Deck 3: Applications of the Derivative

Question 1

Find an equation of a possible function with a local minimum at x = 2 that is continuous but not differentiable at x = 2.

A) $f ( x ) = | x | + 2$
B) $f ( x ) = | x + 2 |$
C) $f ( x ) = | x | - 2$
D) $f ( x ) = | x - 2 |$

Answer 1

$f ( x ) = | x - 2 |$

Question 2

If f has a local minimum at x = 2, then what can you say about $f ^ { \prime } ( 2 )$ ? What if you also know that f is differentiable at x = 2?

Answer 2

Either $f ^ { \prime } ( 2 ) = 0 \text { or } f ^ { \prime } ( 2 )$ is undefined. If we also know that f is differentiable at x = 2, then $f ^ { \prime } ( 2 ) = 0$

Question 3

If a continuous and differentiable function f has zeros at $x = - 2$ , $x = 2$ , and $x = 5$ , what can you say about $f ^ { \prime }$ on [-2, 5]?

Answer 3

$f ^ { \prime }$ has at least two zeros in the interval [-2, 5].

Question 4

If a continuous and differentiable function f is equal to - 3 at $x = - 2$ and $x = 2$ , what can you say about $f ^ { \prime }$ on [-2, 2]?

Question 5

If a function f is continuous and differentiable everywhere, $f ( - 1 ) = 4 , \text { and } f ( 2 ) = 3$ What can you say about $f ^ { \prime }$ on [-1, 2]?

Question 6

A function f that is defined on [-1, 3] with $f ( - 1 ) = f ( 3 ) = 2$ such that f is continuous everywhere, differentiable everywhere except at $x = 1$ but fails the conclusion of Rolle's Theorem. Explain why it doesn't satisfy the Rolle's Theorem?

Question 7

Find the critical points of $f ( x ) = \frac { \ln 4 x } { 2 x }$

A) 0
B) $\frac { 1 } { 2 } e$
C) $\frac { 1 } { 4 } e$
D) 1

Question 8

Find the critical points of $f ( x ) = \csc x$

A) $k \pi$
B) $( 2 k + 1 ) \frac { \pi } { 2 }$
C) $( 2 k + 1 ) \pi$
D) $k \frac { \pi } { 2 }$

Question 9

Find the critical points of $f ( x ) = \sin x$

A) $( 2 k + 1 ) \pi$
B) $k \pi$
C) $k \frac { \pi } { 2 }$
D) $( 2 k + 1 ) \frac { \pi } { 2 }$

Question 10

Find the critical points of $f ( x ) = ( 2 x + 1 ) ^ { 3 }$

A) $1 / 2$
B) - $1 / 2$
C) 1
D) 2

Question 11

Find the critical points of $f ( x ) = 6 x ^ { 4 } - 16 x ^ { 3 } - 24 x ^ { 2 } + 24 x$

Question 12

Find the critical points of $f ( x ) = e ^ { x } \left( x ^ { 2 } - 2 x \right)$

A) $\sqrt { 2 }$
B) - 2, 2
C) - , $\sqrt { 2 }$ $\sqrt { 2 }$
D) 2

Question 13

Determine whether or not the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 2 x$ satisfies the hypothesis of Rolle's Theorem on the interval [0, 2]. If it does, find the exact values of all $c \in ( 0,2 )$ that satisfy the conclusion of Rolle's Theorem.

Question 14

Determine whether or not the function $f ( x ) = \cos 2 x$ satisfies the hypothesis of Rolle's Theorem on the interval $\left[ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } \right]$ If it does, find the exact values of all values of $c \in \left( \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } \right)$ that satisfy the conclusion of Rolle's Theorem.

