Deck 4: Applications of the Derivative

Question 1

Using Newton's method with , determine the third iteration when approximating the root to at least six-digit accuracy of the following equation. x³ + x² - 3x + 5 = 0

A) -2.777778
B) -2.751101
C) -2.751145
D) -2.751446

Answer 1

-2.751101

Question 2

A certain company estimates that it can sell f(x)-thousand video game consoles at the price of $x as given in the table. Use a linear approximation to estimate the number of consoles that can be sold at $62.

A) approximately 106 thousand consoles
B) approximately 109 thousand consoles
C) approximately 131 thousand consoles
D) approximately 139 thousand consoles

Answer 2

B

Question 3

Find the limit.

A) 0
B)
C)
D) limit does not exist

Answer 3

C

Question 4

Use a linear approximation to estimate Note that 16.9 is near 16 which is

A) 2.92755
B) 2.02755
C) 2.02812
D) 3.00155

Question 5

Using Newton's method with , determine the second iteration when approximating the root to at least six-digit accuracy of the following equation.

A) 0.885396
B) 0.783253
C) 0.784946
D) 0.783163

Question 6

Find the linear approximation, L(x), to f(x) at x = x₀. f(x) = e^7x, x₀ = 0

A) L(x) = 7x
B) L(x) = -7x + 1
C) L(x) = 7ex - 1
D) L(x) = 7x + 1

Question 7

Use Newton's method with x₀ = 2.3 to compute x₁ and x₂ without the use of a calculator.
f(x) = x³ - 3x - 5

Question 8

Explain why Newton's method fails for the following equation with the specified initial guess.

A) Using we see that f is not differentiable at
B) Using we obtain .
C) Using we obtain which will give
D) Using we see that f does not have any real roots.

Question 9

Given the graph of f(x), draw in the tangent lines used in Newton's method to determine x₁ and x₂ after starting at x₀ = 1.

Question 10

The impedance Z of a simple RL (resistive inductive) circuit is given by , where R is the resistance of the resistor, L is the inductance of the inductor, and $\omega$ is the angular frequency of the applied voltage. Find a linear approximation of Z(L) for small values of L assuming all other quantities are constant. Show all of your work.

Question 11

Using Newton's method, approximate the root of the following equation to at least six-digit accuracy.

A) 0.817880
B) 0.839580
C) 0.807714
D) 0.827880

Question 12

Use a linear approximation to estimate

A) 0
B) 0.23
C) 0.15
D) 0.07

Question 13

Find the linear approximation at x = 0 to show that the following commonly used approximation is valid for "small" x.

Question 14

Use Newton's method to find an approximate root of the equation (accurate to six decimal places).

A) 0.186548
B) 0.136548
C) 0.056548
D) 0.126548

Question 15

Find the linear approximation, L(x), to f(x) at x = x₀.

A)
B)
C)
D)

Question 16

Use linear approximation of at to estimate the quantity of to the nearest ten thousandth.

A)
B)
C)
D)

Question 17

Approximate accurate to 3 decimal places using Newton's method. State the function used.

A) 2.714 using
B) 2.814 using
C) 2.664 using
D) 2.731 using

Question 18

Find the linear approximation at x = 0 to show that the following commonly used approximation is valid for "small" x.

Question 19

Using Newton's method, approximate the root of the following equation to at least six-digit accuracy. x³ + x² - 5x + 5 = 0

A) -3.125000
B) -3.141392
C) -3.117942
D) -3.123872

Question 20

Find the linear approximation, L(x), to f(x) at x = x₀.

A)
B)
C)
D)

Question 21

Find the limit.

A) 0
B)
C)
D) limit does not exist

Question 22

Find the error in the following incorrect calculation.

Question 23

Given the graph of , locate the absolute extrema (if they exist) on the interval .

A) absolute max:
Absolute min:
B) absolute min:
C) absolute max:
Absolute min:
D) no absolute extrema

Question 24

Find the limit.

A) 1
B) 3
C)
D)

Question 25

Find the limit.

A)
B)
C) 0
D) limit does not exist

Question 26

Determine which function "dominates", where we say that the function f(x) dominates the function g(x) as if and either Show all work.

Question 27

Given the graph of , locate the absolute extrema (if they exist) on the interval .

A) absolute max: approximately
Absolute min:
B) absolute max:
Absolute min:
C) absolute max:
Absolute min:
D) no absolute extrema

Question 28

Find the limit.

A) 0
B) 1
C) -3
D) 3

Question 29

Find the limit.

A) 1
B)
C)
D)

Question 30

Find the limit.

