Deck 2: Functions, Limits, and the Derivative

Question 1

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) ​

Answer 1

​

Question 2

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. ​ at ​

A)
B)
C)
D)

Answer 2

Question 3

The following graph shows the volume of wood produced in a single-species forest. Here f (t) 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. ​ ​

A) cubic meters/hectare per year in the year 10, cubic meters/hectare per year in the year 40
B) cubic meters/hectare per year in the year 10, cubic meters/hectare per year in the year 40
C) cubic meters/hectare per year in the year 10, cubic meters/hectare per year in the year 40
D) cubic meters/hectare per year in the year 10, cubic meters/hectare per year in the year 40

Answer 3

cubic meters/hectare per year in the year 10, cubic meters/hectare per year in the year 40

Question 4

The position of car A and car B, starting out side by side and traveling along a straight road, is given by s = f(t) and s = g(t), respectively, where s is measured in feet and t is measured in seconds (see the accompanying figure). ​ ​
Which car is traveling faster at t​₁, t​₂ and t​₃?
​

A) Cars are traveling at the same speed at t​₁, car B is traveling faster at t​₂, car A is traveling faster at t​₃
B) Car A is traveling faster at , car B is traveling faster at t​₂, cars are traveling at the same speed at t​₃
C) Car A is traveling faster at t​₁, car B is traveling faster at t​₂, car A is traveling faster at t​₃
D) Car A is traveling faster at t​₁, cars are traveling at the same speed at t​₂, car B is traveling faster at t​₃

Question 5

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) 67 bacteria per minute
B) 55 bacteria per minute
C) 72 bacteria per minute
D) 47 bacteria per minute

Question 6

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. ​ at (-2, -3)
​

A) ​ ;
B) ​ ;
C) ;
D) ​ ;

Question 7

In the following figure, f(t) gives the population P​₁ of a certain bacteria culture at time t after a portion of bactericide A was introduced into the population at t = 0. The graph of g gives the population P​₂ of a similar bacteria culture at time t after a portion of bactericide B was introduced into the population at t = 0. ​ ​
Which population is decreasing faster at t​₁ and at t​₂?
​

A) The populations are decreasing at the same rate at t​₁, P​₂ is decreasing faster at ​t​₂
B) ​P​₁ is decreasing faster at t​₁, the populations are decreasing at the same rate at t​₂
C) ​P​₂ is decreasing faster at t​₁, ​P​1 is decreasing faster at t​₂
D) P₁ is decreasing faster at t​₁, ​P​₂ is decreasing faster at t​₂

Question 8

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 9

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) 48 ft/sec
C) 768 ft/sec
D) 96 ft/sec

Question 10

Let ​ Find the equation of the tangent line to the curve at the point (2, 15).
​

A)
B) ​
C) ​
D) ​

Question 11

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, f(x), as a function of the density of aphids x. (Aphids are small insects that suck plant juices.) Here, f(x) 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. ​ ​

A) - 0.5 when it is 200 aphids/bean stem, - 1.333 when it is 800 aphids/bean stem
B) 00.5 when it is 200 aphids/bean stem, 01.333 when it is 800 aphids/bean stem
C) 02 when it is 200 aphids/bean stem, 00.75 when it is 800 aphids/bean stem
D) - 2 when it is 200 aphids/bean stem, - 0.75 when it is 800 aphids/bean stem

Question 12

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?
​

A) $10 billion/yr
B) $13 billion/yr
C) $15 billion/yr
D) $11 billion/yr

Question 13

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. ​ at ​

A)
B) ​
C)
D)

Question 14

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) 0

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. ​ ​
​

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

Question 16

​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 17

At a temperature of 20°C, the volume V (in liters) of 1.33 g of O₂ 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 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

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

Question 20

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

Question 21

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 22

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 23

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 1 yr from now?
​
$ __________ billion/yr
​
What will be the rate of change of the country's GDP 4 yr from now?
​
$ __________ billion/yr

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, f(x), as a function of the density of aphids x. (Aphids are small insects that suck plant juices.) Here, f(x) 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 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 26

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

Question 27

Find the indicated one-sided limit. ​ ​

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

Question 28

Let . ​
Find the zeros of f in [-1, 1] by solving the equation .
​

A)
B)
C)
D)
E)

