance traveled by the car using 8 sub-intervals of length 1 with right-end point values.
Time
(sec)
Velocity
(in./sec)
0
1
2
3
4
5
6
7
8
0
8
23
30
29
27
30
25
4
A) 176 in. B) 166 in. C) 172 in. D) 182 in.
1
3) Joe wants to find out how far it is across the lake. His boat has a speedometer but no odometer. The table
shows the boats velocity at 10 second intervals. Estimate the distance across the lake using right-end
point values.
Time
(sec)
Velocity
(ft/sec)
0
10
20
30
40
50
60
70
80
90
100
0
12
30
53
50
55
52
55
45
15
0
A) 3670 ft B) 367 ft C) 3770 ft D) 5500 ft
4) A piece of tissue paper is picked up in gusty wind. The table shows the velocity of the paper at 2 second
intervals. Estimate the distance the paper travelled using left-endpoints.
Time
(sec)
Velocity
(ft/sec)
0
2
4
6
8
10
12
14
16
0
8
12
6
21
26
16
10
2
A) 182 ft B) 202 ft C) 101 ft D) 179 ft
5) The velocity of a projectile fired straight into the air is given every half second. Use right endpoints to
estimate the distance the projectile travelled in four seconds.
Time
(sec)
Velocity
(m/sec)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
139
134.1
129.2
124.3
119.4
114.5
109.6
104.7
99.8
A) 935.6 m B) 974.8 m C) 487.4 m D) 467.8 m
2
In the following problems, f(x) represents the velocity of an object in ft/sec at time x seconds moving on a straight
track. Estimate the distance the object has traveled during the given time interval using the given number of
rectangles and using left or right sums as instructed.
6) f(x) = x2 between x = 0 and x = 4 using a left sum with two rectangles of equal width.
A) 8 B) 38.75 C) 20 D) 40
7) f(x) = x2 between x = 0 and x = 1 using a right sum with two rectangles of equal width.
A) .625 B) .75 C) .125 D) .3145
8) f(x) = 1
x between x = 1 and x = 9 using a right sum with two rectangles of equal width.
A)56
45B)
24
5C)
8
15D)
56
5
9) f(x) = x2 between x = 2 and x = 6 using a left sum with four rectangles of equal width.
A) 86 B) 54 C) 69 D) 62
10) f(x) = 1
x between x = 1 and x = 6 using an left sum with two rectangles of equal width.
A)95
14B)
15
28C)
45
14D)
95
84
11) f(x) = x2 between x = 1 and x = 5 using a right sum with four rectangles of equal width.
A) 69 B) 41 C) 54 D) 30
3
Graph the function f(x) over the given interval. Partition the interval into 4 sub-intervals of equal length. Then add
to your sketch the rectangles associated with the left sums or right sums as directed. NOTE: These sums are called
Riemann sums, f(x) Δx∑ , where x is the left endpoint of a sub interval for a "left Riemann sum" and x is the right
endpoint for a "right Riemann sum". And Δx = right endpoint of interval - left endpoint of interval
number of sub intervals =
b - a
n
12) f(x) = 2x + 4, [0, 2], left-hand endpoint
x0.5 1 1.5 2
y8
7
6
5
4
3
2
1
x0.5 1 1.5 2
y8
7
6
5
4
3
2
1
A)
x0.5 1 1.5 2
y
8
7
6
5
4
3
2
1
x0.5 1 1.5 2
y
8
7
6
5
4
3
2
1
B)
x0.5 1 1.5 2
y
8
7
6
5
4
3
2
1
x0.5 1 1.5 2
y
8
7
6
5
4
3
2
1
C)
x0.5 1 1.5 2
y
8
7
6
5
4
3
2
1
x0.5 1 1.5 2
y
8
7
6
5
4
3
2
1
D)
x0.5 1 1.5 2
y
8
7
6
5
4
3
2
1
x0.5 1 1.5 2
y
8
7
6
5
4
3
2
1
4
13) f(x) = -2x - 1, [0, 2], left-hand endpoint
x0.5 1 1.5 2
y
-1
-2
-3
-4
-5
-6
-7
-8
x0.5 1 1.5 2
y
-1
-2
-3
-4
-5
-6
-7
-8
A)
x0.5 1 1.5 2
y
-1
-2
-3
-4
-5
-6
-7
-8
x0.5 1 1.5 2
y
-1
-2
-3
-4
-5
-6
-7
-8
B)
x0.5 1 1.5 2
y
-1
-2
-3
-4
-5
-6
-7
-8
x0.5 1 1.5 2
y
-1
-2
-3
-4
-5
-6
-7
-8
C)
x0.5 1 1.5 2
y
-1
-2
-3
-4
-5
-6
-7
-8
x0.5 1 1.5 2
y
-1
-2
-3
-4
-5
-6
-7
-8
D)
x0.5 1 1.5 2
y
-1
-2
-3
-4
-5
-6
-7
-8
x0.5 1 1.5 2
y
-1
-2
-3
-4
-5
-6
-7
-8
14) f(x) = x2 - 2, [0, 8], left-hand endpoint
x2 4 6 8
y646056524844403632282420161284
x2 4 6 8
y646056524844403632282420161284
5
A)
x2 4 6 8
y646056524844403632282420161284
x2 4 6 8
y646056524844403632282420161284
B)
x2 4 6 8
y646056524844403632282420161284
x2 4 6 8
y646056524844403632282420161284
C)
x2 4 6 8
y646056524844403632282420161284
x2 4 6 8
y646056524844403632282420161284
D)
x2 4 6 8
y646056524844403632282420161284
x2 4 6 8
y646056524844403632282420161284
15) f(x) = x2 - 2, [0, 8], right-hand endpoint
x2 4 6 8
y646056524844403632282420161284
x2 4 6 8
y646056524844403632282420161284
6
A)
x2 4 6 8
y646056524844403632282420161284
x2 4 6 8
y646056524844403632282420161284
B)
x2 4 6 8
y646056524844403632282420161284
x2 4 6 8
y646056524844403632282420161284
C)
x2 4 6 8
y646056524844403632282420161284
x2 4 6 8
y646056524844403632282420161284
D)
x2 4 6 8
y646056524844403632282420161284
x2 4 6 8
y646056524844403632282420161284
16) f(x) = -3x2, [0, 4], left-hand endpoint
x1 2 3
y4
-4-8
-12-16-20-24-28-32-36-40-44-48-52-56-60
x1 2 3
y4
-4-8
-12-16-20-24-28-32-36-40-44-48-52-56-60
7
A)
x1 2 3
y4
-4-8
-12-16-20-24-28-32-36-40-44-48-52-56-60
x1 2 3
y4
-4-8
-12-16-20-24-28-32-36-40-44-48-52-56-60
B)
x1 2 3
y4
-4-8
-12-16-20-24-28-32-36-40-44-48-52-56-60
x1 2 3
y4
-4-8
-12-16-20-24-28-32-36-40-44-48-52-56-60
C)
x1 2 3
y4
-4-8
-12-16-20-24-28-32-36-40-44-48-52-56-60
x1 2 3
y4
-4-8
-12-16-20-24-28-32-36-40-44-48-52-56-60
D)
x1 2 3
y4
-4-8
-12-16-20-24-28-32-36-40-44-48-52-56-60
x1 2 3
y4
-4-8
-12-16-20-24-28-32-36-40-44-48-52-56-60
8
17) f(x) = cos x + 2, [0, 2π], left-hand endpoint
xπ2
π 3π2
2π
y
5
4
3
2
1
xπ2
π 3π2
2π
y
5
4
3
2
1
A)
xπ2
π 3π2
2π
y
5
4
3
2
1
xπ2
π 3π2
2π
y
5
4
3
2
1
B)
xπ2
π 3π2
2π
y
5
4
3
2
1
xπ2
π 3π2
2π
y
5
4
3
2
1
C)
xπ2
π 3π2
2π
y
5
4
3
2
1
xπ2
π 3π2
2π
y
5
4
3
2
1
D)
xπ2
π 3π2
2π
y
5
4
3
2
1
xπ2
π 3π2
2π
y
5
4
3
2
1
9
18) f(x) = cos x + 3, [0, 2π], right-hand endpoint
xπ2
π 3π2
2π
y
5
4
3
2
1
xπ2
π 3π2
2π
y
5
4
3
2
1
A)
xπ2
π 3π2
2π
y
5
4
3
2
1
xπ2
π 3π2
2π
y
5
4
3
2
1
B)
xπ2
π 3π2
2π
y
5
4
3
2
1
xπ2
π 3π2
2π
y
5
4
3
2
1
C)
xπ2
π 3π2
2π
y
5
4
3
2
1
xπ2
π 3π2
2π
y
5
4
3
2
1
D)
xπ2
π 3π2
2π
y
5
4
3
2
1
xπ2
π 3π2
2π
y
5
4
3
2
1
Using the "Approximating Distance Traveled Using Tables and Graphs" applets
ESSAY.
