Deck 4: Graphing and Optimization

Question 1

Provide an appropriate response.
Identify the intervals where f‛(x) > 0.

Answer 1

(b, d), (g, h)

Question 2

Provide an appropriate response.
Use the first derivative test to determine the local extrema, if any, for the function:

A) local min at x = 0 and local max at x = -1 and x = 1
B) local max at x = 0 and local min at x = -1 and x = 1
C) local max at x = 1 and local min at x = 0
D) local max at x = -1 and local min at x = 0 and x = 1

Answer 2

local max at x = 0 and local min at x = -1 and x = 1

Question 3

Provide an appropriate response.
Identify the intervals where f(x) is decreasing.

Answer 3

(a, b), (d, g)

Question 4

Provide an appropriate response.
Find the critical values and determine the intervals where f(x) is increasing and f(x) is decreasing if

A) decreasing on
B)
C)
D)

Question 5

Provide an appropriate response.
Write the sign chart that corresponds to the following graph of f(x)::

Question 6

Provide an appropriate response.
Find the critical values and determine the intervals where f(x) is decreasing and the intervals where f(x) is increasing for

A)
B)
C)
D)

Question 7

Provide an appropriate response.
Find the critical values and determine the intervals where f(x) is decreasing for f(x) = 3(x - 4)2/3 + 6.

A)
B)
C)
D)

Question 8

Provide an appropriate response.
Use a graphing utility to approximate where the local extrema of the function are to two decimal places.

A)
B)
C)
D)

Question 9

Provide an appropriate response.

A)
B)
C)
D)

Question 10

Provide an appropriate response.
Determine the interval(s) where is decreasing.

A) (0, 6)
B) (0, 3) and (3, 6)
C)
D) (0, 3) and (6, ∞)

Question 11

Provide an appropriate response.
Write the sign chart that corresponds to the following graph of f(x):

Question 12

Provide an appropriate response.
Given find the values of x corresponding to local maxima and local minima.

A) local minimum at x = -4 (no local maximum)
B) local maximum at x = -4, local minimum at x = 4
C) no local maximum or minimum
D) local maximum at x = -4 (no local minimum)

Question 13

Provide an appropriate response.
The critical values of are x = -2 and x = 2. Use the first derivative test to determine which of the critical values correspond to a local maximum.

A) x = - 2
B) x = 2
C) x = 0 and x = 2
D) x = 2 and x = - 2

Question 14

Provide an appropriate response.
Use the first derivative test to determine the local extrema, if any, for the function:

A) local min at x = 2
B) local max at x = - 4 and local min at x = 2
C) local max at x = 2 and local min at - 4
D) local max at x = - 4

Question 15

Provide an appropriate response.
Find the critical values and determine the intervals where f(x) is increasing and the intervals where f(x) is decreasing for the function

A) decreasing on (-4, 2)
B) decreasing on (
C) decreasing on (-4, 2)
D) increasing on (-4, 2)

Question 16

Provide an appropriate response.
Use a graphing utility to approximate the intervals where f(x) is decreasing and intervals where f(x) is increasing for the function Round your answer to two decimal places.

A) decreasing on (-0.82, 0.62)
B) and (0.62, 2.45); increasing on (-0.82, 0.62) and (2.45, ∞)
C) ) and (0.62, 2.45); decreasing on (-0.82, 0.62) and (2.45, ∞)
D) increasing on (-0.82, 0.62)

Question 17

Provide an appropriate response.
The critical values of are x = -2 and x = 2. Use the first derivative test to determine which of the critical values correspond to a local minimum.

A) x = 2 and x = -2
B) neither x = 2 nor x = -2 correspond to a local minimum
C) x = -2
D) x = 2

Question 18

Provide an appropriate response.
Use the first derivative test to determine the local extrema, if any, for the function:

A) f(x) has a local maximum at x = 4.
B) f(x) has a local minimum at x = 4.
C) f(x) has no local extrema
D) f(x) has a local minimum at 6

Question 19

Provide an appropriate response.
Determine the intervals for which the function is decreasing.

A)
B) (0, 12) and (12, ∞)
C) (-12, 0)
D)

Question 20

Provide an appropriate response.

A)
B)
C)
D)

Question 21

Solve the problem.
The percent of concentration of a certain drug in the bloodstream x hr after the drug is administered is given by How long after the drug has been administered is the concentration a maximum? Round answer to the nearest tenth, if necessary.

A) 7 hr
B) 2 hr
C) 2.5 hr
D) 4.9 hr

Question 22

Solve the problem.
With x representing the water temperature in degrees Celsius, approximation to the number of salmon swimming upstream to spawn. Find the temperature that produces the maximum number of salmon.

