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AP Exam Review (Chapter 2)
has a vertical tangent at 2,0 and horizontal tangents at
1,-1 and 3,1 . For what values of x, in the open interval
-2,4 , is not differentiable?
f
f
(A) 0 only
(B) 0 and 2 only
(C) 1 and 3 only
(D) 0, 1, and 3 only
(E) 0, 1, 2, and 3
Differentiability
AP Exam Review (Chapter 2)
Let and be differentiable functions with the following
properties:
f g
(i) ( ) 0 for all
(ii) (0) 1
g x x
f
If ( ) ( ) ( ) and '( ) ( ) '( ), then ( )h x f x g x h x f x g x f x
(A) '( ) (B) ( ) (C) (D) 0 (E) 1 xf x g x e
Product Rule
is a constant functionf
' ' 'h f g fg ' 0f g
AP Exam Review (Chapter 2)
2
What is the instantaneous rate of change at 2 of the
2function given by ( ) ?
1
x
xf f x
x
Quotient Rule
1 1(A) 2 (B) (C) (D) 2 (E) 6
6 2
2 2
2
1 2'
1
4 1 2' 2 2
1
x x xf x
x
f
AP Exam Review (Chapter 2)
(A) 2 1 (B) 1 (C) (D) 1 (E) 0y x y x y x y x y
An equation of the line tangent to the graph of cos
at the point 0,1 is
y x x
Derivative of Cosine
Point-Slope Form
' 1 sin
' 0 1
1
y x
f
y x
AP Exam Review (Chapter 2)
( ) tan(2 ) '6
f x x f
(A) 3 (B) 2 3 (C) 4 (D) 4 3 (E) 8
Chain Rule
Derivative of Tangent
22
2' 2sec 2
cos 22
' 86 1/ 4
f x xx
f
AP Exam Review (Chapter 2)
Let be the function given by ( ) . Which of the
following statements about are true?
f f x x
f
I. is continuous at 0.f x II. is differentiable at 0.f x
III. has an absolute minimum at 0.f x
(A) I only (B) II only (C) III only (D) I and III only (E) II and III only
ContinuityDifferentiability
Minimum Extrema
T
FT
AP Exam Review (Chapter 2)
4 2
Which of the following is an equation of the line tangent to
the graph of ( ) 2 at the point where '( ) 1?f x x x f x
(A) 8 5
(B) 7
(C) 0.763
(D) 0.122
(E) 2.146
y x
y x
y x
y x
y x
Point-Slope Form
3' 4 4 1
0.2367
0.2367 0.1152
' 0.2367 0.9998
0.1152 0.9998 0.2367
f x x x
x
f
f
y x
(A) is always increasing.
(B) is always decreasing.
(C) is decreasing only when .
(D) is decreasing only when .
(E) remains constant.
A
A
A b h
A b h
A
AP Exam Review (Chapter 3)
If the base of a triangle is increasing at a rate of
3 inches per minute while its height is decreasing
at a rate of 3 inches per minute, which of the following
must be true about the area of the
b
h
A triangle?
3; 3
1
21
2
13 3
2
3
2
db dh
dt dt
A bh
dA db dhh b
dt dt dt
dA dAh b
dth b
dt
AP Exam Review (Chapter 2)
2
A particle moves along the -axis so that its position at time
is given by ( ) 6 5. For what value of is the
velocity of the particle zero?
x
t x t t t t
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
Power Rule
2( ) 6 5
' 2 6 0
2 3 0
x t t t
x t t
t
AP Exam Review (Chapter 2)
2If 10, then when 2, dy
x xy xdx
7 2 3 7
(A) (B) 2 (C) (D) (E) 2 7 2 2
Quotient Rule
2 ' 0
' 2
2 2 4 3'
22 4 2 10 3
x y xy
xy x y
x y x yy
x xx y y
Which of the following could be the derivative of the graph
shown above?
Relative Extrema
Location where
' 0 & where
' goes from to
f
f
2
The function is differentiable for all real numbers. The
1point 3, is on the graph of ( ), and the slope of
4
each point , on the graph is given by 6 2 .
f
y f x
dyx y y x
dx
2
2
2
1(a) Find and evaluate it at the point 3, .
4
(b) Find ( ) by solving the differential equation
1 6 2 with the initial condition (3) .
4
d y
dx
y f x
dyy x f
dx
2
23 22
12 6 2 2
8
d yy x y
dx
2
1
6 13y
x x
Product Rule
Differential Equations
Separation of Variables
2
22
2
22 2
23 2
6 2
2 6 2 2
2 6 2 2
2 6 2 2
dyy x
dx
d y dyy x y
dx dx
y y x y
y x y
2
2
2
2
6 2
6 2
6 2
16
dyy x
dxdy
x dxy
dyx dx
y
x x Cy
4 18 9 13C C
2
2
2
2
2
6 2
6 2
6 2
1
1
6 13
6
4 9 13
dyy x
dx
y dy x dx
y dy x dx
x x Cy
C C
yx x
Problem
2
Water in this container is evaporating so that the depth
-3 1is changing at the constant rate of / .
10 3
(a) Find the volume of water in the container when
5 cm. Indicate units
h
cm hr V r h
V
h
of measure.
(b) Find the rate of change of the volume of water in the
container, with respect to time, when 5 cm. Indicate
units of measure.
(c) Show that the rate of change of the volume of w
h
ater
in the container due to evaporation is directly
proportional to the exposed surface area of the water.
