Applications of Derivatives Maximum and Minimum Class Work Find the extreme values and the x at which they occur 1. a b c d 2. = 2 2 + 6 − 7 3. = 3 3 + 6 2 − 8 4. = √4 − 8 5. = √2 + 1 3 6. = 1 2 −4 7. ={ 2 − 4; ≤ 2 3 + 6; > 2 Homework Find the extreme values and the x at which they occur 8. a b c d 9. = −3 2 + 4 − 7 10. = 2 4 − 6 2 + 4 − 6 11. = √8 − 2 12. = √2 − 2 3 13. = 1 2 14. ={ −2 2 − 2; ≤ 4 −2 − 6; > 4 Mean Value Theorem Class Work Find the value of c that satisfies the Mean Value Theorem for the interval. If the Mean Value Theorem does not apply, state why. 15. = 3 2 − 2 [1,2] 16. = √ [1,4] 17. = 1 [ −1 2 , 1 2 ] 18. = 3 [2,5] 19. = ln [1,3] 20. ={ ||; ≤ 0 sin ; > 0 [– , ]
Transcript
Applications of Derivatives Maximum and Minimum
Class Work
Find the extreme values and the x at which they occur
1. a b c d
2. 𝑦 = 2𝑥2 + 6𝑥 − 7
3. 𝑦 = 3𝑥3 + 6𝑥2 − 8𝑥
4. 𝑦 = √4𝑥 − 8
5. 𝑦 = √2𝑥 + 13
6. 𝑦 =1
𝑥2−4
7. 𝑦 = {𝑥2 − 4𝑥; 𝑥 ≤ 23𝑥 + 6; 𝑥 > 2
Homework
Find the extreme values and the x at which they occur
8. a b c d
9. 𝑦 = −3𝑥2 + 4𝑥 − 7
10. 𝑦 = 2𝑥4 − 6𝑥2 + 4𝑥 − 6
11. 𝑦 = √8 − 2𝑥
12. 𝑦 = √2 − 𝑥23
13. 𝑦 =1
𝑥2
14. 𝑦 = {−2𝑥2 − 2𝑥; 𝑥 ≤ 4
−2𝑥 − 6; 𝑥 > 4
Mean Value Theorem
Class Work
Find the value of c that satisfies the Mean Value Theorem for the interval. If the Mean Value
Theorem does not apply, state why.
15. 𝑦 = 3𝑥2 − 2 𝑜𝑛 [1,2]
16. 𝑦 = √𝑥 𝑜𝑛 [1,4]
17. 𝑦 =1
𝑥 𝑜𝑛 [
−1
2,
1
2]
18. 𝑦 = 𝑥3𝑜𝑛 [2,5]
19. 𝑦 = ln 𝑥 𝑜𝑛 [1,3]
20. 𝑦 = {|𝑥|; 𝑥 ≤ 0
sin 𝑥; 𝑥 > 0𝑜𝑛 [– 𝜋, 𝜋]
Find the local extrema, interval of increase, and interval of decrease.
21. 𝑦 = 4𝑥2 − 6𝑥 + 2
22. 𝑦 = 8𝑥3 + 4𝑥2 − 5
23. 𝑦 = 𝑒𝑥2
24. 𝑦 =𝑥
𝑥2−1
Draw a graph of the differentiable function that satisfies the following
25. defined on [1,10], local max at (2,0), (7,7) local min at (1,-1), (4,-2), and (10,-2), and
f ‘(5)=0 but (5, 4) is not a local max or min.
26. Continuous function with the following characteristics
x<1 x=1 1<x<2 x=2 2<x<3 x=3 x>3
f(x) 2 4 0
f ‘(x) + 1 + 0 - undefined +
Homework
Find the value of c that satisfies the Mean Value Theorem for the interval. If the Mean Value
Theorem does not apply, state why.
27. 𝑦 = 3𝑥2 − 2 𝑜𝑛 [3,6]
28. 𝑦 = √𝑥 𝑜𝑛 [4,9]
29. 𝑦 =1
𝑥 𝑜𝑛 [
1
4,
1
2]
30. 𝑦 = 𝑥3𝑜𝑛 [1,6]
31. 𝑦 = ln|𝑥| 𝑜𝑛 [−1,3]
32. 𝑦 = {cos 𝑥 ; 𝑥 ≤
𝜋
4
sin 𝑥; 𝑥 >𝜋
4
𝑜𝑛 [0,𝜋
2]
Find the local extrema, interval of increase, and interval of decrease.