19) Go to http://www.mathopenref.com/calcinttable.html Click "open in resizeable window" to enlarge the applet. Choose "#5 You Try" from the drop down menu.
Put in v(t) = 2t and select velocity to use: left Put in intervals = 4 Then press Enter.
a) Write down the table of t and v(t) values generated. (Note: t is in sec. and v(t) is in ft/sec.) Calculate the
the estimated total distance traveled using the left velocity values for each time sub interval.
b) What does the applet indicate as the total distance traveled? Did it match your answer?
10
20) Go to http://www.mathopenref.com/calcinttable.html Click "open in resizeable window" to enlarge the applet. Choose "#5 You Try" from the drop down menu.
Put in v(t) = 2t and select velocity to use: left Put in intervals = 8 Then press Enter.
a) Write down the table of t and v(t) values generated. (Note: t is in sec. and v(t) is in ft/sec.) Calculate the
the estimated total distance traveled using the left velocity values for each time sub interval.
b) What does the applet indicate as the total distance traveled? Did it match your answer?
c) Put in intervals = 16 . What does the applet indicate as the total distance traveled?
d) Put in intervals = 64 . What does the applet indicate as the total distance traveled?
e) Put in intervals = 128 . What does the applet indicate as the total distance traveled?
f) Put in intervals = 1000 . What does the applet indicate as the total distance traveled?
g) Do the total distance values seem to be converging to a value as the number of intervals gets larger?
What value?
21) Go to http://www.mathopenref.com/calcinttable.html Click "open in resizeable window" to enlarge the applet. Choose "#5 You Try" from the drop down menu.
Put in v(t) = 2t and select velocity to use: right Put in intervals = 8 Then press Enter.
a) Write down the table of t and v(t) values generated. (Note: t is in sec. and v(t) is in ft/sec.) Calculate the
the estimated total distance (ft) traveled using the right velocity values for each time sub interval.
b) What does the applet indicate as the total distance traveled? Did it match your answer?
c) Put in intervals = 16 . What does the applet indicate as the total distance traveled?
d) Put in intervals = 64 . What does the applet indicate as the total distance traveled?
e) Put in intervals = 128 . What does the applet indicate as the total distance traveled?
f) Put in intervals = 1000 . What does the applet indicate as the total distance traveled?
g) Do the total distance values seem to be converging to a value as the number of intervals gets larger?
What value?
h) In the previous problem the distance values increased toward a limit value, in this problem the values
decreased toward a limit value, why?
22) Go to http://www.mathopenref.com/calcinttable.html Click "open in resizeable window" to enlarge the applet. Choose "#5 You Try" from the drop down menu.
a) Put in various functions, starting and ending t values, number of intervals and experiment until you
really know what is going on with the applet. Show one such experimentation and comment on it.
b) Put in v(t) = sin(t) , start = 0 and end = 2pi , select velocity to use: right , intervals = 100 . What does the
applet show as the total distance traveled? (hint: E-17 means 10-17 a very small number!)
c) How can an object travel and end up having zero distance traveled? Explain what is really going on
here - what does the applet actually do in its calculations?
11
23) Go to http://www.mathopenref.com/calcintgraph.html This is a graphical applet. Click "open in resizeable window" to enlarge the applet. Choose "#5 You Try"
from the drop down menu.
Put in v(t) = t^2 and select velocity to use: left Put in intervals = 4 Then press Enter.
a) Copy the graph generated. (Note: t is in sec. and v(t) is in ft/sec.)
b) What does the applet indicate as the total distance traveled? (it says "pink area =")
c) Put in intervals = 16 . What does the applet indicate as the total distance traveled?
d) Put in intervals = 64 . What does the applet indicate as the total distance traveled?
e) Put in intervals = 128 . What does the applet indicate as the total distance traveled?
f) Put in intervals = 1000 . What does the applet indicate as the total distance traveled?
g) Do the total distance values seem to be converging to a value as the number of intervals gets larger?
What value?
24) Go to http://www.mathopenref.com/calcintgraph.html This is a graphical applet. Click "open in resizeable window" to enlarge the applet. Choose "#5 You Try"
from the drop down menu.
Put in v(t) = t^2 and select velocity to use: right Put in intervals = 4 Then press Enter.
a) Copy the graph generated. (Note: t is in sec. and v(t) is in ft/sec.)
b) What does the applet indicate as the total distance traveled? (it says "pink area =")
c) Put in intervals = 16 . What does the applet indicate as the total distance traveled?
d) Put in intervals = 64 . What does the applet indicate as the total distance traveled?
e) Put in intervals = 128 . What does the applet indicate as the total distance traveled?
f) Put in intervals = 1000 . What does the applet indicate as the total distance traveled?
g) Do the total distance values seem to be converging to a value as the number of intervals gets larger?
What value?
h) In the previous problem the distance values increased toward a limit value, in this problem the values
decreased toward a limit value, why? Does the function being increasing or decreasing on the interval
have anything to do with whether the left estimates are over or under?
i) Use the applet to experiment with increasing functions and decreasing functions until you know enough
to state a rule concerning the relationship between these 6 concepts: v(t) increasing, v(t) decreasing, Left
estimate, Right estimate, over estimate, under estimate.
FROM THE BOOK
25) Read and take notes on section 5.1
26) Do section 5.1 #1 - 23 odd
The Definite Integral
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Evaluate the sum.
