Deck 4: Applications of Differentiation

Question 1

A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut for the square so that the total area enclosed is a minimum? Round your answer to the nearest hundredth.

A)4.35 m
B)3.25 m
C)0 m
D)5.35 m
E)4.4 m

Answer 1

4.35 m

Question 2

The sum of two positive numbers is . What is the smallest possible value of the sum of their squares?

Answer 2

Question 3

Find the point on the line that is closest to the origin.

A)
B)
C)
D)
E)

Answer 3

Question 4

Find the most general antiderivative of the function.

Question 5

A car braked with a constant deceleration of 40 , producing skid marks measuring 60 ft before coming to a stop. How fast was the car traveling when the brakes were first applied?

Question 6

Find the most general antiderivative of the function.

A)
B)
C)
D)
E)

Question 7

A woman at a point A on the shore of a circular lake with radius wants to arrive at the point C diametrically opposite on the other side of the lake in the shortest possible time. She can walk at the rate of and row a boat at . How should she proceed? (Find ). Round the result, if necessary, to the nearest hundredth.

A) radians
B)She should row from point A to point C radians
C) radians
D) radians
E)She should walk around the lake from point A to point C.

Question 8

Given that the graph of f passes through the point (4, 69) and that the slope of its tangent line at is , find f (1) .

A)11
B)0
C)6
D)12
E)1

Question 9

Use Newton's method with the specified initial approximation to find , the third approximation to the root of the given equation. (Round your answer to four decimal places.)

A)
B)
C)
D)
E)

Question 10

Find the position function of a particle moving along a coordinate line that satisfies the given conditions. , s (0) = 5, v (0) = 0

Question 11

Find
f.

Question 12

Use Newton's method to approximate the given number correct to eight decimal places.

A)
B)
C)
D)
E)

Question 13

Use Newton's method with the specified initial approximation to find , the third approximation to the root of the given equation. (Round your answer to four decimal places.)

A)
B)
C)
D)
E)

Question 14

A particle is moving with the given data. Find the position of the particle. ,

A)
B)
C)
D)
E)

Question 15

What constant acceleration is required to increase the speed of a car from 20 ft/s to 45 ft/s in s?

Question 16

Use Newton's method with the specified initial approximation to find , the third approximation to the root of the given equation. (Give your answer to four decimal places.)

Question 17

Find the position function of a particle moving along a coordinate line that satisfies the given condition. , s(1) = -1

Question 18

Use Newton's method to approximate the indicated root of in the interval , correct to six decimal places.Use as the initial approximation.

Question 19

Find the most general antiderivative of the function.

Question 20

Find two positive numbers whose product is and whose sum is a minimum.

A)
B)3, 48
C)
D)6, 24
E)2, 72

Question 21

Use the guidelines of this section to sketch the curve. Select the graph of the curve.

A)
B)
C)
D)
E)

Question 22

What is the minimum vertical distance between the parabolas and ?

Question 23

Use the guidelines of this section to sketch the curve. Select the graph of the curve.

A)
B)
C)
D)
E)

Question 24

A steel pipe is being carried down a hallway 14 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner?

Question 25

Given .(a) Find the intervals on which f is increasing or decreasing.(b) Find the relative maxima and relative minima of f.

A)(a) Increasing on ,
Decreasing on

(b) Rel. max.
B)(a) Increasing on
(b) None
C)(a) Decreasing on
(b) None
D)(a) Increasing on ,
Decreasing on

(b) Rel. min.

Question 26

The graph of the derivative of a continuous function f is shown. On what intervals is f decreasing? .

A)
B)
C)
D)
E)None of these

Question 27

Use the guidelines of this section to sketch the curve. Select the graph of the curve.

A)
B)
C)
D)
E)

Question 28

Determine where the graph of the function is concave upward and where it is concave downward. Also, find all inflection points of the function.

A)CU on , CD on
,
IP

B)CU on , CD on
,
IP

C)CU on , CD on
,
IP

D)CU on , CD on
,
IP

Question 29

Use the guidelines of this section to sketch the curve. Select the graph of the curve.

A)
B)
C)
D)
E)

Question 30

Find an equation of the line through the point that cuts off the least area from the first quadrant.

Question 31

Find the maximum or minimum point(s) of the function.

A)
B)
C)
D)
E)

Question 32

Determine where the graph of is concave upward and where it is concave downward. Also, find all inflection points of the function.

A)concave downward: and
, concave upward:
and
; inflection points:
and

B)concave downward: and
, concave upward:
and
; inflection points:
and

C)concave downward: nowhere, concave upward: and
; inflection points: none
D)concave downward: and
, concave upward: nowhere; inflection points: none

Question 33

A farmer with 710 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens?

Question 34

Given .(a) Find the intervals on which f is increasing or decreasing.(b) Find the relative maxima and relative minima of f.