33. 𝑦 = 3𝑥2 − 5𝑥 + 1
34. 𝑦 = 𝑥3 + 6𝑥2 − 5𝑥
35. 𝑦 = 2𝑥2
36. 𝑦 =𝑥
𝑥2−1𝑥−2
Draw a graph of the differentiable function that satisfies the following
37. defined on [1,10], local max at (3,0), (8,7) local min at (1,-4), (6,-2), and (10,-1), and
f ‘(4)=0 but (4,-1) is not a local max or min.
38. Continuous function with the following characteristics
x<1 x=1 1<x<2 x=2 2<x<3 x=3 x>3
f(x) 0 4 0
f ‘(x) + DNE + 0 - DNE -
Derivatives and Graphs
Class Work
Given f(x), identify any local extrema, the intervals of increase and decrease, concavity, and
points of inflection.
39. f(x) = x3 − 6x2 + 4x − 5
40. f(x) =x
x2+1
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
f(x)
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
f(x)
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
f '(x)
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
f '(x)
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
f "(x)
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
f "(x)
41. a. b.
Given f ‘(x), identify any local extrema, the intervals of increase and decrease, concavity, and
points of inflection.
42. 𝑓′(𝑥) = 𝑥2 − 6𝑥 + 5
43. 𝑓′(𝑥) = 4𝑥 + 6
44. a. b.
Given f “(x), identify concavity and points of inflection.
45. 𝑓′′(𝑥) = 4𝑥 + 5
46. 𝑓′′(𝑥) = 𝑥2 − 1
47. a. b.
Use the following information to identify any local extrema, the intervals of increase and
decrease, concavity, and points of inflection. Graph f(x)
48.
x<1 x=1 1<x<2 x=2 2<x<3 x=3 x>3
f(x) 0 4 6
f ‘(x) - 0 + + + 0 +
f “ (x) + DNE + 0 - - -
49.
x<1 x=1 1<x<2 x=2 2<x<3 x=3 x>3
f ‘(x) + 3 + 0 - -4 -
f “(x) + 0 - - - 0 -
50.
x<1 x=1 1<x<2 x=2 2<x<3 x=3 x>3
f ‘(x) - ? ? 0 ? ? +
f “(x) + 0 - - - 0 +
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
f(x)
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
f(x)
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
f '(x)
-π -π/2 π/2 π
-4
-3
-2
-1
1
2
3
4
x
f '(x)
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
f "(x)
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
f "(x)
Homework
Given f(x), identify any local or extrema, the intervals of increase and decrease, concavity, and
points of inflection.
51. f(x) = 2x3 + 4x − 5
52. f(x) = √x2 + 1
53. a. b.
Given f ‘(x), identify any local or extrema, the intervals of increase and decrease, concavity, and
points of inflection.
54. 𝑓′(𝑥) = 3𝑥2 − 4𝑥
55. 𝑓′(𝑥) = 5𝑥
56. a. b.
Given f “(x), identify concavity and points of inflection.
57. 𝑓′′(𝑥) = 4
58. 𝑓′′(𝑥) = √𝑥23
59. a. b.
Use the following information to identify any local or extrema, the intervals of increase and
decrease, concavity, and points of inflection. Graph f(x)
60.
x<1 x=1 1<x<2 x=2 2<x<3 x=3 x>3
f(x) 3 2 1
f ‘(x) + 0 - DNE - 0 +
f “ (x) - -3 - DNE - -3 -
61.
x<1 x=1 1<x<2 x=2 2<x<3 x=3 x>3
f ‘(x) + 4 + 0 - -4 -
f “(x) + 0 + + + 0 -
62.
x<1 x=1 1<x<2 x=2 2<x<3 x=3 x>3
f ‘(x) - ? ? 0 ? ? +
f “(x) + 0 + + + 0 -
Optimization
Class Work
63. An 8” by 12” sheet has 4 equal squares cut from its corners so that when the sides are
folded the sheet forms an open top box. What is the greatest volume possible?