27)7
k = 1
k∑
A) 7 B) 14 C) 56 D) 28
12
28)14
k = 1
k3∑
A) 11,025 B) 1015 C) 2744 D) 3150
29)25
k = 4
6∑
A) 146 B) 132 C) 150 D) 126
30)6
k = 1
k2 - 8∑
A) 28 B) 83 C) 43 D) 91
FROM THE BOOK
31) Read section 5.2 and take notes on the Definite Integral, The Definite Integral as an Area, and More
general Riemann Sums
ESSAY.
32) a) Go to the internet and find out info on Bernhard Riemann and write a brief paragraph about him and hisRiemann sums.
b) Go to http://mathworld.wolfram.com/RiemannSum.html to see a Riemann sum applet(similar idea to the applet used above but has a different layout). You can use this applet in the some ofthe problems below.
33) There is a great Riemann sum program for the TI-84 . Go tohttp://www.calcblog.com/riemann-sum-program-ti83-ti84/ There you see a nice tutorial
about Riemann sums then you scroll down to the Using the RIEMANN Program section. It shows how to
download the program into your TI-84 from your computer, and various links show you how to get the
program into your computer in the first place. (I did it and it worked!) Then it shows how to use the
program. You can use this program in the some of the problems below. You must have this program or a simplerversion of it in your TI-84.
FROM THE BOOK
34) Do section 5.2 #1, 3, 7, 9; For [11-21 odd, 27] you either use your TI Riemann program or use the wolfram
Riemann sum program mentioned above; #23, 25, 29, 31, 33
13
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Graph the integrand and use areas to evaluate the integral. Then run a Riemann program or applet with n = 100 sub
intervals to verify your answer. (n = 100 is a large enough amount of intervals to give a pretty close estimate for
these problems below because the widths of the intervals are not too large. (Only the problem with 16 - x2 may
have an answer that is significantly far off from the Riemann estimate with n=100 sub intervals - learn why in class)
35)7
-1
4 dx∫
A) 32 B) 8 C) 24 D) 16
36)10
6
x dx∫
A) 8 B) 16 C) 64 D) 32
37)4
0
8x dx∫
A) 128 B) 8 C) 32 D) 64
38)5
-5
(2x + 10) dx∫
A) 100 B) 20 C) 50 D) 200
39)9
-2
x dx∫
A)85
2B) 85 C) 11 D)
77
2
40)4
-4
16 - x2 dx∫
A) 4π B) 16π C) 16 D) 8π
The Fundamental Theorem of Calculus
14
FROM THE BOOK
41) Read section 5.3 and takes notes on The Fundamental Theorem of Calculus, and on the Definite Integral as
an Average. AND know the proof steps of The Fundamental Theorem of Calculus for a test . The proof
steps follow after the statement of the theorem.
The F.T.O.C says that if you can find an antiderivative F(t) for your function f(t), then instead of
computing area estimates or limits of sums to find b
a
f(t)∫ dt you can just subtract F(b) - F(a) . A very
simple subtraction! And this theorem ties the derivative ( F '(t) = f(t) ) to sums ( f(t) △t∑ ) in a very slick
way!
ESSAY.
42) Go to http://calculusapplets.com/fundtheorem.html a) Read and take notes on the first paragraph.
b) In the "Try the following" section, do and answer #3 Hit "launch the presentation" to get a large
re-sizeable view.
c) For the #3 Parabola example, we have f(x) = x2 and its derivative f '(x) = 2x . On the interval [1,3] as
shown, compute the change in f(x) , that is, compute f(3) - f(1).
d) Now, using your Riemann sum program or an applet, find 3
1
2x dx∫ using n = 100 sub intervals. Do
you get the same (or nearly the same) answer as in (c)?
e) What is the f(b) - f(a) total change value the applet shows?
f) What is the 3
1
2x dx∫ area value the applet shows? Are the answers to e and f equal?
g) Do your own function example f(x) and make up your own interval and verify that f(b) - f(a) equals
b
a
f '(x) dx∫
43) The rate at which the world's oil is being consumed is r(t) = 32e.05t billion barrels per year where t = years
after 2004.
a) Set up a definite integral that will give the total amount of barrels of oil consumed from 2004 to 2012
(assuming the rate functions remains accurate over that time)
b) Find(estimate) the value of the integral by using technology methods with n = 100 intervals
c) Now you want to answer the problem by using the Fundamental Theorem of Calculus but you need an
antiderivative (a function whose derivative is 32e^(.05t) ) , call it f(t), so you can find f(b) - f(a). Go to
http://www.wolframalpha.com and put in: antiderivative 32e^(.05t) and an antiderivative shows up.
15
44) A spill of radioactive iodine occurs and the rate of decay is given by r(t) = 2.4e-0.004t millirems/hour
where t = 0 hours is the time of the spill.
a) An acceptable rate of radiation is 0.6 millirems/hour. How many hours will it take to reach that rate?
b) Set up a definite integral that will give the total amount of radiation (in millirems) that will be emitted
from t = 0 hours to the time of the acceptable rate level?
c) Find(estimate) the value of the integral by using technology methods with n = 500 intervals
c) Now you want to answer the problem by using the Fundamental Theorem of Calculus but you need an
antiderivate (a function whose derivative is 2.4e-0.004t ) , call it f(t), so you can then find f(b) - f(a). Go to
http://www.wolframalpha.com and put in: antiderivative 2.4e^(-.004t) and an antiderivative shows
up.
45) A runner is jogging on a straight track and is speeding up and slowing down with velocity function v(t) =
cos(t) + 5 miles/hr where t is the hours after he/she started.
a) Set up a definite integral that will give the total distance he/she ran from the time period of t = 1 hour to
t = 2 hours.
b) Find(estimate) the value of the integral by using technology methods with n = 100 intervals
c) Now you want to answer the problem by using the Fundamental Theorem of Calculus but you need an
antiderivate (a function whose derivative is cos(t)+5 , call it f(t), so you can find f(b) - f(a). Go to
http://www.wolframalpha.com and put in: antiderivative cos(t)+5 and an antiderivative shows up.
FROM THE BOOK
46) Do section 5.3 #15, 17
ESSAY.
47) Go to http://calculusapplets.com/aveval.html and click "hide answer". Put in f(x) = x^2 and put in
ymax = 5 and hit enter.
You want to find the average value of f(x) on a = 0 to b = 2. Think of the function as a velocity of a man
walking in ft/sec. Think of the average value of the function as the constant velocity he could have walked
to cover the same ground as if he walked according to the function.
a) Click on the black square at the origin and move it up to what you think the average value would be.
(Another way to think of it is: Move the black square up to a point where the area under the resulting
rectangle equals the yellow area under the f(x) curve.) What is your average value guess?
b) Click "show answer" and see what the applet says. Write that down.
c) Compute the average value of f(x) = x2 on x = 0 to 2 by using the Average Value of f formula in section
5.3 in the book: Average value of f on a to b = 1
b-a
b
a
f(x) dx∫ . Use F(x) = x3
3 as an antiderivative of f(x)
= x2 . Did the answer match with the applet's answer for the average value?
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the average value of the function over the given interval. If you can think of an antiderivative for the function
then use it, if not, go to http://www.wolframalpha.com and put in: definite integral and then put in the function
to integrate and the limits of integration given.