A) No
B) Yes, $c = \frac { \pi } { 2 } , c = \frac { 3 \pi } { 2 }$
C) Yes, $c = - \frac { \pi } { 2 } , c = \frac { \pi } { 2 }$
D) Yes, $c = \frac { \pi } { 2 }$

Question 15

Determine whether or not the function $f ( x ) = e ^ { x } \left( x ^ { 2 } - 3 x \right)$ satisfies the hypothesis of Rolle's Theorem on the interval [0, 3]. If it does, find the exact values of all $c \in ( 0,3 )$ that satisfy the conclusion of Rolle's Theorem.

Question 16

Does $f ( x ) = 2 x ^ { 2 } + \frac { 1 } { x }$ satisfy the hypothesis of the Mean Value Theorem on the interval [-1, 2]. If it does, then find the exact values of all $c \in ( - 1,2 )$ that satisfy the conclusion of the Mean Value Theorem.

Question 17

Does $f ( x ) = \sqrt { x + 4 }$ satisfy the hypothesis of the Mean Value Theorem on the interval [0, 5]. If it does, then find the exact values of all $c \in ( 0,5 )$ that satisfy the conclusion of the Mean Value Theorem.

A) 9/2
B) $3 / 4$
C) 9/4
D) Does not satisfy M.V.T.

Question 18

Does $f ( x ) = x ^ { 3 } - 2 x ^ { 2 } + 1$ satisfy the hypothesis of the Mean Value Theorem on the interval [0,3]. If it does, then find the exact values of all $c \in ( 0,3 )$ that satisfy the conclusion of the Mean Value Theorem.

Question 19

If two functions $f ( x ) \text { and } g ( x )$ have the same derivatives, then what can you say about the function $f ( x ) - g ( x )$ ?

Question 20

Determine the intervals on which $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 1$ is increasing and decreasing.

Question 21

Determine the intervals on which $f ( x ) = e ^ { x } ( x - 3 )$ is increasing and decreasing.

Question 22

Determine the intervals on which $f ( x ) = \frac { 2 x + 1 } { x ^ { 2 } - 4 }$ is increasing and decreasing.

Question 23

Determine the intervals on which $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ is increasing and decreasing.

Question 24

Determine the intervals on which $f ( x ) = \sin ^ { 2 } x$ is increasing and decreasing.

Question 25

Determine the intervals on which $f ( x ) = \ln \left( x ^ { 2 } + 2 \right)$ is increasing and decreasing.

Question 26

Determine the intervals on which $f ( x ) = \frac { e ^ { x } } { 2 + e ^ { x } }$ is increasing and decreasing.

Question 27

Use the first derivative test to determine the local extrema of $f ( x ) = ( x - 1 ) ^ { 2 } ( x + 2 )$

A) f has a local maximum at x = 2 and a local minimum at x = 1.
B) f has a local maximum at x = 1 and a local minimum at x = -1.
C) f has a local maximum at x = 1 and has no local minimum.
D) f has a local maximum at x = -1 and a local minimum at x = 1.

Question 28

Use the first derivative test to determine the local extrema of $f ( x ) = \frac { ( x - 2 ) ^ { 2 } } { x + 1 }$

A) f has a local maximum at x = 2 and a local minimum at x = 4.
B) f has a local maximum at x = 2 and a local minimum at x = -4.
C) f has a local maximum at x = -4 and a local minimum at x = 2.
D) f has a local maximum at x = 4 and a local minimum at x = 2.

Question 29

Use the first derivative test to determine the local extrema of $f ( x ) = e ^ { x } \left( x ^ { 2 } - 3 x + 2 \right)$

Question 30

Use the first derivative test to determine the local extrema of $f ( x ) = \arctan 2 x$

Question 31

Sketch the graph of a continuous function, if possible, such that $f < 0 \text { on } ( - 2 , \infty )$ , $f > 0 \text { on } ( - \infty , - 2 )$ , $f ^ { \prime } < 0 \text { on } ( - \infty , \infty )$ and $f ^ { \prime \prime } > 0 \text { on } ( - \infty , - 1 )$ , but $f ^ { \prime \prime } < 0 \text { on } ( - 1 , \infty )$

