Deck 2: Functions, Limits and the Derivative

Question 1

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point.

A)
B)
C)
D)

Answer 1

Question 2

The position of car A and car B, starting out side by side and traveling along a straight road, is given by and , respectively, where s is measured in feet and t is measured in seconds (see the accompanying figure). Which car is traveling faster at ?

A) Car A is traveling faster at , car A is traveling faster at
, cars are traveling at the same speed at

B) Cars are traveling at the same speed at , car A is traveling faster at
, car B is traveling faster at

C) Car B is traveling faster at , car A is traveling faster at
, car A is traveling faster at

D) Car B is traveling faster at , cars are traveling at the same speed at
, car A is traveling faster at

Answer 2

Car B is traveling faster at , cars are traveling at the same speed at
, car A is traveling faster at

Question 3

Find the slope of the tangent line to the graph of each function at the given point and determine an equation of the tangent line.

A)
B)
C)
D)

Answer 3

Question 4

In the following figure, gives the population of a certain bacteria culture at time t after a portion of bactericide A was introduced into the population at . The graph of g gives the population of a similar bacteria culture at time t after a portion of bactericide B was introduced into the population at . Which population is decreasing faster at and at ?

A) The populations are decreasing at the same rate at ,
is decreasing faster at

B) is decreasing faster at
,
is decreasing faster at

C) is decreasing faster at
, the populations are decreasing at the same rate at

D) is decreasing faster at
,
is decreasing faster at

Question 5

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point.

A)
B) 6
C)
D) 7
E)

Question 6

Under a set of controlled laboratory conditions, the size of the population of a certain bacteria culture at time t (in minutes) is described by the function Find the rate of population growth at min.

A) 148 bacteria per minute
B) 113 bacteria per minute
C) 125 bacteria per minute
D) 120 bacteria per minute

Question 7

Let Find the equation of the tangent line to the curve at the point .

A)
B)
C)
D)

Question 8

Find the slope of the tangent line to the graph of each function at the given point and determine an equation of the tangent line.

A)
B)
C)
D)

Question 9

Let . Find the derivative of and the point on the graph of where the tangent line to the curve is horizontal.

A)
B)
C)
D)

Question 10

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point.

A)
B)
C)
D)

Question 11

The gross domestic product (GDP) of a certain country is projected to be billion dollars t yr from now. What will be the rate of change of the country's GDP 5 yr from now?

A) $15 billion/yr
B) $13 billion/yr
C) $7 billion/yr
D) $12 billion/yr

Question 12

Under a set of controlled laboratory conditions, the size of the population of a certain bacteria culture at time t (in minutes) is described by the function Find the rate of the population growth at t = 30 min.

A) 3 bacteria per minute
B) 30 bacteria per minute
C) 180 bacteria per minute
D) 2 bacteria per minute
E) 123 bacteria per minute

Question 13

Let . Find the derivative of and the point on the graph of where the tangent line to the curve is horizontal.

A) ; (
)
B) ; (
)
C) ; (
)
D) ; (
)
E) ; (
)

Question 14

Let . Find the average rate of change of y with respect to x in the interval from to , from to , from to , and the (instantaneous) rate of change of y at .

A)
B)
C)
D)

Question 15

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. f(x) = 7

A) 0
B) -7x
C) 7x
D) 7

Question 16

At a temperature of , the volume V (in liters) of 1.33 g of is related to its pressure p (in atmospheres) by the formula . What is the average rate of change of V with respect to p as p increases from to ? What is the rate of change of V with respect to p when

A) average rate of change: liter/atmosphere, instantaneous rate of change:
liter/atmosphere
B) average rate of change: liter/atmosphere, instantaneous rate of change:
liter/atmosphere
C) average rate of change: liter/atmosphere, instantaneous rate of change:
liter/atmosphere
D) average rate of change: liter/atmosphere, instantaneous rate of change:
liter/atmosphere

Question 17

Find the slope of the tangent line to the graph of each function at the given point and determine an equation of the tangent line.

A)
B)
C)
D)

Question 18

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point.

A)
B)
C)
D)

Question 19

The following graph shows the volume of wood produced in a single-species forest. Here is measured in cubic meters/hectare and t is measured in years. By computing the slopes of the respective tangent lines, estimate the rate at which the wood grown is changing at the beginning of year 10 and at the beginning of year 35.