48) f(x) = 4x on [5, 7]
A) 12 B) 48 C) 24 D) 96
16
49) f(x) = 4 - x on [0, 4]
A) 32 B) 2 C) 8 D) 4
50) f(x) = x on [-6, 6]
A) 6 B)3
2C) 36 D) 3
51) y = x2 - 2x + 4; [0, 2]
A) 3 B)22
3C) 4 D)
10
3
52) y = 6 - x2; [-5, 2]
A) - 1
3B) -
13
3C) 3 D)
81
7
FROM THE BOOK
53) Do section 5.3 #9, 11, 19, 27, 31, 35, 37, 39, 41 (for 19, 27, 39, 41 use http://www.wolframalpha.com to
evaluate the definite integral involved, or, use your Riemann program with n = 100 sub-intervals))
54) Read and take notes on section 5.4 Properties of Limits of Integration, Properties of Sums and Constant
Multiples of the Integrand, Area Between Curves, Using the Fundamental Theorem to Compute Integrals
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use your technology methods to estimate the area under the graph of the given function on the stated interval as
instructed. If you can think of an antiderivative, F(x), then use that and the FTOC.
55) f(x) = x2 between x = 0 and x = 2 using a lower sum with two rectangles of equal width.
A) 4.5 B) 1 C) 5 D) 2.5
56) f(x) = x2 between x = 0 and x = 1 using an upper sum with two rectangles of equal width.
A) .125 B) .75 C) .3145 D) .625
57) f(x) = 1
x between x = 1 and x = 8 using a lower sum with two rectangles of equal width.
A)2359
20736B) -
1
3C) -
1
6D)
2359
6912
58) f(x) = 1
x between x = 2 and x = 5 using a upper sum with two rectangles of equal width.
A)117
980B) -
29
70C)
39
980D) -
29
35
59) f(x) = x2 between x = 4 and x = 8 using an upper sum with four rectangles of equal width.
A) 174 B) 149 C) 126 D) 165
17
Solve the problem.
60) Suppose that 2
1
f(x) dx∫ = -2. Find 2
1
6f(u) du∫ and 2
1
- f(u) du∫ .
A) 4; -2 B) 6; 2 C) -12; - 1
2D) -12; 2
61) Suppose that -1
-4
g(t) dt∫ = -12. Find -1
-4
g(x)
-12 dx∫ and
-4
-1
- g(t) dt∫ .
A) 1; 12 B) 1; -12 C) 0; -12 D) -1; 12
62) Suppose that f and g are continuous and that 8
4
f(x) dx∫ = -5 and 8
4
g(x) dx∫ = 9.
Find 8
4
4f(x) + g(x) dx∫ .
A) 13 B) -11 C) 16 D) 31
63) Suppose that f and g are continuous and that 6
2
f(x) dx∫ = -2 and 6
2
g(x) dx∫ = 7.
Find 6
2
f(x) - 2g(x) dx∫ .
A) -18 B) 12 C) -16 D) -9
64) Suppose that f and g are continuous and that 6
2
f(x) dx∫ = -5 and 6
2
g(x) dx∫ = 7.
Find 2
6
g(x) - f(x) dx∫ .
A) 12 B) -2 C) -12 D) 2
65) Suppose that h is continuous and that 5
-4
h(x) dx∫ = 7 and 6
5
h(x) dx = -9.∫ Find 6
-4
h(t) dt∫ and
-4
6
h(t) dt.∫
A) -2; 2 B) 2; -2 C) 16; -16 D) -16; 16
66) Suppose that f is continuous and that 4
-4
f(z) dz = 0∫ and 5
-4
f(z) dz = 4.∫ Find -5
4
2f(x) dx∫ .
A) -2 B) 8 C) -8 D) -4
18
Evaluate the integral. Think of antiderivative or use Wolframalpha to get one, then use the FTOC.
67)13
1
x dx∫
A) 13 - 1 B) 12 C) - 6 D) 6
68) 2π
3π/2
θ dθ∫
A)9π2
8B)
7π2
8C)
π2
2D)
π2
8
69)1/8
0
t2 dt∫
A)1
1536B) -
1
1536C) -
1
8D) 1536
70)
313
0
x2 dx∫
A) 169 B)3
13
3C)
13 13
3D)
13
3
71)2π
π
θ2 dθ∫
A)π3
3B)
27π3
24C)
7π3
3D)
π3
24
72)12
8
7 dx∫
A) 84 B) 28 C) -76 D) 0
73)15
2
z- 15 dz∫
A) - 19
2 15 B) -
19
2 + 2 15 C) -
15
2 15 D) - 15
FROM THE BOOK
74) Do section 5.4 #3-15 odd (for 5-15 odd, set up the integral, then evaluate using technology methods such
as the calculator program with n= 100 sub-intervals, or http://www.wolframalpha.com ,
19-25 odd, 37, 49; Review for Chapter 5 #5-13 odd, 17, 19, 21, 33, 47
19
75) Read and take notes on section 6.1 Visualizing antiderivatives using slopes, Computing values of an
Antiderivative using the FTOC.
76) Do section 6.1 #1-8 ALL, 15, 19, 21, 22, 23
Now we learn how to find Antiderivatives by hand (analytically, or,
algebraically)
77) Read and take notes on section 6.2 (all of it!)
78) Do section 6.2 #1 - 77 odd, 81
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Evaluate the integral.
79)16
0
2 x∫ dx
A) 128 B) 192 C) 16 D)256
3
80)6
-2
6x5∫ dx
A) 1280 B) 279,552 C) -46,592 D) 46,592
81)4
1
t2 + 1
t dt∫
A)92
5B)
77
5C)
72
5D) 32
82)π/2
0
18 sin x dx∫
A) 0 B) 1 C) 18 D) -18
Find the total area of the region between the curve and the x-axis.
83) y = 2x + 7; 1 ≤ x ≤ 5
A) 52 B) 18 C) 26 D) 9
84) y = 2x - x2; 0 ≤ x ≤ 2
A)2
3B)
4
3C)
7
3D)
5
3
20
85) y = 3
x3; 1 ≤ x ≤ 3
A)1
3B)
4
3C)
1
2D) 3
86) y = -x2 + 9; 0 ≤ x ≤ 5
A)98
3B)
5
9C)
10
9D)
10
3
87) y = 1
x; 1 ≤ x ≤ 4
A) 2 B) 4 C)1
4D)
1
2
Find the average value of the function over the given interval.
88) f(x) = 10x on [1, 3]
A) 40 B) 10 C) 20 D) 80
89) f(x) = 4 - x on [0, 4]
A) 2 B) 32 C) 4 D) 8
90) y = x2 - 2x + 6; [0, 2]
A)16
3B) 5 C) 6 D)
34
3
91) y = 3 - x2; [-2, 2]
A) 3 B)5
3C) -
1
3D) 0
Construction of Antiderivatives for functions we can't use any techniques to obtain.
(There are still more techniques of integration to learn but still there will be functions forwhich we just cannot obtain a closed form antiderivative - we use the construction method
to find values of the antiderivative function in that case.)
FROM THE BOOK
92) Read and take notes on section 6.4 The Second Fundamental Theorem of Calculus. Know the proof for a
test of the statement F(x) = x
a
f(t)dt∫ is an antiderivative of f(x)
21
ESSAY.