A) 0
B) 1
C)
D)

Question 31

Given the graph of , locate the absolute extrema (if they exist) on the interval .

A) absolute max:
B) absolute min: and
C) absolute min:
D) no absolute extrema

Question 32

Find the limit.

A) 0
B)
C)
D) does not exist

Question 33

Suppose the size of a particular animal's pupils expand and contract depending on the amount of light available. Let be the size in mm of the pupils at light intensity x. Find the smallest possible size of the pupils by finding

A) 1 mm
B) 5 mm
C) 30 mm
D) mm

Question 34

Given the graph of , locate the absolute extrema (if they exist) on the interval .

A) absolute max:
B) absolute min:
C) absolute min:
D) no absolute extrema

Question 35

Find the error in the following incorrect calculation.

Question 36

Given the graph of , locate the absolute extrema (if they exist) on the interval .

A) absolute max:
B) absolute min: and
C) absolute min:
D) no absolute extrema

Question 37

Given the graph of , locate the absolute extrema (if they exist) on the interval .

A) absolute max:
Absolute min:
B) absolute max:
Absolute min:
C) absolute max:
Absolute min:
D) no absolute extrema

Question 38

Find the limit.

A) 0
B) 1
C) 10
D) -10

Question 39

Given the graph of , locate the absolute extrema (if they exist) on the interval .

A) absolute max:
Absolute min:
B) absolute min:
C) absolute max:
Absolute min:
D) no absolute extrema

Question 40

Find the indicated limit.

A) 0
B) 1
C)
D) limit does not exist

Question 41

Find the absolute extrema of the given function on the indicated interval. on

A) absolute maximum: , absolute minimum:
B) absolute maximum: , absolute minimum:
C) absolute maximum: , absolute minimum:
D) absolute maximum: , absolute minimum:

Question 42

Given , on the interval , determine the critical number(s).

A)
B)
C)
D)

Question 43

Find all critical numbers. Then use a graph to determine whether the critical numbers represent a local maximum, local minimum, or neither.

A) (local maximum), (neither), (local minimum)
B) (local maximum), (local maximum), (local minimum)
C) (local maximum), (local minimum)
D) (local minimum), (local maximum)

Question 44

Given , determine the absolute extrema on the interval .

A) absolute max: ; absolute min:
B) absolute max: and ; absolute min: and
C) absolute max: ; absolute min:
D) absolute max: ; absolute min:

Question 45

Using a calculator or computer, estimate the absolute extrema of on the interval . Round answer to the nearest hundredth.

A) absolute max: ; absolute min:
B) absolute max: ; absolute min:
C) absolute max: ; absolute min:
D) absolute max: ; absolute min:

Question 46

Find all critical numbers. Then use the graph to determine whether the critical numbers represent a local maximum, local minimum, or neither.

A) , local maximum
B) , local minimum
C) , local minimum
D) , neither

Question 47

Given the graph of , locate the absolute extrema (if they exist) on the interval .

A) absolute max: approximately
B) absolute max: approximately
Absolute min:
C) absolute max: approximately
Absolute min:
Absolute min:
D) no absolute extrema

Question 48

Given the graph of , locate the absolute extrema (if they exist) on the interval .

A) absolute max: approximately
B) absolute max:
Absolute min:
C) absolute max:
Absolute min:
D) no absolute extrema

Question 49

Using a calculator or computer, estimate the absolute extrema of , on the interval .

A) absolute max: ; absolute min:
B) absolute max: ; absolute min:
C) absolute max: ; absolute min:
D) absolute max: ; absolute min:

Question 50

Given , on the interval , determine if the critical number represents a local maximum, local minimum or neither.

A) local maximum
B) local minimum
C) neither

Question 51

Determine all critical numbers of .

A)
B)
C)
D)

Question 52

Find all critical numbers.

A)
B) ,
C)
D)

Question 53

Given , determine the critical number(s).

A)
B)
C)
D)

Question 54

Given determine if the critical number represents a local maximum, local minimum or neither.

A) local maximum
B) local minimum
C) neither

Question 55

Given determine the critical number(s).

A)
B)
C)
D)

Question 56

Using a calculator or computer, estimate the absolute extrema of on the interval . Round answer to the nearest hundredth.

A) absolute max: ; absolute min:
B) absolute max: ; absolute min:
C) absolute max: ; absolute min:
D) absolute max: ; absolute min:

Question 57

Given determine if the critical number represents a local maximum, local minimum or neither.

A) local maximum
B) local minimum
C) neither

Question 58

Determine all critical numbers of .