What is the constant of proportionality?
315 cm /
8
dVhr
dt
circledAdVk
dt dt
Problem_Worked_Out_Completely
3
10k
AP Exam Review (Chapter 2)2 3
2 2
3
Consider the curve given by 6.
3(a) Show that .
2
(b) Find all points on the curve whose -coordinate is 1,
and write an equation for the tangent line at each of these
poi
xy x y
dy x y y
dx xy x
x
nts.
(c) Find the -coordinate of each point on the curve where
the tangent line is vertical.
x2,0m 2 & 3y x y
0 &???x
Implicit Differentiation
Point-Slope Form
Vertical Tangent
2 2 3
2 2
3
2 3 0
3
2
dy dyy x y x y x
dx dx
dy x y y
dx xy x
2 6 3 2 0 1,3 & 1, 2y y y y
3 232 0
2 2
x xxy x y
x
22 3
22 23
62
62 2
xy xy x y
x xx x
5 55 5
5 5
6 2 244 2
24 25
x xx x
x x
AP Exam Review (Chapter 3)
3 2
What is the -coordinate of the point of inflection on the
1graph of 5 24?
3
x
y x x
10(A) 5 (B) 0 (C) (D) 5 (E) 10
3
Inflection Point
AP Exam Review (Chapter 3)
The graph of a twice-differentiable function is shown in
the figure above. Which of the following is true?
f
(A) (1) '(1) ''(1)
(B) (1) ''(1) '(1)
(C) '(1) (1) ''(1)
f f f
f f f
f f f
(D) ''(1) (1) '(1)
(E) ''(1) '(1) (1)
f f f
f f f
Graphical Behavior
AP Exam Review (Chapter 3)
2If ''( ) ( 1)( 2) , then the graph of has inflection
points when
f x x x x f
x
(A) -1 only (B) 2 only (C) 1 and 0 only
(D) 1 and 2 only (E) 1, 0, and 2 only
Inflection Point
AP Exam Review (Chapter 3)
4 2The function is given by ( ) 2. On which of
the following intervals is increasing?
f f x x x
f
1(A) ,
2
1 1(B) ,
2 2
(C) 0,
(D) ,0
1(E) ,
2
Increasing and Decreasing Functions
AP Exam Review (Chapter 3)
3 2
The maximum acceleration attained on the interval 0 3
by the particle whose velocity is given by
( ) 3 12 4 is
t
v t t t t
(A) 9 (B) 12 (C) 14 (D) 21 (E) 40
Maximum Extrema
Position, Velocity, and Acceleration FCTNS
AP Exam Review (Chapter 3)
The function is continuous on the closed interval 0,2 and
1has values that are given in the table above. If ( ) , then
2( ) ( ) must have at least two solutions in the interval 0,2
if
f
g x
f x g x
k
1(A) 0 (B) (C) 1 (D) 2 (E) 3
2Intermediate Value THM
0,1
1,0
2,2
1/ 2y
f x
g x
AP Exam Review (Chapter 3)
The graph of the function is shown above. Which of the
following statements about is false?
f
f
(A) is continuous .
(B) has a relative maxima at .
(C) is in the domain of .
f at x a
f x a
x a f
(D) lim ( ) lim ( )f x f xx a x a
(E) lim ( ) existsf xx a
AP Exam Review (Chapter 3)
2
The first derivative of the function is given by
cos 1'( ) . How many critical values does have
5on the open interval 0,10 ?
f
xf x f
x
(A) One
(B) Three
(C) Four
(D) Five
(E) SevenCritical Number
AP Exam Review (Chapter 3)
The graphs of the derivatives of the function , , and are
shown above. Which of the functions , , or have a
relative maximum on the open interval ?
f g h
f g h
a x b (A) only (B) only (C) only
(D) and only (E) , , and
f g h
f g f g h Relative Extrema
AP Exam Review (Chapter 3)
2
If is a differentiable function such that ( ) 0 for all real
numbers and if '( ) 4 ( ), which of the
following are true?
g g x
x f x x g x
(A) has a relative maximum at 2 and a relative minimum at 2.
(B) has a relative minimum at 2 and a relative maximum at 2.
(C) has a relative minima at 2 and at 2.
(D) has a relat
f x x
f x x
f x x
f
ive maxima at 2 and at 2.
(E) It cannot be determined if has any relative extrema.
x x
f
Relative Extrema
'f x
AP Exam Review (Chapter 3)
Let be a function that is differentiable on the open interval
1,10 . If (2) -5, (5) 5, and (9) -5, which of the
following must be true?
f
f f f
I. has at least 2 zeros.fII. The graph of has at least one horizontal tangent.fIII. For some , 2 5, ( ) 3.c c f c
(A) None (B) I only (C) I and II only (D) I and III only (E) I, II and III
Differentiability
2, 5
5,5
9, 5
AP Exam Review (Chapter 3)
2
Let be a function defined for all 0 such that (4) 3
2and the derivative of is given by '( ) for all 0.
h x h
xh h x x
x
(a) Find the values of for which the graph of has ax h
horizontal tangent, and determine whether has a local
maximum, a local minimum, or neither at each of these
values. Justify your answers.
h
(b) On what intervals, if any, is the graph of concave up?
Justify your answer.
(c) Write an equation for the line tangent to the graph of
at 4.