64. A farmer has 1000’ of fence to make a new pen. If he uses an existing pen as one side
what is the maximum area he can enclose?
65. A 7’ sofa is place at an angle to the corner of a room forming a right triangle behind the
sofa. What is the greatest area that the sofa can block off.
66. A pizza box is formed by cutting 6 equal squares from a rectangular piece of cardboard.
One square from each corner and a square from the middle of the longest sides of the
48” by 20” sheet. When folded, the box has a lid with over-lapping sides. If the box is
made to maximize volume, what is the largest diameter pizza that will fit in the box?
67. An underground storage vault needs to be built with a square base and have a volume of
600 sq ft. If the top of the vault cost $20 sq ft, the sides $18 sq ft, and the bottom $10 sq
ft. what dimensions will minimize the cost?
68. A cylindrical can needs to hold 500 cc of soda. Recalling 𝑆𝐴 = 2𝜋𝑟2 + 2𝜋𝑟ℎ, find the
height that minimizes the material used.
Homework
69. An 7” by 11” sheet has 4 equal squares cut from its corners so that when the sides are
folded the sheet forms an open top box. What is the greatest volume possible?
70. A farmer has 2000’ of fence to make a new pen. He wants to use an existing pen, that is
50’ long, as part of one side what is the maximum area he can enclose?
71. A 60” TV is place at an angle to the corner of a room forming a right triangle behind the
TV. What is the greatest area that the sofa can block off.
72. A pizza box is formed by cutting 6 equal squares from a rectangular piece of cardboard.
One square from each corner and a square from the middle of the longest sides of the
54” by 30” sheet. When folded, the box has a lid with over-lapping sides. If the box is
made to maximize volume, what is the largest height pizza that will fit in the box?
73. An underground storage vault needs to be built with a square base and have a volume of
1000 sq ft. If the top of the vault cost $40 sq ft, the sides $30 sq ft, and the bottom $15
sq ft. what dimensions will minimize the cost?
74. A right conical storage facility is built for the road department to hold ice melt. The depot
needs to hold 10000 cu ft of materials. What is the height of the cone that minimizes
materials for the walls if 𝑟 ≤ 20 (no floor so 𝐿𝐴 = 𝜋𝑟ℎ)?
Linear Approximation
Class Work
Find the approximate value at the given point by first writing the linear approximation of the
equation near that point. Determine whether approximation is an over or under estimate.
75. 𝑓(5) 𝑓𝑜𝑟 𝑓(𝑥) = √𝑥
76. 𝑔(8) 𝑓𝑜𝑟 𝑔(𝑥) =1
√𝑥
77. ℎ (4𝜋
9) 𝑓𝑜𝑟 ℎ(𝑥) = cos 𝑥
78. 𝑓( 59) 𝑓𝑜𝑟 𝑓(𝑥) = √𝑥3
79. 𝑔(7)𝑓𝑜𝑟 𝑔(𝑥) =𝑥2+8
𝑥
80. ℎ(. 1) 𝑓𝑜𝑟 ℎ(𝑥) = (1 + 𝑥)𝑘
a. use your answer to approximate (1.003)50
b. use your answer to approximate √1.0064
c. use your answer to write a linear approximation for
ℎ(𝑥) =3
1−𝑥 𝑛𝑒𝑎𝑟 𝑧𝑒𝑟𝑜
d. use your answer to write a linear approximation for
ℎ(𝑥) =1
√1−𝑥3 𝑛𝑒𝑎𝑟 𝑧𝑒𝑟𝑜
Homework
Find the approximate value at the given point by first writing the linear approximation of the
equation near that point. Determine whether approximation is an over or under estimate.