A)(a) Increasing on ,
Decreasing on

(b) Rel. max.
B)(a) Increasing on ,
Decreasing on

(b) Rel. min.
C)(a) Increasing on ,
Decreasing on

(b) Rel. max.
D)(a) Increasing on ,
Decreasing on

(b) Rel. min.

Question 35

Given .(a) Find the intervals on which f is increasing or decreasing.(b) Find the relative maxima and relative minima of f.

A)(a) Increasing on and
, decreasing on

(b) Rel. max. ,
Rel) min.
B)(a) Increasing on and
, decreasing on
and

(b) Rel. max. ,
Rel) min.
C)(a) Increasing on , decreasing on
and

(b) Rel. max. ,
Rel) min.
D)(a) Increasing on and
decreasing on
and

(b) Rel. max. ,
Rel) min.

Question 36

Use the guidelines of this section to sketch the curve. Select the graph of the curve.

A)
B)
C)
D)
E)

Question 37

Determine where the graph of the function is concave upward and where it is concave downward. Also, find all inflection points of the function.

A)CU on , CD on
,
IP

B)CU on , CD on
and

IP
and

C)CU on , CD on
,
IP

D)CU on , CD on
,
IP

Question 38

Use the Second Derivative Test to find the relative extrema, if any, of the function .

A)Rel. max. , rel. min.
B)Rel. max. , rel. min.
C)Rel. max. , rel. min.
D)Rel. max. , rel. min.

Question 39

Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length L = 8 and width W = 3.

Question 40

Given .(a) Find the intervals on which f is increasing or decreasing.(b) Find the relative maxima and relative minima of f.

A)(a) Increasing on and
decreasing on

(b) Rel. max. ,
Rel) min.
B)(a) Increasing on , decreasing on
and

(b) Rel. max. ,
Rel) min.
C)(a) Increasing on and
decreasing on

(b) Rel. max. ,
Rel) min.
D)(a) Increasing on , decreasing on
and

(b) Rel. max. ,
Rel) min.

Question 41

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. ,

A)None of these
B) ,
C) ,
D) ,
E) ,

Question 42

Find the absolute maximum and absolute minimum values, if any, of the function on .

A)Abs. max. abs. min.
B)Abs. max. abs. min.
C)Abs. max. abs. min.
D)Abs. max. abs. min.

Question 43

Find the absolute maximum and absolute minimum values, if any, of the function on [0, 25]

A)Abs. max. abs. min.
B)Abs. max. abs. min.
C)Abs. max. abs. min.
D)Abs. max. abs. min.

Question 44

Estimate the extreme values of the function. Round the answers to the nearest hundredth.

Question 45

The function satisfies the hypotheses of the Mean Value Theorem on the interval . Find all values of c that satisfy the conclusion of the theorem.

A)-7
B)-8, -6
C)-8, -7
D)-6

Question 46

Consider the function . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of F.

Question 47

Find the critical number(s), if any, of the function

A)
B)
C)
D)

Question 48

Find the critical numbers of .

A)0, 3,
B)0, 3,
C)
D)0, 2,
E)0, 2,

Question 49

Determine where the graph of is concave upward and where it is concave downward. Also, find all inflection points of the function.

Question 50

The graph of the first derivative of a function f is shown below. At what values of x does f have a local maximum or minimum?

Question 51

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem.

A)
B)
C)
D)
E)

Question 52

Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. ,

A)
B)
C)
D)
E)None of these

Question 53

The function satisfies the hypotheses of Rolle's Theorem on the interval . Find all values of c that satisfy the conclusion of the theorem.

A)-6, -4
B)-5, -4
C)-6
D)-5

Question 54

At 4:00 P.M. a car's speedometer reads 25 . At 4:15 it reads 72 . At some time between 4:00 and 4:15 the acceleration is exactly x . Find x.

Question 55

Find the critical number(s), if any of the function

A)0,
B)
C)
D)0,

Question 56

Find a cubic function that has a local maximum value of at 1 and a local minimum value of -1,184 at 7.

Question 57

Find the critical number(s), if any, of the function .

A)
B)
C)None of these
D)

Question 58

Find the critical number(s), if any, of the function

A)
B)
C)-4
D)

Question 59

The function satisfies the hypotheses of Rolle's Theorem on the interval . Find all values of c that satisfy the conclusion of the theorem.

A)
B)
C)
D)

Question 60

Find the critical number(s), if any, of the function .

A) and

B) and

C)
D)

Question 61

Find the absolute minimum value(s) of on the interval [0, 6].

Question 62

Sketch the graph of on and find its absolute maximum and absolute minimum values, if any.

Question 63

Find the absolute maximum and absolute minimum values, if any, of the function on

Question 64

Find the maximum and minimum points of the function.

Question 65

Find the critical numbers of the function.

Question 66

Find all the critical numbers of the function.

Question 67

The graph below is the graph of function f on the interval Find the absolute maximum and absolute minimum values of f (if they exist) and where they are attained.

Question 68

Find the absolute maximum and absolute minimum values, if any, of the function on .