(d) Does the tangent line to the graph of
h
h
x
h
at 4 lie above or
below the graph of for 4? Why?
x
h x
2, both are rel. minsx
is concave up on its entire domain.h
717
2y x
Relative ExtremaConcavity
Below Point-slope Form
Tangent Lines
Two runners, and , run on a straight racetrack for 0 10
seconds. The graph above, which consists of two line segments,
shows the velocity, in meters per second, of Runner . The
velocity, in meters
A B t
A
per second, of runner is given by the
24function defined by ( ) .
2 3
B
tv v t
t
(3,10) (10,10)
Time in seconds
Velocity of Runner A(meters per second)
Question
(a) Find the velocity of runner and the velocity of Runner at
time 2 seconds. Indicate units of measure.
A B
t
(b) Find the acceleration of Runner and the acceleration
of Runner at time 2 seconds. Indicate units of
measure.
(c) Find the total distance run by Runner and the total
distanc
A
B t
A
e run by Runner over the time interval
0 10 seconds. Indicate units of measure.
B
t
Graph & Function
(2) 6 meters per second
(2) 48 / 7 meters per secondA
B
V
V
(2) 10 / 3 meters per second per second
(2) 72 / 49 meters per second per secondA
B
A
A
A
B
Distance 85 meters
Distance 83 meters
The graph above shows ', the derivative of , for 7 7.
The graph of 'has horizontal tangent lines at 3, 2,
and 5, and a vertical tangent line at 3.
f f x
f x x
x x
(a) Find all values of , for 7 7, at which attains a
relative minimum. Justify your answer.
(b) Find all values of , for 7 7, at which attains a
relative maximum. Justify your answe
x x f
x x f
r.
(c) Find all values of , for 7 7, at which ''( ) 0.
(d) At what value of , for 7 7, does attain its
absolute maximum? Justify your answer.
x x f x
x x f
1x
5x
7, 3 & 2,5
7, on -1,7 , ' 0x f
Definition
Relative ExtremaConcavity
Maximum Extrema
AP Exam Review (Chapter 4)2
1
2e xdx
x
2 22 21 1 5
(A) (B) (C) (D) 2 (E) 2 2 2 2
e ee e e e e
e
If is a linear function and 0 , then ''( )b
af a b f x
2 2
(A) 0 (B) 1 (C) (D) (E) 2 2
ab b ab a
Rewrite the integrand
AP Exam Review (Chapter 4)
2
3What are the values of for which 0
kk x dx
(A) 3 (B) 0 (C) 3 (D) 3 and 3 (E) 3,0 and 3
3
0If ( ) 1 , then '(2)
xF x t dt F
(A) -3 (B) 2 (C) 2 (D) 3 (E) 18
Special Definite Integrals
Second Fundamental THM of Calculus
AP Exam Review (Chapter 4)
0sin
xtdt
(A) sin (B) cos (C) cos (D) cos 1 (E) 1 cosx x x x x
Fundamental THM of Calculus
AP Exam Review (Chapter 4)
2
A particle moves along the -axis with velocity given by
( ) sin for 0.
y
x t t t t
(a) In which direction (up or down) is the particle moving
at time 1.5? Why?
(b) Find the acceleration of the particle at time 1.5. Is the
velocity of the particle increasing at 1.5?
t
t
t
Why or why
not?
(c) Given that ( ) is the position of the particle at time and
that (0) 3, find (2).
(d) Find the total distance traveled by the particle from
0 2.
y t t
y y
t to t
Up because ( ) 0x t
( ) 2.05, ( ) is decreasing
because acceleration is neg.
a t v t
(2) 3.83y
0.83 units
Direction of a particle
v(x) increasing or decreasing?
s(t), v(t), and a(t)
Area under v(t)
AP Exam Review (Chapter 4)
3 2
A cubic polynomial function is defined by
( ) 4
where , , and are constants. The function has a local
minimum at 1, and the graph of has a point of inflection
at 2
f
f x x ax bx k
a b k f
x f
x
1
0
.
(a) Find the values of and .
(b) If ( ) 32, what is the value of ?
a b
f x dx k24, 36a b
5k Relative Extrema
Inflection PointFundamental THM of Calculus
2' 12 2
' 1 12 2 0
'' 24 2
f x x ax b
f a b
f x x a
'' 2 48 2 0
24, 36
f
b
a
a
1
13 2 4 3 2
00
4 24 36 24 / 3 18
1 24 / 3 18 32
x x x k dx x x x kx
k
AP Exam Review (Chapter 4)
1
The graph of the function , consisting of three line segments,
is given below. Let ( ) ( ) .x
f
g x f t dt
(a) Compare (4) and (-2).
(b) Find the instantaneous rate of change of , with respect to
, at 1.
(c) Find the absolute minimum value of on the closed
interval -2,4 . Justify your answer
g g
g
x x
g
.
(d) The second derivative of is not defined at 1 and 2.
How many of these values are -coordinates of points of
inflection of the graph of ? Justify your answer.
g x x
x
g
(4) 5 / 2 ( 2) 6g g
4
2
Both
Special Definite Integral Rules
Inflection Point
An object moves along the -axis with initial position (0) 2.
The velocity of the object at time 0 is given by ( ) sin .3
(a) What is the acceleration of the object at time 4?
(b) Consider t
x x
t v t t
t
he following two statements.
Statement I: For 3 4.5, the velocity is decreasing.
Statement II: For 3 4.5, the speed is increasing.
Are either or both of these statements correct
t
t
? Justify
(c) What is the total distance traveled by the object over the time
interval 0 4?