81. 𝑓(14) 𝑓𝑜𝑟 𝑓(𝑥) = √1 + 𝑥
82. 𝑔(−23) 𝑓𝑜𝑟 𝑔(𝑥) =1
√1−𝑥
83. ℎ (𝜋
25) 𝑓𝑜𝑟 ℎ(𝑥) = sin 𝑥
84. 𝑓( 22) 𝑓𝑜𝑟 𝑓(𝑥) = √6𝑥3
85. 𝑔(7)𝑓𝑜𝑟 𝑔(𝑥) =𝑥+8
𝑥
86. ℎ(. 1) 𝑓𝑜𝑟 ℎ(𝑥) = (1 + 𝑥)𝑘
a. use your answer to approximate (1.0008)70
b. use your answer to approximate √1.0075
c. use your answer to write a linear approximation for
ℎ(𝑥) =5
4(1−𝑥) 𝑛𝑒𝑎𝑟 𝑧𝑒𝑟𝑜
d. use your answer to write a linear approximation for
ℎ(𝑥) =5
√1−𝑥4 𝑛𝑒𝑎𝑟 𝑧𝑒𝑟𝑜
Related Rates
Class Work
87. A 17’ ladder slides down a wall at a rate of 3ft/sec, how fast is the bottom sliding away
when the ladder 8’ up the wall?
88. A rock is dropped in a pond creating a circular splash. The radius grows at a rate of
3ft/sec. How fast does the area change when the splash is 4’ across?
89. An airplane flies over an observation post at 5 mi high. An observer at the post measures
the distance to the plane to be 13 mi and moving at 200 mph. What is the planes actual
speed?
90. A boat is tied to a dock by a 10’ rope. The tide is going down at a rate of 6 in/hr.
Assuming the rope is taut, how is the boat’s distance to the dock changing when the boat
is 6’ below the dock?
91. A cylindrical water cooler is 2’ wide and 3’ high. When the spigot is left on, water runs out
at 20 cu in/ min. How fast is the height of the water changing when there are 18”left in
the cooler?
92. Air is being pumped into a spherical balloon at a rate of 40 cc/min. How fast is the surface
area changing when the radius is 5 cm?
93. An hour glass is an inverted right cone above another right cone. When all of the sand is
in one end it forms a cone 3” high by 2” across. Sand runs from the top cone at 3 cu in/
hr.
a. How fast is the height dropping in the top cone when there are 2” of sand left?
b. How fast is the height growing in the bottom cone when there are 2” left in the
top?
94. A 2 meter tall man walks away from a 4m lamp post at a rate 2m/sec. How fast is the
man’s shadow growing when he is 3m from the post?
Homework
95. A 15’ ladder slides down a wall at a rate of 4ft/sec, how fast is the bottom sliding away
when the ladder 9’ up the wall?
96. A firecracker radiates sound in semispherical pattern. The radius grows at a rate of
5ft/sec. How fast does the volume change 6 seconds after exploding?
97. An airplane flies over an observation post at 6 mi high. An observer at the post measures
the distance to the plane to be 8 mi and moving at 300 mph away from her. What is the
rate of change of the planes angle of depression?
98. A boat is tied to a dock by a rope. The rope is pulled in at 2’ per sec. The level of the
water is 8’ below the rope on the dock. How fast is the boat approaching the dock when
there is 20’ of rope left?
99. A cylindrical water tank 10’ wide and 30’ high. A pipe fills the tank at 20 cu ft/ min. How
fast is the height of the water changing when there after 30 minutes?
100. Air is being pumped into a spherical balloon at a rate of 30 cc/min. How fast is the
surface area changing when the radius is 8 cm?
101. An hour glass is an inverted right cone above another right cone. When all of the sand is
one end it forms a cone 8” high by 4” across. Sand runs from the top cone at 4 cu in/ hr.
a. How fast is the height dropping in the top cone when there are 2” of sand left?
b. How fast is the height growing in the bottom cone when there are 2” left in the
top?
102. A 2 meter tall man walks toward a 5m lamp post at a rate 3m/sec. How fast is the man’s
shadow changing when he is 3m from the post?
Antiderivatives
Class Work
Find the antiderivatives of the following.
103. 𝑓′(𝑥) = 4
104. 𝑔′(𝑥) = 2𝑥
105. ℎ′(𝑥) = 3𝑥 + 5
106. 𝑓′(𝑥) =2
𝑥2
107. 𝑔′(𝑥) = 4 cos 𝑥
108. ℎ′(𝑥) = − sin 𝑥
Given y’ and the initial value, find y.
109. 𝑦′ = 4𝑥 + 2; 𝑦(0) = 3
110. 𝑦′ = 6𝑥2 + 4𝑥 − 1; 𝑦(1) = 6
111. 𝑦′ = cos 𝑥; 𝑦(𝜋
6) = 3
112. 𝑦′ =6
𝑥3 ; 𝑦(. 5) = 4
Homework
Find the antiderivatives of the following.