Question 32

Sketch the graph of a continuous function, if possible, such that $f < 0 \text { on } ( - \infty , 1 )$ , $f > 0 \text { on } ( 1 , \infty )$ , $f ^ { \prime } > 0 \text { on } ( - \infty , \infty )$ , $f ^ { \prime \prime } > 0 \text { on } ( - \infty , 1 )$ and $f ^ { \prime \prime } < 0 \text { on } ( 1 , \infty )$

Question 33

Sketch the graph of a continuous function, if possible, such that $f > 0 \text { on } ( - \infty , - 4 )$ and $( 0 , \infty )$ , $f < 0 \text { on } ( - 4,0 )$ , $f ^ { \prime } > 0 \text { on } ( - 2 , \infty )$ , $f ^ { \prime } < 0 \text { on } ( - \infty , - 2 )$ and $f ^ { \prime \prime } > 0 \text { on } ( - \infty , \infty )$

Question 34

Sketch the graph of a continuous function, if possible, such that $f ^ { \prime } < 0 \text { on } ( - \infty , 2 )$ $f ^ { \prime } > 0 \text { on } ( 2 , \infty )$ , $f ( 2 ) = 0$ and $f ^ { \prime \prime } > 0 \text { on } ( - \infty , \infty )$

Question 35

Sketch the graph of a continuous function, if possible, such that $f < 0 \text { on } ( - \infty , 4 )$ , $f > 0 \text { on } ( 4 , \infty )$ , $f ^ { \prime } > 0 \text { on } ( - \infty , 0 )$ and $( 8 / 3 , \infty )$ , $f ^ { \prime } < 0 \text { on } ( 0,8 / 3 )$ , $f ^ { \prime \prime } > 0 \text { on } ( 4 / 3 , \infty )$ , but $f ^ { \prime \prime } < 0 \text { on } ( - \infty , 4 / 3 )$

Question 36

Sketch the graph of a continuous function, if possible, such that $f < 0 \text { on } ( - \infty , 0 )$ , $f > 0 \text { on } ( 0 , \infty )$ , and $f ^ { \prime } > 0 \text { on } ( - \infty , 1 / 3 ) \text { and } ( 1 , \infty )$ $f ^ { \prime } < 0 \text { on } ( 1 / 3,1 )$ and $f ^ { \prime \prime } > 0 \text { on } ( 2 / 3 , \infty )$ and $f ^ { \prime \prime } < 0 \text { on } ( - \infty , 2 / 3 )$

Question 37

Sketch the graph of a continuous function, if possible, such that $f ^ { \prime }$ does not exist at $x = 0 , f ^ { \prime \prime } > 0$ on $( 0 , \infty )$ , $f ( x ) > 0$ on $( 1,5 ) , f ( x ) < 0$ on $( - \infty , 1 ]$ and on $[ 5 , \infty ) , f ^ { \prime } ( x ) > 0$ on $( - \infty , 3 ) , f ^ { \prime } < 0$ on $( 3 , \infty )$ , and $f ^ { \prime } ( x )$ does not exist at $x = 3$ .

Question 38

Sketch labeled graphs of each function $f ( x ) = x ^ { 2 } + 5 x$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f$

Question 39

Sketch labeled graphs of each function $f ( x ) = ( x - 1 ) ( x + 2 )$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f^{\prime \prime }$

Question 40

Sketch labeled graphs of each function $f ( x ) = x ^ { 3 } + 2 x ^ { 2 }$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f ^ { \prime\prime }$

Question 41

Sketch labeled graphs of each function $f ( x ) = \frac { - 1 } { x ^ { 2 } + 2 }$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f^ { \prime\prime }$

Question 42

Sketch labeled graphs of each function $f ( x ) = \frac { x } { x ^ { 2 } + 4 }$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f^ { \prime\prime }$

Question 43

Sketch labeled graphs of each function $f ( x ) = \ln 2 x + 1$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f^ { \prime \prime}$

Question 44

Sketch labeled graphs of each function $f ( x ) = e ^ { - x ^ { 2 } }$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f^ { \prime\prime }$

Question 45

Use a sign chart to determine the intervals on which $f ( x ) = ( x + 3 ) ^ { 4 }$ is concave up and concave down, and identify the locations of any inflection points.