A)
B)
C)
D)

Question 59

Find the absolute extrema of the given function on the indicated interval. on

A) absolute maximum: , absolute minimum:
B) absolute maximum: , absolute minimum:
C) absolute maximum: , absolute minimum:
D) absolute maximum: , absolute minimum:

Question 60

Find the absolute extrema of the given function on the indicated interval. on

A) absolute maxima: , absolute minimum:
B) absolute maximum: , absolute minimum:
C) absolute maximum: , absolute minimum:
D) absolute maximum: , absolute minimum:

Question 61

Determine, by hand, the interval(s) where is increasing and/or decreasing.

A)
B) increasing ; decreasing
C) increasing ; decreasing
D) increasing ; decreasing

Question 62

Determine, by hand, the interval(s) where is increasing and/or decreasing.

A) increasing and ; decreasing
B) increasing ; decreasing
C) increasing and ; decreasing and
D) increasing ; decreasing and

Question 63

Using a calculator or computer determine all critical numbers of and use the First Derivative Test to classify each as a local minimum, local maximum or neither.

A) local maximum: none
Local minimum: none
Neither at
B) local maximum at
Local minimum at
Neither at
C) local maximum at
Local minimum at
Neither at
D) local maximum: none
Local minimum at
Neither at and

Question 64

A herd of sixty antelope is released onto a small game reserve so that their reproductive habits can be studied. In the beginning the population of the herd increases rapidly in size but eventually slows due to a dwindling food supply. Suppose the population of antelope after t years is given by the function where . Determine when the population begins to decline.

A) 14 years
B) 16 years
C) 17 years
D) 18 years

Question 65

Determine the intervals where is concave up and concave down. Round answers to nearest hundredth.

A) concave up for
Concave down for
B) concave down for
Concave up for
C) concave up for
D) concave down for
Concave up for

Question 66

Estimate critical numbers and sketch graphs showing both global and local behavior.

Question 67

Find all critical numbers and use the First Derivative Test to classify each as a local maximum, local minimum, or neither.

A) critical point: (neither)
B) critical points: (local minimum), (local maximum)
C) critical points: (local minimum), (neither), (local maximum)
D) critical points: (local maximum), (local minimum), (local maximum)

Question 68

Sketch a graph of a function with the following properties. if if

Question 69

Find the x-coordinates of all extrema and sketch the graph of showing global and local behavior. Round answer to nearest thousandth.

Question 70

Find the intervals where the function is increasing and decreasing. Use this information to determine all local extrema.

A) decreasing: no extrema
B) increasing: no extrema
C) decreasing: increasing: local minimum:
D) decreasing: increasing: local minimum:

Question 71

Find the x-coordinates of all extrema and sketch the graph of showing global and local behavior.

Question 72

Find the intervals where the function is increasing and decreasing on the specified interval. Use this information to determine all local extrema. on

Question 73

Determine, by hand, all critical numbers of and use the First Derivative Test to classify each as a local minimum, local maximum or neither.

A) local maximum at
Local minimum at
Neither:
B) local maximum at
Local minimum at
Neither:
C) local maximum at
Local minimum at
Neither:
D) local maximum at
Local minimum at
Neither:

Question 74

Determine, by hand, the interval(s) where is increasing and/or decreasing.

A) increasing ; decreasing
B) increasing and ; decreasing and
C) increasing and ; decreasing and
D) increasing and ; decreasing

Question 75

Find all critical numbers and use the First Derivative Test to classify each as a local maximum, local minimum, or neither.

A) critical point: (local maximum)
B) critical point: (local minimum)
C) critical points: (local minimum), (local maximum)
D) critical points: (local minimum), (neither), (local maximum)

Question 76

Find the intervals where the function is increasing and decreasing. Use this information to determine all local extrema.

A) decreasing: increasing: local maximum
B) decreasing: increasing: local minimum
C) decreasing: no extrema
D) increasing: no extrema

Question 77

Find all asymptotes and extrema and sketch a graph.

Question 78

Find all asymptotes and extrema and sketch a graph.

Question 79

Determine, by hand, all critical numbers of and use the First Derivative Test to classify each as a local minimum, local maximum or neither.

A) local maximum at
Local minimum at
Neither at
B) local maximum at
Local minimum at
Neither at
C) local maximum:
Local minimum:
Neither at
D) local maximum:
Local minimum:
Neither at

Question 80

Determine the intervals where is concave up and concave down.

A) concave down for
Concave up for
B) concave down for
Concave up for
C) concave down for
Cancave up for
D) concave down for and
Concave up for