113. 𝑓′(𝑥) = −5
114. 𝑔′(𝑥) = 8𝑥
115. ℎ′(𝑥) = 3𝑥2 − 7
116. 𝑓′(𝑥) =−6
𝑥4
117. 𝑔′(𝑥) = −3 cos 𝑥
118. ℎ′(𝑥) = 2 sin 𝑥
Given y’ and the initial value, find y.
119. 𝑦′ = 6𝑥 − 9; 𝑦(3) = 4
120. 𝑦′ = 9𝑥2 − 2𝑥 − 3; 𝑦(2) = 7
121. 𝑦′ = sin 𝑥; 𝑦(𝜋
2) = 3
122. 𝑦′ =−4
𝑥2 ; 𝑦(4) = −10
Multiple Choice
Calculator Not Permitted
1. The table shows the velocity of a particle. Estimate the acceleration at t = 5
time 0 2 4 6
velocity 12 9 8 6
a. 7 b. 2 c. 1 d. -1 e. -2
Use the graph of f ‘(t) to answer 2-4 about a particle moving along the x-axis with f(0)= 3.
2. Which of the following is true?
a. f(2) is a point of inflection
b. the particle is speeding up at f(4.5)
c. the particle’s velocity at t=7 is 0
d. f(t) is concave up at t=1
e. f(7)= -3
3. What is the total distance traveled by the particle for 0 ≤ 𝑡 ≤ 8?
a. 1
b. -1
c. 10
d. -10
e. 11
4. What is the particle’s greatest distance from the origin?
a. 1
b. 3
c. 6
d. 11
e. 14
5. Which of the following is a point of inflection for 𝑔(𝑥) =2
5𝑥5 −
27
2𝑥2
a. -1.5
b. -.5
c. 0
d. .5
e. 1.5
Calculator Permitted
6. An inverted conical tank is 15’ deep and 16’ across its top. If water is draining so that
depth of water is dropping at 9” per hour, how fast is the water draining when there is 9’
left in the tank?
a. 1.920 ft3/hr
b. 1.707 ft3/hr
c. 2.560 ft3/hr
d. 3.600 ft3/hr
e. 54.287 ft3/hr
7. g(8)= 15 and 𝑔′(𝑥) = 𝑥2 − 12𝑥 + 26 find the approximate value of g(7.98)
a. 14.88
b. 14.98
c. 15.02
d. 15.12
e. none of the above
8. A rectangle is to be drawn in the first quadrant such that one corner is at the origin and
the opposite lies on the graph of 𝑦 = 4𝑥 + 4 − 𝑥2. Find the area of the rectangle that has
the maximum area.
a. 2.500 u2
b. 3.097 u2
c. 7.797 u2
d. 21.049 u2
e. 48.620 u2
9. Given the graph of f ‘(x) at the right which of the choices could be of f(x)
a. b. c. d. e.
10. 𝑓(𝑥) = 4𝑥3 − 6𝑥 +1
𝑥, find c such that f(c) will equal the average value for f(x) for
1 ≤ 𝑥 ≤ 4.
a. 36.938 b. 12.313 c. 4.464 d. 2.644 e. DNE
Open Ended
Calculator Not Permitted
1. Consider the graph of f ‘ , use the given value s of x to answer
a. when is f the greatest
b. when is f the least
c. when does f have a point of inflection
2. A particle moves along the x-axis with a velocity of 𝑣(𝑡) = 𝑡𝑐𝑜𝑠(𝜋𝑡)
for 0 ≤ 𝑡 ≤ 4.
a. What is v(.5)?
b. What is a(.5)?
c. Is the particle speeding up or slowing down?
d. When does the particle change directions?
Calculator Permitted
3. A conical holding tank 25’ high and 20’ across fills a rectangular tanker truck 8’ high and
wide by 30’ long. Water is being pumped out of the holding tank 5 ft3/ min.
a. How fast is the water dropping in the holding tank when the depth is 20’
b. How fast is surface of the water in the holding tank changing at this instance?
c. How fast is the depth of water in the tanker changing when there is 3’ of water in
the tank?
d. How fast is the surface area of the water in the tanker changing at that instance?