Question 46

Use a sign chart to determine the intervals on which $f ( x ) = \frac { 1 } { x ^ { 2 } + 4 }$ is concave up and concave down, and identify the locations of any inflection points.

Question 47

Use a sign chart to determine the intervals on which $f ( x ) = e ^ { 2 x } \left( 1 - e ^ { x } \right)$ is concave up and concave down, and identify the locations of any inflection points.

Question 48

Use a sign chart to determine the intervals on which $f ( x ) = ( x - 2 ) ^ { 3 } ( x - 1 )$ is concave up and concave down, and identify the locations of any inflection points.

Question 49

Use a sign chart to determine the intervals on which $f ( x ) = x ^ { 3 } - 3 x + 1$ is concave up and concave down, and identify the locations of any inflection points.

Question 50

Use the derivative $f ^ { \prime } ( x ) = \frac { 2 } { x }$ to find the local extrema and inflection points of $f$

Question 51

Use the derivative $f ^ { \prime } ( x ) = e ^ { 2 x } ( x + 2 )$ to find the local extrema and inflection points of $f$

Question 52

Use the derivative $f ^ { \prime } ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 4 x$ to find the local extrema and inflection points of $f$

Question 53

Use the derivative $f ^ { \prime } ( x ) = x ^ { 4 } - 2$ to find the local extrema and inflection points of $f$

Question 54

Find the location and values of any global extrema of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x$ on the intervals: (a) [-2, 3](b) [-2, 4]

Question 55

Find the location and values of any global extrema of $f ( x ) = 3 x ^ { 4 } - 4 x ^ { 3 } - 6 x ^ { 2 } + 12 x$ on the intervals: (a) (-2, 1](b) [-2, 1]

Question 56

Find the location and values of any global extrema of $f ( x ) = x - 2 \cos x$ on $\left[ - \frac { \pi } { 4 } , \frac { \pi } { 2 } \right]$

Question 57

Find the location and values of any global extrema of $f ( x ) = \frac { x - 2 } { x + 1 }$ on [-2, 4).

Question 58

Jen needs to make a flyer for her dog's birthday party. She wants the flyer to contain 40 square inches of printed portion and she wants to use 2 inches of each side as well as two inches of top and bottom of the paper for decoration. What size of paper should Jen choose in order to use the least amount of paper per flyer?

Question 59

Find the point(s) on the curve $y = x ^ { 2 }$ that is closest to the point (3, 0).

A) (2, 2)
B) (-1, 1)
C) (1, 1)
D) (3, 3)

Question 60

Find the point(s) on the curve $y = x ^ { 2 }$ that is closest to the point (0, 3).

Question 61

Find two numbers whose product is 12 and whose sum of squares is minimum?

A) (2, 6) and (-2, -6)
B) $- \sqrt { 12 } , - \sqrt { 12 } \text { and } \sqrt { 12 } , \sqrt { 12 }$
C) (3, 4) and (-3, -4)
D) $- \sqrt { 3 } , \sqrt { 4 } \text { and } \sqrt { 3 } , \sqrt { 4 }$

Question 62

My brother wants to make an open-topped box out of a 4 × 6 square feet piece of cardboard by cutting identical squares from the corners and folding up the sides. What is the dimension of each square he will cut out of each corner in order to maximize the volume of the box he makes?