(d) What is the position of the object at time 4?
t
t
AP Exam Review (Chapter 4)
( ) cos3 3
a t t
, Speed ( )Both v t
1.4 units3.4 units
s(t), v(t), and a(t)
Velocity and Speed
Area under v(t)
AP Exam Review (Chapter 4)If is continuous for and differentiable for
, which of the following could be false?
f a x b
a x b
( ) ( )(A) '( ) for some such that .
(B) '( ) 0 for some such that
(C) has a minimum value on .
(D) has a maximum value on .
(E) ( ) exists.b
a
f b f af c c a c b
b af c c a c b
f a x b
f a x b
f x dx
Instantaneous Velocity & Slope
FalseTrue
True
True
True
AP Exam Review (Chapter 4)
3
1
If is a continuous function and if '( ) ( ) for all
real numbers , then (2 )
f F x f x
x f x dx
(A) 2 (3) 2 (1)
1 1(B) (3) (1)
2 2(C) 2 (6) - 2 (2)
(D) (6) - (2)
1 1(E) (6) (2)
2 2
F f
F F
F F
F F
F F
Antiderivative of a Composite FCTN
The flow of oil, in barrels per hour, through a pipeline on
July 9 is given by the graph. Of the following, which best
approximates the total number of barrels of oil that passed
through the pipeline that day?
(A) 500 (B) 600 (C) 2,400 (D) 3,000 (E) 4,800
Area under v(t)
AP Exam Review (Chapter 4)
x 2 5 7 8
f(x) 10 30 40 20
8
2
The function is continuous on the closed interval 2,8 and
has values that are given in the table. Using the subintervals
2,5 , 5,7 , and 7,8 , what is the trapezoidal
approximation of ( ) ?
f
f x dx
(A) 110 (B) 130 (C) 160 (D) 190 (E) 210
Trapezoidal Approximation
AP Exam Review (Chapter 4)
0
The graph of the function consists of two line segments.
Let be the function given by ( ) ( ) .x
f
g g x f t dt
(a) Find (-1), '(-1), and ''(-1).
(b) For what values of x in the open
interval 2, 2 is increasing? Explain
(c) For what values of x in the open
interval 2, 2 is the graph of concave
g g g
g
g
down? Explain
(d) On the axes provided, sketch the graph
of on the closed interval -2,2 .g
3, 0, 3
2
in 2, 1 & 0,1x
in 0,2 , ''( ) 0x g x
AP Exam Review (Chapter 4)The rate at which water flows out of a pipe,
in gallons per hour, is given by the differential
function of time . The table shows the rate
as measured every three hours for a 24-hour period.
R t
24
0
(a) Use a midpoint Riemann sum with 4
subdivisions of equal length to approximate
( ) . Using correct units, explain the
meaning of your answer in terms of water flow.
(b) Is there s
R t dt
2
ome time , 0 24, such that '( ) 0?
Justify your answer.
(c) The rate of water flow ( ) can be approximated
1 by ( ) 768 23 . Use ( ) to
79 approximate the average rate of water
t t R t
R t
Q t t t Q t
flow during
the 24-hour time period. Indicate units of measure.
255 gallons in a 24 hour period
No, concavity doesn’t change
Approximately 10.8 gallons per hour
Let be a differentiable function. The table gives the values
of and ' for selected points in 1.5,1.5 . ''( ) 0 in
1.5,1.5 .
f
f f x f x x
(d) Let ( )={g x22 7 for 0x x x 22 7 for 0.x x x
The graph of passes through each of the points , ( )
given in the table. Is it possible that and are the same
function? Give a reason for your answer.
g x f x
f g
3/2
0(a) Evaluate (3 '( ) 4) . Show the work that leads
to your answer.
(b) Write an equation of the line tangent to the graph of
at the point where 1. Use this point to approximate the
f x dx
f
x
value of (1.2). Is this approximation greater than or less
than the actual value of (1.2)? Why?
(c) Find a positive real number having the property that
there must exist a value w
f
f
r
c ith 0 0.5 and ''( ) .
Give a reason for your answer.
c f c r
24
5 9y x (1.2) 3f
, because the tangent is below the curve
No, g is not differentiable at x=0
6r
2
The rate at which people enter an amusement park on a
given day is modeled by the function defined by
15600 ( )
24 160
The rate at which people leave the same amusement park on
th
E
E tt t
2
e same day is modeled by the function defined by
9890 ( )
38 370
Both ( ) and ( ) are measured in people per hour and time
is measured in hours after midnight. These functions
L
L tt t
E t L t
t
are
valid for 9 23, the hours during which the park is open.
At time 9, there are no people in the park.
t
t
(a) How many people have entered the park by 5:00 PM
( 17)? Round answer to the nearest whole number.
(b) The price of admission to the park is $15 until 5:00 PM
( 17). After 5:00 PM, the pri
t
t
9
ce of admission is $11.
How many dollars are collected from admissions to the
park on the given day? Round your answer to the nearest
whole number.
(c) Let ( ) ( ) - ( ) for 9 23. Tt
H t E x L x dx t he value of
(17) to the nearest whole number is 3725. Find the
value of '(17) and explain the meaning of (17) and
'(17) in the context of the park.
(d) At what time , for 9 23, doe
H
H H
H
t t s the model predict
that the number of people in the park is a maximum?