A) cubic meters/hectare per year in the year 10,






cubic meters/hectare per year in the year 35
B) cubic meters/hectare per year in the year 10,






cubic meters/hectare per year in the year 35
C) cubic meters/hectare per year in the year 10,






cubic meters/hectare per year in the year 35
D) cubic meters/hectare per year in the year 10,






cubic meters/hectare per year in the year 35

Question 20

During the construction of a high-rise building, a worker accidentally dropped his portable electric screwdriver from a height of 144 ft. After t sec, the screwdriver had fallen a distance of ft. What was the velocity of the screwdriver at the time it hit the ground?

A) 12 ft/sec
B) 96 ft/sec
C) 768 ft/sec
D) 48 ft/sec

Question 21

Find the indicated one-sided limit.

A)
B)
C)
D)
E)

Question 22

Find the values of x for which the function is continuous.

A)
B)
C)
D)
E)

Question 23

During the construction of a high-rise building, a worker accidentally dropped his portable electric screwdriver from a height of 400 ft. After t sec, the screwdriver had fallen a distance of ft.
How long did it take the screwdriver to reach the ground?
__________ sec
What was the average velocity of the screwdriver between the time it was dropped and the time it hit the ground?
__________ ft/sec
What was the velocity of the screwdriver at the time it hit the ground?
__________ ft/sec

Question 24

Productivity and yield of cultivated crops are often reduced by insect pests. The following graph shows the relationship between the yield of a certain crop, , as a function of the density of aphids x. (Aphids are small insects that suck plant juices.) Here, is measured in kilograms/ 4,000 square meters, and x is measured in hundreds of aphids/bean stem. By computing the slopes of the respective tangent lines, estimate the rate of change of the crop yield with respect to the density of aphids when that density is 200 aphids/bean stem and when it is 800 aphids/bean stem.

Question 25

Let .
Find the derivative of . __________
Find the point on the graph of f where the tangent line to the curve is horizontal.
__________
Sketch the graph of f and the tangent line to the curve at the point where the tangent line to the curve is horizontal.
Select the correct graph by entering A, B, or
C.
__________

Question 26

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. __________

Question 27

Under a set of controlled laboratory conditions, the size of the population of a certain bacteria culture at time t (in minutes) is described by the function Find the rate of population growth at min.
rate = __________

Question 28

Find the indicated one-sided limit.

A)
B)
C)
D)
E)

Question 29

Let .
Find the average rate of change of y with respect to x in the interval from to .
__________
Find the average rate of change of y with respect to x in the interval from to .
__________
Find the average rate of change of y with respect to x in the interval from to .
__________
Find the (instantaneous) rate of change of y at .
__________

Question 30

Let Find the value of k that will make f continuous on .

A)
B)
C)
D)
E)

Question 31

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. __________

Question 32

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. __________

Question 33

Determine all values of x at which the function is discontinuous.

A) - 5
B) 0, 5
C) 0, - 5
D) - 3
E) 3

Question 34

In the following figure, gives the population of a certain bacteria culture at time t after a portion of bactericide A was introduced into the population at . The graph of g gives the population of a similar bacteria culture at time t after a portion of bactericide B was introduced into the population at . Which population is decreasing faster at ?
Population __________
Which population is decreasing faster at ?
Population __________
Which bactericide is more effective in reducing the population of bacteria in the short run?
Bactericide __________
Which bactericide is more effective in reducing the population of bacteria in the long run?
Bactericide __________

Question 35

The position of car A and car B, starting out side by side and traveling along a straight road, is given by and , respectively, where s is measured in feet and t is measured in seconds (see the accompanying figure). Which car is traveling faster at ?
Car __________
Which car is traveling faster at ?
Hint: Compare tangent lines.
Car __________
Which car is traveling faster at ?
Car __________

Question 36

The gross domestic product (GDP) of a certain country is projected to be billion dollars t yr from now.
What will be the rate of change of the country's GDP 3 yr from now?
$ __________ billion/yr
What will be the rate of change of the country's GDP 5 yr from now?
$ __________ billion/yr

Question 37

At a temperature of , the volume V (in liters) of 1.33 g of is related to its pressure p (in atmospheres) by the formula .
What is the average rate of change of V with respect to p as p increases from to ? What is the rate of change of V with respect to p when

Question 38

Let Find the derivative of . __________
Find the equation of the tangent line to the curve at the point .
__________
Select the correct graph of by answering A, B, or
C.
__________

Question 39

The graph shows the volume of wood produced in a single-species forest. Here is measured in cubic meters/hectare and t is measured in years. By computing the slopes of the respective tangent lines, estimate the rate at which the wood grown is changing at the beginning of year 10 and at the beginning of year 40.