93) a) Evaluate x
a
2t dt∫ using the FTOC and call the result F(x). ("a" is any constant)
b) Take the derivative of F(x) you found in part (a) and show you get 2x and thus demonstrating that
x
a
2t dt∫ is an antiderivative for 2x.
94) a) Make a table of values of F(x) = x
0
2t dt∫ for x = 0, 1, 2, 3, 4 by using your Riemann program with
n=100 sub-intervals (use left sums) to evaluate the integral at each x value, one at a time.
b) Graph your table of values. Does the graph of F(x) look like it's derivative function would be 2x ?
95) a) Make a table of values of F(x) = x
0
e-t2dt∫ for x = 0, .2, .4, .6, .8, 1 by using your Riemann program
with n=100 sub-intervals (use left sums) to evaluate the integral at each x value.
b) Graph your table of values. Does the graph of F(x) look like it's derivative function would be e-x2 ?
FROM THE BOOK
96) Do section 6.4 #5-19 odd, 25-35 odd
97) Read and take notes on section 6.5 Equations of Motion. Know (for a test) the derivation steps of how we
start with the acceleration of an object due to gravity, -g, and end up knowing the position function of the
object, s(t).
98) Do section 6.5 #3 (use g = 32 ft
sec2 ), 7, 8
ESSAY.
99) On Mars the acceleration due to gravity is 12 ft
sec2 . (On Earth gravity is much stronger at 32
ft
sec2 . ) In
the movie, John Carter, it shows Carter leaping about 100 feet up into the air on Mars. John Carter is an
Earth man who has been transported to Mars so his leg muscles have been built to handle Earth's gravity
while Mars gravity is a lot less. On Earth, Michael Jordan (a famous basketball player) had a vertical jump
velocity of 16 ft/sec. Suppose John Carter could triple that initial jump velocity due to being on Mars, so
his initial velocity would be v0 = 48 ft/sec .
a) How high could he jump on Mars?
b) How long could he stay in the air before he hit the ground?
c) The movie shows Carter jumping about 100 ft. high. Is that about right by the Calculus?
d) What would his speed be when he hit the ground?
22
100) referring to the previous problem, suppose the planet John Carter transported to had an acceleration due
to gravity of 2 ft
sec2 and his initial jump velocity was still v0 = 48 ft/sec .
a) How high could he jump under these conditions?
b) How long could he stay in the air before he hit the ground?
c) What would his speed be when he hit the ground?
More advanced Integration Methods to find antiderivatives by hand (algebraically) -
Integration by Substitution
FROM THE BOOK
101) Read and take notes on section 7.1 but where they use the letter "w" for the substitution, most people
(including me) use the letter "u" , so use "u" . Know the proof steps of why substitution works for a test.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Evaluate the integral using the given substitution.
102) x cos (4x2)∫ dx, u = 4x2
A) sin(4x2) + C B)x2
2 sin (4x2) + C C)
1
8 sin (4x2) + C D)
1
u sin (u) + C
103) 2 - sin t
2
2cos
t
2∫ dt, u = 2 - sin t
2
A)1
32 - sin
t
2
3sin
t
2 + C B)
2
32 - cos
t
2
3 + C
C) 2 2 - sin t
2
3 + C D) -
2
32 - sin
t
2
3 + C
104) x4(x5 - 4)5∫ dx , u = x5 - 4
A)1
6(x5 - 4)6 + C B)
1
30x30 - 4 + C C)
1
30(x5 - 4)6 + C D)
1
20(x5 - 4)4 + C
105)8s3 ds
8 - s4∫ , u = 8 - s4
A) -4s3 8 - s4 + C B)-2
2 8 - s4 + C C)
4s4
8 - s4D) -4 8 - s4 + C
106)dx
7x + 2∫ , u = 7x + 2
A)7
2
1
7x + 2 + C B) 2 7x + 2 + C C)
2
77x + 2 + C D)
1
7(7x + 2)3/2 + C
23
107) 18(6x - 8)-6∫ dx , u = 6x - 8
A) - 6
5(6x - 8)-5 + C B) -
3
5(6x - 8)-5 + C C) -
3
7(6x - 8)-7 + C D) (6x - 8)-5 + C
FROM THE BOOK
108) Do section 7.1 #3-39 odd, 47-75 odd, 91-97 odd, 101, 103, 109
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Evaluate the integral.
109)x dx
(7x2 + 3)5∫
A) - 7
3(7x2 + 3)-4 + C B) -
1
56(7x2 + 3)-4 + C
C) - 7
3(7x2 + 3)-6 + C D) -
1
14(7x2 + 3)-6 + C
110)sin t
(3 + cos t)4 dt∫
A)3
(3 + cos t)3 + C B)
1
(3 + cos t)3 + C C)
1
5(3 + cos t)5 + C D)
1
3(3 + cos t)3 + C
111)ln x7
x dx∫
A)1
ln x7 + C B)
1
2(ln x7)2 + C C)
1
14(ln x7)2 + C D)
1
7(ln x7)2 + C
112)dx
x 9x2 - 6∫
A)6
6 sin-1
1
26 x + C B)
6
6 sec-1
1
26 x + C
C)1
3 sec-1 3x - 6 + C D)
1
3 sec-1 3 x + C
113)1
t2 sin
3
t + 5∫ dt
A)1
3 cos
3
t + 5 + C B) -cos
3
t + 5 + C C) -
1
3 cos
3
t + 5 + C D) 3 cos
3
t + 5 + C
24
114) sin (8x - 4) dx∫A)
1
8 cos (8x - 4) + C B) 8 cos (8x - 4) + C
C) -cos (8x - 4) + C D) - 1
8 cos (8x - 4) + C
115)dx
xln x3∫
A) ln x3+ C B)1
3ln ln x3 + C C)
1
3ln x3 + C D) ln ln x3 + C
116) x3∫ x4 + 2 dx
A)2
3x4 + 2 3/2 + C B) -
1
2x4 + 2 -1/2 + C
C)1
6x4 + 2 3/2 + C D)
8
3x4 + 2 3/2 + C
117) x5(x6 - 6)4∫ dx
A) x6 - 6 5 + C B)x6 - 6 3
18 + C C)
x6 - 6 5
6 + C D)
x6 - 6 5
30 + C
ESSAY.
Solve the problem.
118) Evaluate x2∫ 5x3 + 1 dx.
119) Evaluate t2
16t3 + 5∫ dt.
120) Evaluate ( x + 4)3
3 x∫ dx.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the substitution formula to evaluate the integral.
121)1
0
x + 16∫ dx
A) 17 17 - 64 B)34
317 -
128
3C)
51
217 -
51
2D)
34
317
25
122)0
-1
2t
2 + t2 3 dt∫
A) - 5
72B)
5
72C) -
5
18D) -
5
36
123)0
-1
3t
4 + t2 4 dt∫
A) - 61
8000B)
61
16000C) -
183
8000D) -
61
16000
124)1
0
4 r dr
4 + 2r2∫
A) - 2 6 + 4 B) 6 - 2 C) 2 6 - 4 D)6
2 - 1
125)4
1
4 - x
x dx∫
A) - 5
2B) 5 C) 10 D)
5
2
126)π
0
(1 + cos 5t) 2 sin 5t dt∫
A)1
15B)
8
3C)
8
15D)
1
5
127)π/2
0
cos x
(4 + 2 sin x)3 dx∫
A) - 5
288B)
5
288C)
5
576D) -
15
64
128)π/8
0
(1 + etan 2x)∫ sec2 2x dx
A) e B) - e
2C) 2e D)
e
2
129)3π/2
π
sin θ dθ
2 + cos θ∫
A) -ln 2 B) ln 3 C) 0 D) -ln 3
26
130)3π/4
0
tan x
3∫ dx
A)3 ln 2
2B)
3 2
2C)
-3 ln 2
2D)
-3 2
2
131)ln 3/4
0
4 e4x dx
1 + e8x∫
A)π
12B) -
π
6C) -
π
12D)
π
6
Find the area enclosed by the given curves.