2
A car is traveling on a straight road with velocity 55 ft/sec
at time 0. For 0 18 seconds, the car's acceleration
( ), in ft/sec , is the piecewise linear function defined by
the graph.
t t
a t
(2,15) (18,15)
(10, 15) (10, 15)
2
( )
(ft / sec )
a t
(seconds)t
(a) Is the velocity of the car increasing at 2 seconds? Why
or why not?
(b) At what time in the interval 0 18, other than 0, is
the velocity of the car 55 / sec? Why or why not?
(c) On th
t
t t
ft
e interval 0 18, what is the car's absolute
maximum velocity, in ft/sec, and at what time does it
occur? Justify your answer.
(d) At what times in the interval 0 18, if any, is the car's
t
t
velocity equal to zero? Justify your answer.
Increasing because (2) 0.a
6, a(6)=0 and ( ) goes from + to .t a x (18) (0) (6)v v v
( ) 0, the absolute min is 0v x
(6) 115 ft/sec, (10)=85 ft/sec, (12)=55 ft/secv v v
The temperature, in degrees Celsius, of the water in a pond
is a differentiable function of time . The table shows the
water temperatures as recorded every 3 days over a 15-day
period.
W t
(a) Use data from the table to find an approximation for
(12). Show the computations that lead to your answer.
Indicate units of measure.
(b) Approximate the average temperature, in degrees
'W
Celsius,
of the water over the time interval 0 15 days by
using a trapezoidal approximation with subintervals of
3 days.
(c) A student proposes the function , given by
( ) 2
t
t
P
P t
(- /3)0 10 , as a model for the temperature of the
water in the pond at time , where is measured in days
and ( ) is measured in degrees Celsius. Find '(12).
Using appropriate units
tte
t t
P t P
, explain the meaning of your
answer in terms of water temperature.
(d) Use the function defined in part (c) to find the average
value, in degrees Celsius, of ( ) over the time interv
P
P t al
0 15 days.t
(15) (9) 1'(12) degrees Celsius per day
15 9 2
w ww
Trapezoid Rule
15(20 62 56 48 44 21) 376.5
10
1376.5 25.1 degrees Celsius
15
.549 C/day
25.757 C
2
1
1e xdx
x
2 22 21 1 3
(A) - (B) 2 (C) (D) 2 (E) 2 2 2 2
e ee e e ee
If ( )f x
(A) ln 2 (B) ln8 (C) ln16 (D) 4 (E) DNE
then lim ( ) isf x2x
{ln for 0 2x x
2 ln 2 for 2 4,x x
If and is a nonzero constant, then could bedy
ky k ydt
2
(A) 2 (B) 2 (C) +3
1 1(D) 5 (E)
2 2
kty kt kte e e
kty ky
2
3
Let be the function given by ( ) 3 and let be the
function given by ( ) 6 . At what value of do the
graphs of and have parallel tangent lines?
xf f x e g
g x x x
f g
(A) 0.701 (B) 0.567 (C) 0.391 (D) 0.302 (E) 0.258
Population grows according to the equation ,
where is a constant and is measured in years. If the
population doubles every 10 years, then the value of is
dyy ky
dtk t
k
(A) 0.069 (B) 0.200 (C) 0.301 (D) 3.322 (E) 5.000
If ( ) sin( ), then '( )xf x e f x
(A) cos( )
(B) cos( )
(C) cos( )
(D) cos
(E) cos
x
x x
x x
x x
x x
e
e e
e e
e e
e e
The temperature outside a house during a 24-hour period is
given by
(a) Sketch the graph of . F
( ) 80 10cos , 0 24,12
tF t t
where ( ) is measured in degrees Fahrenheit and is
measured in hours.
F t t
(b) Find the average temperature, to the nearest degree,
between 6 and 14. t t (c) An air conditioner cooled the house whenever the
outside temperature was at or above 78 degrees
Fahrenheit. For what values of was the air conditioner
cooling the house?
t
0,70
24,70
87.1625
5.23<x<18.77
The temperature outside a house during a 24-hour period is
given by
(d) The cost of cooling the house accumulates at a rate of
$0.05 per hour for each degree the outside temperature
exceeds 78 degrees Fahrenheit. What was the total cost,
to the nearest cent, to cool the house for this 24-hour
period?
( ) 80 10cos , 0 24,12
tF t t
where ( ) is measured in degrees Fahrenheit and is
measured in hours.
F t t
18.769
5.231
0.05 80 10cos 7812
$5.10dC t
dtdt
-2
Suppose that the function has a continuous second
derivative for all , and that (0) 2, '(0) 3, and
''(0) 0. Let be a function whose derivative is given by
'( ) [3 ( ) 2 '( )] for all .x
f
x f f
f g
g x e f x f x x
(a) Write an equation for the tangent line to the graph of
at the point where 0.
(b) Is there sufficient information to determine whether or
not the graph of has a point of inflection
f
x
f
-2
where 0?
Explain your answer.
(c) Given that (0) 4, write an equation of the line tangent
to the graph of at the point where 0.
(d) Show that ''( ) [ 6 ( ) '( ) 2 ''( )]. Does
x
x
g
g x
g x e f x f x f x
have a local maximum at 0? Justify your answer.g x
2 3y x
No, I don't know if ''( ) changes sign at 0.f x x
Inflection Points
4y
Yes
2
The function is differentiable for all real numbers. The
1point 3, is on the graph of ( ), and the slope at
4
each point , on the graph is given by 6 2 .
f
y f x
dyx y y x
dx
2
2
1(a) Find and evaluate it at the point 3, .