A) 6°C
B) 20°C
C) 19°C
D) 5°C

Question 23

Find the intervals where f''(x) < 0 ir f''(x) > 0 as indicated.

A) (-5, 2)
B)
C) (-5, 5)
D)

Question 24

Sketch a graph of the function.

A)
B)
C)
D)

Question 25

Find the intervals where the function has the indicated concavity. Give the x coordinates of inflection points.
Concave upward

A)
B)
C)
D)

Question 26

Sketch a graph of the function.

A)
B)
C)
D)

Question 27

Find the intervals where the function has the indicated concavity. Give the x coordinates of inflection points.
Concave downward

A)
B)
C) (-1, 0); no inflection points
D)

Question 28

Sketch a graph of a single function that has these properties.
a) Continuous and differentiable for all real numbers
b)
c)
d)
e)
f)
g) An inflection point at (0,0)

Question 29

Sketch a graph of the function.

A)
B)
C)
D)

Question 30

Sketch a graph of the function.

A)
B)
C)
D)

Question 31

Sketch a graph of the function.

A)
B)
C)
D)

Question 32

Solve the problem.
The cost of manufacturing x electric woks in one day is given by C(x) = 2x3 -16x2 + 4x. Find the average cost per electric wok and the interval where the average cost per electric wok is decreasing.

A)
B)
C)
D)

Question 33

Sketch a graph of a single function that has these properties.
Find

A)
B)
C)
D)

Question 34

Find the intervals where f''(x) < 0 ir f''(x) > 0 as indicated.

A)
B)
C)
D) (-1, 0)

Question 35

Solve the problem.
The annual revenue and cost functions for a manufacturer of grandfather clocks are approximately where x denotes the number of clocks made. What is the maximum annual profit?

A) $553,333
B) $753,333
C) $653,333
D) $853,333

Question 36

Sketch a graph of a single function that has these properties.
a) Continuous and differentiable for all real numbers
b)
c) ) and ( -1 , 2)
d)
e)
f)
g)

Question 37

Find the intervals where f''(x) < 0 ir f''(x) > 0 as indicated.

A) (-3, 3)
B)
C)
D) (0, 3)

Question 38

Find the intervals where the function has the indicated concavity. Give the x coordinates of inflection points.
Concave downward

A)
B) (-5, 2); x = 0
C) (-5, 5); no inflection points
D)

Question 39

Solve the problem.
The Olympic flame at the 1992 Summer Olympics was lit by a flaming arrow. As the arrow moved d feet horizontally from the archer, assume that its height h(d), in feet, was approximated by the function Find the relative maximum of the function.

A) (0, 6.9)
B) (175, 61.25)
C) (350, 129.4)
D) (175, 68.15)

Question 40

Find the intervals where the function has the indicated concavity. Give the x coordinates of inflection points.
Concave upward

A)
B)
C)
D)

Question 41

Sketch a graph of a single function that has these properties.
Find

A)
B)
C)
D)

Question 42

Sketch a graph of a single function that has these properties.
Determine the interval(s) over which is concave downward.

A)
B)
C)
D)

Question 43

Sketch a graph of a single function that has these properties.
Find

A)
B)
C)
D)

Question 44

Sketch a graph of a single function that has these properties.
Find all inflection points for

A)
B) Inflection points at x = 1, x = 4
C) Inflection points at x = 0, x = 1, x = 4
D) This function does not have any inflection points.

Question 45

Decide if the given value of x is a critical number for f, and if so, decide whether the point for x on f is a local minimum,
local maximum, or neither.

A) Critical number; minimum at (-4 , 0)
B) Critical number; maximum at (-4 , 0)
C) Not a critical number.
D) Critical number but not an extreme point.

Question 46

Sketch the graph and show all local extrema and inflection points.

A)
B)
C)
D)

Question 47

Sketch a graph of a single function that has these properties.
Find

A)
B)
C)
D)

Question 48

Sketch the graph and show all local extrema and inflection points.

A)
B)
C)
D)

Question 49

Solve the problem.
A backpack manufacturer is planning to expand its work force. They estimate that the number of backpacks
produced by hiring new workers is given by Determine when the rate of
backpacks is increasing and when it is decreasing. Determine the point of diminishing returns and the
maximum rate of change of backpack production.

Question 50

Sketch a graph of a single function that has these properties.
Find

A)
B)
C)
D)

Question 51

Sketch a graph of a single function that has these properties.
Determine the interval(s) over which is concave upward.