132) Find the area of the region between the curve y = 6x/(1 + x2) and the interval -2 ≤ x ≤ 2 of the x-axis.
A) 6 ln 5 B) 6 e5 C) ln 5 D) 0
133) Find the area of the region between the curve y = 53-x and the interval 0 ≤ x ≤ 2 on the x-axis.
A) 120 ln 5 B)120
ln 5C)
125
ln 5D) 125
27
Answer KeyTestname: INTEGRATION WORKSHEET 1 - UNDERSTANDING THE DEFINITE INTEGRAL
1) B
2) A
3) A
4) B
5) D
6) A
7) A
8) A
9) B
10) C
11) C
12) C
13) D
14) C
15) D
16) C
17) B
18) D
19) The applet shows the answer
20) The applet shows the answers
21) The applet shows the answers. For (h) discuss in class.
22) see applet
23) The applet shows the answers
24) The applet shows the answers. For (i) discuss in class.
25)
26) see answers in the book
27) D
28) A
29) B
30) C
31) section 5.2
32)
33)
34) see answers in the book. See your Riemann program or the Riemann applet
35) A
36) D
37) D
38) A
39) A
40) D
41)
42) see applet
43) a) 8
0
32e.05t∫ dt billion barrels b) 314.1 billion barrels (using a left sum, you could use a right sum or average
the two) c) 314.8 billion barrels
28
Answer KeyTestname: INTEGRATION WORKSHEET 1 - UNDERSTANDING THE DEFINITE INTEGRAL
44) a) 346.6 hours b) 346.6
0
2.4e-0.004t∫ dt millirems c) 450.6 millirems (using a left sum, you could use a right
sum or average the two) d) 450.016 millirems
45) a) 2
1
(cos(t)+5)∫ dt miles b) 5.07 miles (using a left sum, you could use a right sum or average the two) c) 5.068
miles
46) see answers in the book
47) a) your guess b) 1.3333333 c) 4
3 yes it matched exactly.
48) C
49) B
50) D
51) D
52) A
53) see answers in the book
54)
55) B
56) D
57) C
58) B
59) A
60) D
61) B
62) B
63) C
64) C
65) A
66) C
67) D
68) B
69) A
70) D
71) C
72) B
73) B
74) see answers in the book
75)
76) See answers in the book, for even numbers discuss in class.
77)
78) see answers in the book
79) D
80) D
81) C
82) C
83) A
84) B
29
Answer KeyTestname: INTEGRATION WORKSHEET 1 - UNDERSTANDING THE DEFINITE INTEGRAL
85) B
86) D
87) A
88) C
89) A
90) A
91) B
92)
93) discuss in class.
94) discuss in class.
95) discuss in class
96) see answers in the book
97)
98) see answers in the book, for #8 discuss in class
99) a) s = 96 ft. max height b) 8 seconds in the air. c) Yes, the movie was in line with the Calculus!! d) -48 ft/sec ,
that's 48 ft/sec downward = 32.7 miles/hour
100) a) s = 576 ft. max height b) 48 seconds in the air c) -48 ft/sec , that's 48 ft/sec downward = 32.7 miles/hour
101)
102) C
103) D
104) C
105) D
106) C
107) B
108) see answers in the book.
109) B
110) D
111) C
112) B
113) A
114) D
115) B
116) C
117) D
118)2
45(5x3 + 1)3/2 + C
119)16t3 + 5
24 + C
120)( x + 4)4
6 + C
121) B
122) A
123) D
124) C
125) B
126) C
127) C
30
Answer KeyTestname: INTEGRATION WORKSHEET 1 - UNDERSTANDING THE DEFINITE INTEGRAL
128) D
129) A
130) A
131) A
132) A
133) B
31
Integration Worksheet 2 - Using the Definite Integral
Show all work on your paper as described in class. Video links are included throughout for instruction on how to do
the various types of problems. Important: Work the problems to match everything that was shown in the videos. For
example: Suppose a video shows 3 ways to do a problem, (such as algebraically, graphically, and numerically), then
your work should show these 3 ways also. That is , each video is a model for the work I want to see on your paper.
More videos will have been added to the online version of this worksheet by the time you get here!
FROM THE BOOK
1) Read and take notes on section 8.1 Areas and Volumes
2) Do section 8.1 #1-8 ALL For # 1-6 find the area first using Calculus then by simple known geometric area
formulas - see if the answers match. Realize that you can always evaluate a definite integral using your
Riemann program or the wolfram site to verify answers. For #4,5 set up the integral then use
http://www.wolframalpha.com to get the antiderivative function.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the area of the shaded region.
3) f(x) = x3 + x2 - 6x
x-4 -2 2 4
y30
20
10
-10
-20
-30(-4, -24)
(0, 0)
(3, 18)
x-4 -2 2 4
y30
20
10
-10
-20
-30(-4, -24)
(0, 0)
(3, 18)
g(x) = 6x
A)343
12B)
81
12C)
937
12D)
768
12
1
4) f(x) = -x3 + x2 + 16x
g(x) = 4x
x-4 -2 2 4 6
y
30
20
10
-10
-20
-30
(-3, -12)
(0, 0)
(4, 16)
x-4 -2 2 4 6
y
30
20
10
-10
-20
-30
(-3, -12)
(0, 0)
(4, 16)
A)1153
12B)
343
12C)
937
12D) -
343
12
5) y = x2 - 4x + 3
x-3 -2 -1 1 2 3 4 5
y8
6
4
2
-2
-4
-6
-8
x-3 -2 -1 1 2 3 4 5
y8
6
4
2
-2
-4
-6
-8
y = x - 1
A)41
6B) 3 C)
9
2D)
25
6
2
6)
x1 2
y
1
-1
-2
x1 2
y
1
-1
-2
y = x2 - 2x
y = -x4
A) 2 B)76
15C)
7
15D)
22
15
7) y = 2x2 + x - 6 y = x2 - 4
x-3 -2 -1 1 2 3
y5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
(2, 4)
x-3 -2 -1 1 2 3
y5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
(2, 4)
A)11
6B)
9
2C)
19
3D)
8
3
3
8)
x2 4 6 8 10
y6
4
2
-2
-4
-6
x2 4 6 8 10
y6
4
2
-2
-4
-6
y = x - 4
y = 2x
A)32
3B)
64
3C) 32 D)
128
3
9) y = x4 - 32
x-4 -3 -2 -1 1 2 3 4
y5
-5
-10
-15
-20
-25
-30
-35
-40
x-4 -3 -2 -1 1 2 3 4
y5
-5
-10
-15
-20
-25
-30
-35
-40
y = -x4
A)256
5B)
516
5C)
512
5D)
2816
5
4
10)
x1 2 3
y3
2
1
-1
-2
-3
x1 2 3
y3
2
1
-1
-2
-3
y = 2
y = 2 sin(πx)
A) 8 B)4
πC) 4 +
4
πD) 4
11) y = sec2 x
xπ4
π2
y
3
2
1
xπ4
π2
y
3
2
1y = cos x
A)2
2B) 2 - 2 C) 1 -
2
2D) 1 + 2
Find the area enclosed by the given curves.