4
d y
dx
2 2 2
22
2
2 2 2
6 2 6 2
12 2 2
12 6 2 2 2 6 2
dyy x y xy
dx
d y dy dyy y x y
dx dx dx
y y x y xy y x
2
2
d y
dx 13,
4
1
8
2
The function is differentiable for all real numbers. The
1point 3, is on the graph of ( ), and the slope at
4
each point , on the graph is given by 6 2 .
f
y f x
dyx y y x
dx
2
(b) Find ( ) by solving the differential equation
1 6 2 with the initial condition (3) .
4
y f x
dyy x f
dx
2
2
2
22
6 2
6 2
1 16 , 3, 4 18 9
4
16
6 1
1
313
dyy x
dx
y dy xdx
x x C C
x yxy x x
y
2Let be a function given by ( ) 2 .xf f x xe
(a) Find the lim ( ) and lim ( ).
(b) Find the absolute minimum of . Justify your answer.
f x f x
fx x ,0
2 2
2
'( ) 2( 2 )
2 1 2 0
1
2' 1 0
1' 0 0 is the location of an absolute min.
21 1
is the absolute m2
in.
x x
x
f x e x e
e x
x
f
f x
fe
2Let be a function given by ( ) 2 .xf f x xe
(c) What is the range of ?
(d) Consider the family of functions defined by ,
where is a nonzero constant. Show that the absolute
minimum value of is the same for all nonzero
bx
bx
f
y bxe
b
bxe
values of .b
2
2 in [ , )y
e
' ( )
(1 ) 0 1
1
bx bx
bx
b
y b e x be
be x x
bf
e
The mins occur at the same location,
but they are not the same value????
2
Let be a function with (1) 4 such that for all points
3 1, on the graph of the slope is given by .
2
f f
xx y f
y
(a) Find the slope of the graph of at the point where 1.f x dy
dx 1,4
1
2
(b) Write an equation for the tangent line to the graph of
at 1 and use it to approximate (1.2).
f
x f
1.2
14 1
24.. 4 1.5 2
y x
f
2
Let be a function with (1) 4 such that for all points
3 1, on the graph of the slope is given by .
2
f f
xx y f
y
2
(c) Find ( ) by solving the seperable differential equation
3 1 with the initial condition (1) 4.
2
f x
dy xf
dx y
2
2
2 3
2 3
3 1
2
2 3 1
, 1,4 1
14
6 2 14
dy x
dx y
ydy x dx
y x x
y x x
C C C
2
Let be a function with (1) 4 such that for all points
3 1, on the graph of the slope is given by .
2
f f
xx y f
y
(d) Use your solution from part (c) to find (1.2).f
2 3
3
14
1.2 1.2 1.2 4.11414
y x x
f
2
2
3Consider the differential equation .
y
dy x
dx e
(a) Find a solution ( ) to the differential equation
1 satisfying (0) .
2(b) Find the domain and range of the function found in part
(a).
y f x
f
f
2 2
3
2 3
3
3
3
1
2
2
2 ln 2
ln 2 1 1 ln, 0,
2 2 2 2
y
u
y
e dy x dx
e du x C
e x C
y x C
x C Cy C e
2
2
u y
du dy
3ln 2
2
x ey
3: ( , )2
:
eD
R
3
3
3
2 0
2
2
x e
ex
ex
2
Determine the area of the region bounded by the graphs of
4 and 4.y x x y x
(A) 9 / 2
(B) 23/ 6
(C) 9 / 2
(D) 8 / 3
(E) None of these
4 2
1
9
4
2
4A x x x dx
3
Find the volume of the sold formed by revolving the region
bounded by , 1, and 2 about the - .y x y x x axis
(A) 127 / 7
(B) 120 / 7
(C) 240 / 7
(D) 1013 /10
(E) None of these
2 6
1
27
1
120
7
1
7
x dx
xx
2
Identify the definite integral that represents the area of the
surface formed by revolving the graph of ( ) on the
interval 0, 2 about the - .
f x x
x axis
2 2 4
0
2 2 2
0
2 2
0
2
0
(A) 2 1
(B) 2 1 4
(C) 2 1 4
1(D) 2 1
(E) None of these
x x dx
x x dx
x x dx
y dyy
22 1 '
b
aA r f x dx
2 2 2
02 1 4x x dx
2
Find the volume of the solid formed by revolving the region
bounded by the graphs of +1 and 0 about the
-axis.
y x y
x
(A) 2 / 3
(B) 8 /15
(C) 16 /15
(D) 4 / 3
(E) None of these
1
22
0
161
12
5V x dx
Identify the definite integral that represents the arc length of
the curve over the interval 0,3 .y x
3
0
3
0
3
0
3
0
1(A) 1
4
1(B) 1
2
(C) 1
(D)
(E) None of these
dxx
dxx
xdx
xdx
21 '
b
aS f x dx
3
0
11
4dxx
3
Which of the following integrals represents the volume of
the solid formed by revolving the region bounded by ,
1, and 2, about the line 2?
y x
y x x
83
1
2 23 2
1
28 23
1
2 3
1
(A) 2 2 1
(B) 1 1
(C) 1
(D) 2 2 1
(E) None of these
y y dy
x dx
y dy
x x dx
2 3
12 2 1x x dx
3
Identify the definite integral that represents the area of the
surface formed by revolving the graph of ( ) on the
interval 0,1 about the -axis.