A) $\frac { 10 - \sqrt { 7 } } { 6 }$
B) $\frac { 10 + \sqrt { 7 } } { 6 }$
C) $\frac { 5 + \sqrt { 7 } } { 3 }$
D) $\frac { 5 - \sqrt { 7 } } { 3 }$

Question 63

A veterinarian has 90 ft. of fence and he wants to enclose a rectangular dog-run along the 60-feet long back side of his office building. He will not fence the side along the building. What are the dimensions of the dog-run that gives the maximum area he desires?

Question 64

A veterinarian wants to make three identical adjoining dog-runs in the backyard of his office building. He needs each dog-run to be 400 square feet. What are the dimensions of each dog-run that requires the minimum amount of fencing material?

Question 65

Melissa wants to make a rectangular box with a square base and cover its top and bottom faces by velvet which will cost her $3 per square inch and the sides by silk which will cost her $5 per square inch. The box should have a volume of 1600 cubic inches. Find the dimensions of the box that will cost her the least amount of money.

Question 66

The cost of the material for the top and bottom of a cylindrical can is 10 cents per square inch. The material for the rest of the can costs 5 cents per square inch. If the can must hold 500 cubic inches of liquid, what dimensions should be chosen to make the cheapest can?

Question 67

Given $u = u ( t ) , \quad v = v ( t ) , \quad w = w ( t )$ are functions of t, calculate the derivative of the functions:
(a) $f ( t ) = u ^ { 2 } + 3 v$
(b) $f ( t ) = 2 u \sqrt { v - w }$

Question 68

Given $u = u ( t ) , \quad v = v ( t ) , \quad w = w ( t )$ are functions of t, calculate the derivative of the functions:
(a) $f ( t ) = t u + 5 u ^ { 3 } v$
(b) $f ( t ) = \frac { 2 u } { v w }$

Question 69

Given $u = u ( t ) , \quad v = v ( t ) , \quad w = w ( t )$ are functions of t, calculate the derivative of the functions:
(a) $f ( t ) = ( u + v ) ^ { 2 } + 2 t w$
(b) $f ( t ) = \frac { 3 } { u ^ { 2 } v }$

Question 70

Find $\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } + x - 6 } { x - 2 }$

A) 2
B) -5
C) 5
D) DNE

Question 71

Find $\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } - 3 x + 2 } { x - 1 }$

A) 1
B) -1
C) 0
D) DNE

Question 72

Find $\lim _ { x \rightarrow - \infty } \frac { x + 2 } { 3 - 2 x ^ { 2 } }$

A) 1
B) - $1 / 2$
C) $\infty$
D) 0

Question 73

Find $\lim _ { x \rightarrow \infty } \frac { e ^ { 2 x } } { 1 + e ^ { 5 x } }$

A) DNE
B) 0
C) $\infty$
D) 2/5

Question 74

Find $\lim _ { x \rightarrow \infty } \frac { 2 ^ { x } } { 1 + 3 ^ { x } }$

A) 1
B) -1
C) 0
D) DNE

Question 75

Find $\lim _ { x \rightarrow 0 } \frac { 2 \cos x - 2 } { \sin 2 x }$

A) 1
B) 2
C) -1
D) 0

Question 76

Find $\lim _ { x \rightarrow 0 ^ { + } } x ^ { 3 x }$

A) $e ^ { 3 }$
B) 0
C) 1
D) e

Question 77

Find $\lim _ { x \rightarrow \infty } x ^ { 2 / x }$

A) $\infty$
B) 1
C) 0
D) e

Question 78

Find $\lim _ { x \rightarrow 0 ^ { + } } x ^ { \sin x }$

A) $\infty$
B) 1
C) e
D) 0

Question 79

Find $\lim _ { x \rightarrow 0 ^ { + } } \left( e ^ { 3 x } - 1 \right) ^ { x }$

A) 1
B) e
C) $e ^ { 2 }$
D) 0

Question 80

Find $\lim _ { x \rightarrow 0 } \left( e ^ { x } + x \right) ^ { 1 / x }$

A) 1
B) e
C) $e ^ { 2 }$
D) 0