Question 40

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point.
f(x) = -10 = __________

Question 41

Find the values of x for which the function is continuous.

A)
B)
C)
D)

Question 42

Find the values of x for which the function is continuous.

A)
B)
C)
D)

Question 43

Use the graph of the function f to find and if the limit exists.

A)
B)
C)
D)

Question 44

Determine all values of x at which the function is discontinuous.

A) 4, 5, and 6
B) 4 and 5
C) 0 and 6
D) 0
E) 6

Question 45

Find and , where .

A)
B)
C)
D)
E)

Question 46

A function f is continuous at the point if the following conditions are satisfied. 1. is defined.
2) exists.
3) .
Determine the values of x, if any, at which the function is discontinuous. At each point of discontinuity, state the condition(s) for continuity that are violated.

A)
B)
C)
D)
E) Continuous everywhere

Question 47

Use the graph of the function f to find and if the limit exists.

A)
B)
C)
D)

Question 48

A function f is continuous at the point if the following conditions are satisfied. 1. is defined.
2) exists.
3) .
Determine the values of x, if any, at which the function is discontinuous. At each point of discontinuity, state the condition(s) for continuity that are violated.

A)
B)
C)
D)
E) Continuous everywhere.

Question 49

Find the one-sided limit, if it exists.

A)
B)
C)
D)
E) The limit does not exist

Question 50

Find and , where

A)
B)
C)
D)
E) The limits do not exist

Question 51

The base salary of a salesman working on commission is $13,000. For each $50,000 of sales beyond $100,000 he is paid a $1,000 commission. Sketch a graph showing his earnings as a function of the level of his sales x.

A)
B)
C)

Question 52

Find the indicated one-sided limit.

A) 81
B) 0
C) -18
D)
E)

Question 53

Let . Find the zeros of f in by solving the equation .

A)
B)
C)
D)
E)

Question 54

Refer to the graph of the function f and determine whether the statement below is true or false.

A) true
B) false

Question 55

Determine all values of x at which the function is discontinuous.

A)
B)
C)
D)

Question 56

Determine all values of x at which the function is discontinuous.

A)
B)
C)
D)

Question 57

The function P, whose graph follows, gives the prime rate (the interest rate banks charge their best corporate customers) as a function of time for the first 32 wk during a year. Determine the values of t for which P is discontinuous.

A)
B)
C)
D)

Question 58

Find the one-sided limit, if it exists.

A)
B)
C)
D)
E) The limit does not exist

Question 59

Find the one-sided limit, if it exists.

A)
B)
C)
D)
E) The limit does not exist

Question 60

The function that gives the cost of a certain commodity is defined by where x is the number of pounds of a certain commodity sold and C(x) is measured in dollars. Sketch the graph of the function C. Determine the values of x for which the function C is discontinuous.

A)

B)

C)

Question 61

Find the values of x for which the function is continuous.

Question 62

Let Find the value of k that will make f continuous on . __________

Question 63

Find the one-sided limit.

Question 64

Find the one-sided limit.

Question 65

Find the one-sided limit, if it exists. If it does not, answer none.

Question 66

Let Find the value of k that will make f continuous on . __________

Question 67

Use the method of bisection to find the root of the equation , accurate to two decimal places if necessary. __________

Question 68

Use the intermediate value theorem to find the value of c such that .

A)
B)
C)
D)

Question 69

Find the values of x for which the function is continuous.

Question 70

Find the indicated one-sided limit.

Question 71

Find the indicated one-sided limit.

Question 72

Use the method of bisection to find the root of the equation , accurate to two decimal places.

A)
B)
C)
D)

Question 73

Use the graph of the function f to find and if the limit exists. If it does not, answer none. __________ __________ __________

Question 74

A function f is continuous at the point if the following conditions are satisfied.
1. is defined.
2. exists.
3. .
Determine the value of x, if any, at which the function is discontinuous. __________
State the condition(s) for continuity that are violated.
Conditions: __________

Question 75

Find the values of x for which the function is continuous.

Question 76

A function f is continuous at the point if the following conditions are satisfied.
1. is defined.
2. exists.
3. .
Determine the value of x, if any, at which the function is discontinuous. __________
State the condition(s) for continuity that are violated.
Conditions: __________

Question 77

Use the intermediate value theorem to find the value of c such that . __________

Question 78

Find and , where __________ __________

Question 79

Use the graph of the function f to find and if the limit exists. If it does not, answer none. __________ __________ __________

Question 80

Let Find the value of k that will make f continuous on .

A)
B)
C)
D)