f x x
y
1 4
0
1 4
0
1 3 2
0
1 2
0
(A) 2 1 9
(B) 2 1 9
(C) 2 1 3
(D) 2 1 3
(E) None of these
x x dx
x dx
x x dx
x x dx
1 4
02 1 9x x dx
22 1 '
b
ar f x dx
Find the volume of the solid generated by revolving the
region bounded by the graphs of ( ) 6 , 6, and the
- axis about the - axis.
f x x x
x y
(A) 864 / 5
(B) None of these
(C) 432
(D) 108
6
0
2 6V x xdx
6
6
6
u x
du dx
ux
363/ 2
018
864
5V u du
2
Find the volume of the solid formed by revolving the
region bounded by the graphs of 2 , 0, and 2
about the - axis.
y x x y
y
(A) / 4
(B) 2 /3
(C)
(D) 16 / 3
(E) None of these
1
2
0
2 2 2V x x dx
Use your calculator to approximate the volume of the solid
formed by revolving the region bounded by , 0,
0 and 1 about the -axis. Round your answer to three
decimal places.
xy e y
x x y
(A) 1.359
(B) 6.283
(C) 8.540
(D) 9.870
(E) None of these
1
0
2 xV xe dx
Find the area of the region bounded by the graphs of
( ) sin and ( ) cos , for / 4 5 / 4.f x x g x x x
2
1
1
2
3
2
5 / 4
5 / 4
/ 4/ 4
sin cos cos sin
2 2 2 22 2
2
2
2
2
2 2
x x dx x x
Consider the surface formed by revolving the graph of
( ) sin on the closed interval 0, about the -axis.f x x x
(a) Write the integral that computes the surface area of the
solid described.
(b) Use your calculator to approximate the surface area.
Round your answer to three decimal places.
2
02 sin 1 cosx xdx
14.424
22 1 '
b
a
S r f x dx
2
Find the volume of the solid formed by revolving the region
bounded by the graphs of and 4 about the -axis.y x y x
256
5
24
0
2 16V x dx
Find the volume of the solid formed by revolving the region
bounded by the graphs of 2, 0, and 6 about
the -axis.
y x y x
y
6
2
2 2V x x dx 2
2
u x
du dx
x u
41/ 2
0
43/ 2 1/ 2
0
45/ 2 3/ 2
0
2 2
2 2
2 42
704
1
5 3
5
V u u du
u u du
u u
2 2x y
2
22
0
706
13 2
4
5V y dy
Use the shell method to set up the integral that represents
the volume of the solid formed by revolving the region
1bounded by the graphs of and 2 2 5 about the
1line . (Do not evaluate the i
2
y x yx
y
ntegral.)
2
1/ 2
1 5 2 12
2 2
yy dy
y
2
2
1 5 22 5 2
2
2 5 2 0
2 1 2 0
1 1,2 2,
2 2
xx x
x
x x
x x
x y
1 1
5 22 2 5
2
y xx y
yx y x
is differentiable on a,b
is a continuous,
smooth curve on , and
does not have a vertical
tangent.
f
f
a b
f
Question 1
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2.3 The Product and Quotient Rules & Higher-Order Derivatives
( ) ( ) '( ) ( ) ( ) '( )d
f x g x f x g x f x g xdx
The Product Rule:
Question 1
Question 2
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2.3 The Product and Quotient Rules & Higher-Order Derivatives
2( ) '( ) ( ) ( ) '( )
( ) ( )
d f x f x g x f x g x
dx g x g x
The Quotient Rule:
Question 1
Question 2
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you read this definition.
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Basic Differentiation Rules for Elementary Functions
Question 2
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1 1
The equation for the line that has slope and goes through
the point , is:
m
x y
1 1y y m x x
Point-Slope:
Question 1
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Question 3
Question 4
2.4 The Chain Rule
The Chain Rule:
( ( )) '( ( )) '( )d
f g x f g x g xdx
Question 1
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THM 3.8 Points of Inflection
( , ( )) is a point of inflection of
''( ) 0 or ''( ) DNE
c f c f
f c f c
Question 1
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Question 3
Question 4
:
'' must change sign at .
Note
f x c
Question 5
Question 6
Question 7
Question 8
2.2 Basic Differentiation Rules& Rates of Change
1
The Power Rule
n ndx nx
dx
Question 1
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3.3 Increasing and Decreasing Functions and the First Derivative Test
Test for Increasing and Decreasing Functions
Let f be a continuous on [a,b] and differentiable on (a,b).
1. '( ) 0 in ( , ) is increasing on [ , ].f x x a b f a b
2. '( ) 0 in ( , ) is decreasing on [ , ].f x x a b f a b
3. '( ) 0 in ( , ) is constant on [ , ].f x x a b f a b
Question 1
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Question 3
Question 4
3.1 Extrema on an Interval Guidelines for Finding Extrema on a Closed Interval
To find the extrema of a continuous funtion f
on a closed interval [a,b], use the following steps.