A)
B)
C)
D)

Question 52

Sketch a graph of a single function that has these properties.
Find the inflection point(s) for

A)
B) (0, 6) and (2, 2)
C) (0, 0)
D) (0, 0) and (2, 2)

Question 53

Sketch the graph and show all local extrema and inflection points.

A)
B)
C)
D)

Question 54

Sketch a graph of a single function that has these properties.
Find

A)
B)
C)
D)

Question 55

Sketch a graph of a single function that has these properties.
Find the inflection point(s) for

A) (0, -6)
B) (1, -1)
C) (-1, 6)
D) (0, -1)

Question 56

Sketch a graph of a single function that has these properties.
Find

A)
B)
C)
D)

Question 57

Solve the problem.
A company estimates that it will sell N(t) hair dryers after spending $t thousands on advertising as given by: For which values of t is the rate of sales N'(t) in creasing?

A) 40 < t < 50
B) 50 < t < 60
C) 40 < t < 60
D) t > 40

Question 58

Sketch the graph and show all local extrema and inflection points.

A)
B)
C)
D)

Question 59

Sketch a graph of a single function that has these properties.
Find

A)
B)
C)
D)

Question 60

Sketch a graph of a single function that has these properties.
Find the inflection point(s) for

A) (-3, 2)
B) (-6, 1)
C) (-7, 0)
D) There are no points of inflection.

Question 61

Find the limit, if it exists.
Find

A)
B) 0
C)
D) 1

Question 62

Sketch a graph of a single function that has these properties.

A) Domain: All real numbers except -9; y intercept: 0; x intercept: 0
B) y intercept: 0; x intercept: 0
C) Domain: All real numbers except 9; y intercept: 0; x intercept: 0
D) Domain: All real numbers except -9; y intercept: 0; no x intercept

Question 63

Sketch a graph of the function.

A)
B)
C)
D)

Question 64

Find the limit, if it exists.

A) -3
B)
C) 3
D)

Question 65

Find the limit, if it exists.
Find

A)
B)
C) 0
D)

Question 66

Sketch a graph of a single function that has these properties.
a) Continuous for all real numbers
b) Differentiable everywhere except x = 0
c) d) e) f) f(-2) = f (2) = 5
g) y-intercept and x-intercept at (0, 0)

Question 67

Sketch a graph of a single function that has these properties.

A) y intercept: 8; x intercept: -64
B) y intercept: -64; x intercept: 8
C) y intercept: 8; x intercept: -64
D) y intercept: 8; x intercept: -64

Question 68

Solve the problem.
Because of material shortages, it is increasingly expensive to produce 6.2L diesel engines. In fact, the profit in millions of dollars from producing x hundred thousand engines is approximated by Find the inflection point of this function to determine the point of
Diminishing returns.

A) (7.50, 2048)
B) (10, 1958)
C) (10, 2048)
D) (10, 114.67)

Question 69

Find the limit, if it exists.
Find:

A)
B)
C)
D)

Question 70

Find the limit, if it exists.

A) -6
B) 0
C)
D)

Question 71

Provide an appropriate response.
Find horizontal asymptotes, if any, for

A)
B) y = 0
C)
D)

Question 72

Find the limit, if it exists.
Find

A)
B) 0
C) 2
D)

Question 73

Find the limit, if it exists.
Find

A) 0
B)
C)
D)

Question 74

Find the limit, if it exists.
Find .

A) 3
B) 108
C) 0
D) 81

Question 75

Provide an appropriate response.
Sketch the graph of Include sketch of all asymptotes.

Question 76

Provide an appropriate response.
Consider the function Determine the intervals where f(x) is increasing and decreasing,
concave up and concave down and all local extrema. Use that information to obtain a sketch of the function.

Question 77

Provide an appropriate response.
Find vertical asymptotes for

A) x = 1, x = -4
B) x = -1, x = -4
C) x = 1, x = 4
D) x = -1, x = 4

Question 78

Graph the function and locate intervals on which the function is increasing or decreasing, open intervals on which the
function is concave up or concave down, and all inflection points.

A)
B)
C)
D)

Question 79

Sketch a graph of a single function that has these properties.
f(x) = 4x + 8

A) y intercept: 8; x intercept: -2
B) y intercept: -2; x intercept: 8
C) y intercept: 8; x intercept: -2
D) y intercept: 8; x intercept: 2

Question 80

Graph the function and locate intervals on which the function is increasing or decreasing, open intervals on which the
function is concave up or concave down, and all inflection points.

A)
B)
C)
D)