12) y = 2x - x2, y = 2x - 4
A)34
3B)
31
3C)
37
3D)
32
3
13) y = x, y = x2
A)1
3B)
1
2C)
1
6D)
1
12
14) y = x3, y = 4x
A) 8 B) 16 C) 2 D) 4
5
15) y = 1
2 x2, y = -x2 + 6
A) 32 B) 4 C) 8 D) 16
16) y = - 4sin x, y = sin 2x, 0 ≤ x ≤ π
A) 4 B)1
2C) 16 D) 8
17) Find the area of the region in the first quadrant bounded by the line y = 8x, the line x = 1, the curve y = 1
x,
and the x-axis.
A)5
4B)
3
2C) 6 D)
3
4
18) Find the area of the region in the first quadrant bounded on the left by the y-axis, below by the line y = 1
3x,
above left by y = x + 4, and above right by y = - x2 + 10.
A)73
6B)
39
2C) 15 D)
39
4
19) Find the area between the curves y = ln x and y = ln 2x from x = 1 to x = 5.
A) ln 2 B) ln 32 C) ln 16 - 8 D) ln 16
20) Find the area of the "triangular" region in the first quadrant that is bounded above by the curve y = e2x,
below by the curve y = ex, and on the right by the line x = ln 4.
A) 4 ln 4 B) 4 C)9
2D)
27
2
21) Find the area of the region between the curve y = 2x/(1 + x2) and the interval -3 ≤ x ≤ 3 of the x-axis.
A) 2 e10 B) ln 10 C) 2 ln 10 D) 0
FROM THE BOOK
22) Do section 8.1 #9-14 ALL, 23-28 ALL
23) Read and take notes on section 8.2 Volumes of Revolution, Volumes of regions with a known
cross-section, Arc Length (not parametric)
24) Do section 8.2 #1-15 odd, 21-41 odd, 45, 47, 49
6
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the volume of the described solid.
25) The solid lies between planes perpendicular to the x-axis at x = 0 and x = 6. The cross sections
perpendicular to the x-axis between these planes are squares whose bases run from the parabola y = - 3 x
to the parabola y = 3 x.
A) 630 B) 648 C) 108 D) 324
26) The solid lies between planes perpendicular to the x-axis at x = -2 and x = 2. The cross sections
perpendicular to the x-axis between these planes are squares whose bases run from the semicircle
y = - 4 - x2 to the semicircle y = 4 - x2.
A)64
3B)
32
3C)
128
3D)
16
3
27) The base of the solid is the disk x2 + y2 ≤ 25. The cross sections by planes perpendicular to the y-axis
between y = - 5 and y = 5 are isosceles right triangles with one leg in the disk.
A)1000
3B)
250
3C)
1250
3D)
2000
3
28) The solid lies between planes perpendicular to the x-axis at x = - 2 and x = 2. The cross sections
perpendicular to the x-axis are semicircles whose diameters run from y = - 4 - x2 to y = 4 - x2.
A)16
3π B)
64
3π C)
32
3π D)
8
3π
29) The solid lies between planes perpendicular to the x-axis at x = - 4 and x = 4. The cross sections
perpendicular to the x-axis are circular disks whose diameters run from the parabola y = x2 to the
parabola y = 32 - x2.
A)256
3π B)
8192
5π C)
16384
15π D)
8192
15π
30) The base of a solid is the region between the curve y = 3cos x and the x-axis from x = 0 to x = π/2. The
cross sections perpendicular to the x-axis are squares with bases running from the x-axis to the curve.
A) 2π B)9
2π C)
3
2π D)
9
4π
7
Find the volume of the solid generated by revolving the shaded region about the given axis.
31) About the x-axis
x1 2 3
y10987654321
x1 2 3
y10987654321
y = - 2x + 4
A) 12π B)224
3π C)
32
3π D)
64
3π
32) About the x-axis
x1 2 3 4 5
y20
16
12
8
4
x1 2 3 4 5
y20
16
12
8
4 y = 9 - x2
A)1053
5π B)
648
5π C)
3159
5π D) 18π
33) About the y-axis
x1 2 3 4 5 6 7 8
y8
7
6
5
4
3
2
1
x1 2 3 4 5 6 7 8
y8
7
6
5
4
3
2
1
x = 6y/7
A) 98π B) 21π C) 168π D) 84π
8
34) About the y-axis
x1 2 3 4 5 6
y6
5
4
3
2
1
x1 2 3 4 5 6
y6
5
4
3
2
1
y = 5x
A)25
3π B) 25π C) 50π D) 625π
35) About the y-axis
x1 2 3 4 5 6
y6
5
4
3
2
1
x1 2 3 4 5 6
y6
5
4
3
2
1
x = y2
3
A)27
5π B)
108
5π C)
45
2π D) 18π
36) About the x-axis
x1 2 3 4
y2018161412108642
x1 2 3 4
y2018161412108642
y = 4 - x2
A)64
15π B)
8
3π C)
224
15π D)
256
15π
9
37) About the x-axis
xπ2
y4
2
xπ2
y4
2
y = 3 sin x
A)9
2π2 - 9π B)
9
2π2 C)
9
2π2 - 3π D)
9
2π2 + 9π
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the
x-axis.
38) y = x, y = 0, x = 2, x = 6
A) 16π B) 20π C)208
3π D)
4
3π
39) y = x2, y = 0, x = 0, x = 6
A)7776
5π B) 72π C) 324π D) 1944π
40) y = 2x + 3, y = 0, x = 0, x = 1
A) 2π B) π C) 4π D)3π
2
41) y = 1
x, y = 0, x = 1, x = 8
A)7
16π B) πln 8 C)
3
8π D)
7
8π
42) y = 2x, y = 2, x = 0
A) 1π B)8
3π C)
4
3π D) 6π
43) y = - 5x + 10, y = 5x, x = 0
A) 50π B) 150π C) 25π D) 10π
44) y = 3x, y = 3, x = 0
A) 9π B) 18π C)27
4π D)
27
2π
10
45) y = x2, y = 16, x = 0
A)6144
5π B)
4096
5π C)
1024
5π D)
128
3π
46) y = 9
x, y = - x + 10
A)512
3π B) 72π C)
1000
3π D) 128π
Find the volume of the solid generated by revolving the region about the given line.
47) The region bounded above by the line y = 16, below by the curve y = 16 - x2, and on the right by the line
x = 4, about the line y = 16
A)7168
15π B)
64
3π C)
8192
15π D)
1024
5π
48) The region in the second quadrant bounded above by the curve y = 4 - x2, below by the x-axis, and on the
right by the y-axis, about the line x = 1
A)256
15π B)
32
3π C)
56
3π D) 8π
Solve the problem.