4. The least of these values is the minimum.5. The greatest is the maximum.
1. Find the critical numbers of in ( , ).f a b
2. Evaluate at each critical number in ( , ).f a b
3. Evaluate at each endpoint of [ , ].f a b
Question 2Question 1
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Question 3
Definition of a Critical NumberLet be defined at .f c '( ) 0f c or '( ) is undefinedf c is a critical number of .c f
Question 4
Question 3
Question 2
Question 1
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a
'y f
is the location of a relative minimuma
b 'y f
is the location of a relative maximumb
Question 2
Question 3
Question 4
Question 1
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Question 5
Question 6
Question 7
5.6 Differential Equations:Growth & Decay
2Solve '
xy
y
' 2yy x' 2yy dx xdx
2ydy xdx 2
21 22
yC x C
2 22y x C You can use implicit
differentiation to check.
dy=y’dx
Multiply by 2Question
Click through this example if you need help.
Definition of Continuity on a Closed Interval
A function is continuous on a closed interval , if it is
continuous on the opoen interval , and
f a b
a b
x a x blim ( ) ( ) and lim ( ) ( ).f x f a f x f b
The function is continuous from the right at and
continuous from the left at .
f a
b
a b
Question 1
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Guidelines for Implicit Differentiation
1. Differentiate both sides of the equation with respect to .
2. Collect all terms involving / on the left and move
all other terms to the right.
3. Fact
x
dy dx
or / out of the left side of the equation.
4. Solve for / .
dy dx
dy dx
Question 1
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Vertical Tangent Line
( ) ( )lim
( ) ( )lim
The vertical tangent is .
f c x f c
xor
f c x f c
xx c
Question 1
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0x
0x
. . The derivative is undefinedi e
Question 1
Question 4
Question 3
Question 2' is decreasing is concave down
' is increasing is concave up
f f
f f
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Question 4
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you read this definition.
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Question 3
Question 2
Question 1
Intermediate Value THM:
is continuous on [ , ] and is any number between ( )
and ( ) there exists at least one number in [ , ] such
that ( ) .
f a b k f a
f b c a b
f c k
Question 4
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Question 3
Question 2
Question 1
Question 4
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Question 3
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Question 1
Position ( ), velocity ( ), and Acceleration ( ) Functions
( ) = "Position of an object"
'( ) ( ) "Velocity of an object"
''( ) '( ) ( ) "Acceleration of an object"
s t v t a t
s t
s t v t
s t v t a t
Question 4
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Question 3
Question 2
Question 1
Special Definite Integral Rules:
is defined at ( ) 0a
a
f x a f x dx
a b
b a
is integrable on a,b ( ) ( )f f x dx f x dx is integrable on three closed intervals determined by
, , and ( ) ( ) ( )b c b
a a c
f
a b c f x dx f x dx f x dx
Question 4
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Question 3
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Question 1
The Second Fundamental THM of Calculus
is continuous on an open interval containing f I a
( ) ( ).x
a
df t dt f x
dx
Question 4
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Question 3
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Question 1
THM 4.9 The Fundamental THM of Calculus
is continuous on , and and is an antiderivative of
on , ( ) ( ) ( ).b
a
f a b F f
a b f x dx F b F a
Question 4
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Question 3
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Question 1
Direction of a Particle:
'( ) 0 The particle is moving up
'( ) 0 The particle is moving down
f x
f x
Question 4
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Question 3
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Question 1
Is velocity increasing or decreasing?
( ) 0 ( ) is increasing
( ) 0 ( ) is decreasing
a x v x
a x v x
Question 4
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Question 3
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Question 1
THE AREA UNDER ( ) EQUALS THE
DISTANCE A PARTICLE HAS TRAVELED
v t
Question 4
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Question 1
( )Speed v t
Question 4
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Question 3
Question 2
Question 1
a bc
Instantaneous Velocity and Slope( ) ( )f b f a
mb a
'( )f c m
Question 4
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Question 3
Question 2
Question 1
( ( )) '( ) ( ( ))f g x g x dx F g x C If ( ), then '( ) and u g x du g x dx
( ) ( ) .f u du F u C
Integration by Substitution
Question 4
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Question 3
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Question 1
Trapezoid Rule
THM 4.16 The Trapezoidal Rule
Let f be continuous on a,b .
b
0 1 1a( ) ( ) 2 ( ) ... 2 ( ) ( )
2 n n
b af x dx f x f x f x f x
n
Question 4
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Question 3
Question 2
Question 1
Trapezoidal Approximation
1
10
?Area
h
r10 cm
10 cm2
Water in this container is evaporating so that the depth
-3 1is changing at the constant rate of / .
10 3
h
cm hr V r h
3
10
dh
dt
(a) Find the volume of water in the container when
5 cm. Indicate units of measure.
V
h
21
3V r h
Use similar right 's
to write in terms of .V h
5
10 2
r
h
hr
2 31
3 2 12
hh
hV
125
5
12V
(b) Find the rate of change of the volume of water in the
container, with respect to time, when 5 cm. Indicate
units of measure.
h3
12
hV
2' 312
dhV t h
dt
33' 5 3 25
12 10
15 cm / hr
8V
3cm
h
r10 cm
10 cm2
Water in this container is evaporating so that the depth
-3 1is changing at the constant rate of / .
10 3
h
cm hr V r h
3
10
dh
dt
(a) Find the volume of water in the container when
5 cm. Indicate units of measure.
V
h
(c) Show that the rate of change of the volume of water
in the container due to evaporation is directly
proportional to the exposed surface area of the water.
What is the constant of proportionality?
(b) Find the rate of change of the volume of water in the
container, with respect to time, when 5 cm. Indicate
units of measure.
h
2Let 5 :
12
1
5 3.5
8 0
h
k k
33 cm
12
hV
315' 5 cm / hr
8V