Question 29

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

Question 30

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

Question 31

In the following figure, f(t) gives the population P​₁ of a certain bacteria culture at time t after a portion of bactericide A was introduced into the population at t = 0. The graph of g gives the population P​₂ of a similar bacteria culture at time t after a portion of bactericide B was introduced into the population at t = 0.
​ ​
Which population is decreasing faster at t​₁?
​
Population __________
​
Which population is decreasing faster at ​t​₂?
​
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 32

The graph shows the volume of wood produced in a single-species forest. Here f(t) 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.
​
​

Question 33

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

A)
B)
C)
D)
E)

Question 34

Let .
​
Find the derivative f '(x) of f(x).
​
f '(x) = __________
​
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.
​
__________
​
A​
​ ​
B C ​
Let represent the point on the graph of f where the tangent line to the curve is horizontal. What is the rate of change of f at this point?
​

Question 35

The position of car A and car B, starting out side by side and traveling along a straight road, is given by s = f(t) and s = g(t), respectively, where s is measured in feet and t is measured in seconds (see the accompanying figure).
​ ​
Which car is traveling faster at t​₁?
​
Car __________
​
Which car is traveling faster at t​₂?

Question 36

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

A) 0, 4
B) 0, -4
C) -5
D) 5
E) -4

Question 37

Find the indicated one-sided limit. ​ ​

A) 3
B) -3
C) 4
D) -4
E) 0

Question 38

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

A)
B)
C)
D)
E)

Question 39

At a temperature of 20°C, the volume V (in liters) of 1.33 g of O₂ 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 40

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

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

Question 41

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) ​ x = 20, 30, 40
B) ​ x = 50, 60
C) ​ x = 20, 30, 40

Question 42

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

A) false
B) true

Question 43

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) t = 11
B) t = 3, 2, 5
C) t = 8, 2, 5
D) t = 8, 16, 20

Question 44

Find the one-sided limit, if it exists. ​ ​

A) -36
B) -6
C) 81
D) 9
E) The limit does not exist

Question 45

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

A)
B)
C)
D)

Question 46

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

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

Question 47

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

A)
B)
C)
D)

Question 48

Find the indicated one-sided limit.
​

Question 49

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

A)
B)
C)
D)

Question 50

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) ; conditions 1, 2
B) ; conditions 1, 2
C) ; condition 3
D) ; condition 2
E) Continuous everywhere

Question 51

The base salary of a salesman working on commission is $11,000. For each $100,000 of sales beyond $100,000 he is paid a $2,000 commission. Sketch a graph showing his earnings as a function of the level of his sales x. ​

A) ​
B) ​
C) ​

Question 52

Find the one-sided limit, if it exists. ​ ​

A) -15
B)
C) -6
D) 9
E) The limit does not exist

Question 53

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

A) -4 and 4
B) 9
C) -7 and 7
D) -6

Question 54

Find the one-sided limit, if it exists. ​ ​

A) 4
B) -2
C) -10
D) 5
E) The limit does not exist

Question 55

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

A) -3 and -2
B) 2
C) 3 and 2
D) 3

Question 56

Find and , where ​ ​

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

Question 57

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

A)
B)
C)
D)

Question 58

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

A)
B)
C)
D)

Question 59

Find the indicated one-sided limit.
​

Question 60

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) ; conditions 1, 2
B) ; conditions 2, 3
C) ; conditions 1, 2
D) ; condition 2
E) Continuous everywhere.

Question 61

Find the one-sided limit.
​

Question 62

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

Question 63

Find the one-sided limit.
​

Question 64

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

Question 65

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

Question 66

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. Select a graph showing his earnings as a function of the level of his sales x.
​ ​
​
​ ​ ​
Select the correct graph. Answer A, B, or C. __________
​
Determine the values of x for which the function f is discontinuous. Give the least three values of x.
​
x = __________

Question 67

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

Question 68

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

Question 69

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.
​ ​ ​
x = __________
​
State the condition(s) for continuity that are violated.
​
Conditions: __________

Question 70

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

Question 71

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

Question 72

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

Question 73

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

Question 74

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

Question 75

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

Question 76

Find and , where
​ ​ = __________
​ = __________

Question 77

Joan is looking straight out a window of an apartment building at a height of 20 ft from the ground. A boy throws a tennis ball straight up by the side of the building where the window is located. Suppose the height of the ball (measured in feet) from the ground at time t is
​ ​
At what time(s) does the ball cross Joan's line of sight?

Question 78

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

Question 79

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

Question 80

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.
​ ​ ​
x = __________
​
State the condition(s) for continuity that are violated.
​
Conditions: __________