49) The disk (x - 6)2 + y2 ≤ 4 is revolved about the y-axis to generate a torus. Find its volume. (Hint:
2
-2
4 - y2 dy∫ = 2π, since it is the area of a semicircle of radius 2.)
A) 24π2 B) 48π2 C) 12π2 D) 24π
50) The hemispherical bowl of radius 5 contains water to a depth 1. Find the volume of water in the bowl.
A)7
3π B)
14
3π C)
139
3π D) 88π
51) A water tank is formed by revolving the curve y = 5x4 about the y-axis. Find the volume of water in the
tank as a function of the water depth, y.
A) V(y) = 3π
2 5y3/2 B) V(y) =
2π
3 5y3/2 C) V(y) =
π
2 5y1/2 D) V(y) =
π
9y9
52) A right circular cylinder is obtained by revolving the region enclosed by the line x = r, the x-axis, and the
line y = h, about the y-axis. Find the volume of the cylinder.
A) πrh B) πrh2 C) 2πr2h D) πr2h
53) Find the volume that remains after a hole of radius 1 is bored through the center of a solid sphere of radius
2.
A)5
3π B)
8
3π C)
10
3π D)
32
3π
11
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated
axis.
54) About the y-axis
x1 2 3 4 5
y7
6
5
4
3
2
1
x1 2 3 4 5
y7
6
5
4
3
2
1
y = 2x - x2
A) 2π B) 4π C)8
3π D)
4
3π
55) About the x-axis
x1 2 3 4 5
y
4
3
2
1
x1 2 3 4 5
y
4
3
2
1
y = 1 - x2
A)2
3π B)
3
2π C)
1
3π D) 1π
56) About the y-axis
x1.8
y
4
3
2
1
x1.8
y
4
3
2
1
y =3sin(x2)
A) 12π B) 6π C) 9π D) 3π
12
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves
and lines about the y-axis.
57) y = 7x, y = - x
7, x = 1
A)33
7π B)
100
21π C)
50
21π D) 50π
58) y = x2, y = 4 + 3x, for x ≥ 0
A) 32π B) 192π C) 96π D) 64π
59) y = 2
x, y = 0, x = 1, x = 4
A)56
3π B)
64
3π C) 24π D)
28
3π
60) y = 4e-x2, y = 0, x = 0, x = 1
A) 8 1 - 1
e π B) 4(e - 1) π C) 4 1 -
1
e π D) 8 1 +
1
e π
61) y = 6ex2, y = 0, x = 0, x = 1
A) 6(e - 1) π B) 12e π C) 3(e - 1) π D) 12(e - 1) π
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves
and lines about the x-axis.
62) x = 3 y, x = - 3y, y = 1
A)22
5π B) 6π C) 8π D)
11
5π
63) x = 7y - y2, x = 0
A)2401
3π B)
2401
6π C)
2401
12π D)
343
6π
64) y = x, y = 0, y = x - 6
A)225
2π B)
63
2π C) 27π D)
63
4π
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves
about the given lines.
65) y = 4 - x2, y = 4, x = 2; revolve about the line y = 4
A)256
15π B)
224
15π C)
8
3π D)
32
5π
66) y = 2x, y = 0, x = 2; revolve about the line x = -3
A)52
3π B)
104
3π C) -
40
3π D)
52
3
13
Find the length of the curve.
67) y = 2x3/2 from x = 0 to x = 5
4
A)9
4B)
335
3C)
335
72D)
335
108
68) y = 1
6x3 +
1
2x from x = 1 to x = 5
A)316
15B)
79
5C)
632
15D)
127
6
Set up an integral for the length of the curve.
69) y = x4, 0 ≤ x ≤ 1
A)1
0
1 + 16x8 dx∫ B)1
0
1 + 4x6 dx∫ C)1
0
1 + 4x3 dx∫ D)1
0
1 + 16x6 dx∫
FROM THE BOOK
70) Read and take notes on section 8.5 Work (not force and pressure)
71) Do section 8.5 # 1, 5-17 odd, and Examples in the section #1-6 ALL
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
72) How much work is required to move an object from x = 0 to x = 3 (measured in meters) in the presence of a
constant force of 7 N acting along the x-axis?
A) 32 J B) ∞ C) 21 J D) 2 J
73) A 20-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the
system and that the chain has a density of 5 kg/m. How much work is required to wind the entire chain
onto the cylinder using the winch?
A) 1000g J B) 100g J C) 2000g J D) 1950g J
74) A swimming pool has the shape of a box with a base that measures 20 m by 19 m and a depth of 2 m. How
much work is required to pump the water out of the pool when it is full?
A) 28,302,400 J B) 14,896,000 J C) 760,000 J D) 7,448,000 J
14
75) A cylindrical water tank has height 10 m and radius 3 m (see figure). If the tank is full of water, how much
work is required to pump the water to the level of the top of the tank and out of the tank? Express the
answer in terms of π.
10 m
3 m
A) 490,000 J B) 44,100,000π J C) 4,410,000π J D) 1,455,300π J
76) A spherical water tank with an inner radius of 4 m has its lowest point 3 m above the ground. It is filled by
a pipe that feeds the tank at its lowest point (see figure). Neglecting the volume of the inflow pipe, how
much work is required to fill the tank if it is initially empty? Express the answer in terms of π.
4 m
3 m
A)1,254,400
3π J B)
627,200
3π J C)
5,017,600
3π J D)
2,508,800
3π J
77) A water trough has a semicircular cross section with a radius of 0.5 m and a length of 5 m (see figure).
How much work is required to pump water out of the trough when it is full? Round to two decimal places
when appropriate.
5 m
0.5 m
A) 2041.67 J B) 4083.33 J C) 60,331.25π J D) 24,500 J
15
78) A glass has circular cross sections that taper (linearly) from a radius of 8 cm at the top of the glass to a
radius of 7 cm at the bottom. The glass is 14 cm high and full of lemonade. How much work is required to
drink all the lemonade through a straw if your mouth is 6 cm above the top of the glass? Assume the
density of lemonade equals the density of water. Round to two decimal places when appropriate.
A) 3.08 J B) 4.7 J C) 0.98 J D) 53.88 J
THE END !!!
16
Answer KeyTestname: INTEGRATION WORKSHEET 2 - USING THE DEFINITE INTEGRAL
1)
2) see answers in the book. For the evens, discuss in class.
3) C
4) C
5) C
6) C
7) C
8) B
9) C
10) D
11) C
12) D
13) C
14) A
15) D
16) D
17) A
18) A
19) D
20) C
21) C
22) see answers in the book. For the evens, discuss in class.
23)
24)
25) B
26) C
27) A
28) A
29) C
30) D
31) C
32) B
33) D
34) B
35) B
36) C
37) A
38) C
39) A
40) C
41) D
42) B
43) A
44) D
45) B
46) A
47) D
48) C
17
Answer KeyTestname: INTEGRATION WORKSHEET 2 - USING THE DEFINITE INTEGRAL
49) B
50) B
51) B
52) D
53) C
54) C
55) A
56) B
57) B
58) D
59) A
60) C
61) A
62) A
63) B
64) B
65) D
66) B
67) D
68) A
69) D
70)
71) See answers in the book.
72) C
73) A
74) D
75) C
76) D
77) B
78) A
18