Deck 2: Rate of Change: the Derivative

Question 1

The height of an object in feet above the ground is given in the following table. If the height of the object is doubled, the average velocity over any interval

A)doubles also.
B)stays the same.
C)is cut in half.

Answer 1

doubles also.

Question 2

Given the following data about the function f, estimate .

Answer 2

5

Question 3

Given the following data about the function f, the equation of the tangent line at x=3.2 is approximately y = _____x+_____. Use the nearest right-hand value to make your estimate.

Answer 3

Part A: 5
Part B: -6.5

Question 4

Let . What is the change in between t=2 and t=5?

Question 5

The following figure is the graph of , the cumulative number of customers served in a certain store during business hours one day, as a function of the hour of the day. About when was the store the busiest?

A)11am
B)1pm
C)3pm
D)5pm

Question 6

Let be the length of the side of a cube in terms of its volume. As V decreases, does the rate of change of x increase or decrease?

Question 7

The graph of is shown below. Arrange the following values in order from smallest to largest by placing a "1" by the smallest, a "2" by the next smallest, and so forth.
A. B. C. D. slope AB E. 1 F. 0

Question 8

A certain function f is decreasing and concave down. In addition, and . Which of the following are possible zeroes of f? Select all that apply.

A)3
B)5
C)7
D)9

Question 9

From the following graph, estimate .

A)-3.25
B)-2.25
C)-1.25
D)-0.25

Question 10

The height of an object in feet above the ground is given in the following table. The average velocity over the interval is _____ feet/sec.

Question 11

The growth graph in the following figure shows the height in inches of a bean plant during 30 days. On the 15^th day, the plant was growing about _____ inches/day. Round to 2 decimal places.

Question 12

Given the following data about the function f, give the average rate of change of f between x=3.2 and x=3.8. Round to 2 decimal places.

Question 13

Estimate when . Take smaller and smaller intervals until your estimate is accurate to 3 decimal places.

Question 14

If is the length of the side of a cube in terms of its volume, then calculate the average rate of change of x with respect to V over the interval 1

Question 15

Using a difference quotient, compute to 2 decimal places for .

Question 16

Recently Esther swam a lap in an Olympic swimming pool (the length of the pool is 50 meters, and the length of a lap is 100 meters); her times for various positions s (in meters from her starting point) during the lap are given in the following table. Her approximate velocity at time t=3.2 seconds was _____ m/sec. Round to 3 decimal places.

Question 17

A certain function f is decreasing and concave down. In addition, and . Which of the following are possible values for ? Select all that apply.

A)3
B)4
C)5
D)6

Question 18

Given the graph below of , is positive, negative, zero or undefined?

Question 19

An amount of $500 was invested in 1970 and the investment grew as shown in the following table. (Amounts are given for the beginning of the year.) The average rate of increase of the investment between 1980 and 1990 is _____ per year.

Question 20

The graph of in the figure gives the position of a particle at time t. Arrange the following values in order from smallest to largest by placing a "1" by the smallest, a "2" by the next smallest, and so forth.
A. average velocity on .
B. average velocity on .
C. instantaneous velocity at t=1.
D. instantaneous velocity at t=3.
E. instantaneous velocity at t=10.

Question 21

Using the graph of , at x=B is positive ?

Question 22

Suppose is the height in inches of a person who is t years old. Is it reasonable that ?

Question 23

A certain bacterial colony was observed for several hours and the following conditions were reported. Let be the number of bacteria present after t hours. There were 1000 bacteria after 5 hours. The growth rate was never negative and never exceeded 100 per hour. The growth rate was decreasing for the first 5 hours. At 7 hours, the growth rate was zero.
Is it possible that ?

Question 24

Consider the two functions shown below. A. B.

A)The function in graph A is the derivative of the function in graph B.
B)The function in graph B is the derivative of the function in graph A.
C)Neither function is the derivative of the other.

Question 25

Estimate the value of using the following table. Use the nearest right-hand value to make your estimate.

Question 26

A certain bacterial colony was observed for several hours and the following conditions were reported. Let be the number of bacteria present after t hours. There were 1000 bacteria after 5 hours. The growth rate was never negative and never exceeded 100 per hour. The growth rate was decreasing for the first 5 hours. At 7 hours, the growth rate was zero.
Is it possible that ?

Question 27

Let be the time, in minutes, that it takes for an oven to heat up to F. What are the units of ?

A)degrees per minute
B)minutes per degree

Question 28

Using the following table, tell whether is likely greater than 0, likely less than 0, or might be equal to 0. Type "<",">", or "=".

Question 29

The distance that a bird flies is measured by y miles for x minutes, and is given by the function . What are the units of

A) and
B) ?

Question 30

Considering the graphs below, could f(x) be the derivative of g(x)?

Question 31

Using the graph of , at x=C is positive ?

Question 32

Suppose is the height in inches of a person who is t years old. Would you expect to be

A)greater than 0
B)less than 0
C)equal to 0

Question 33

Consider the two functions shown below. A. B.

A)The function in graph A is the derivative of the function in graph B.
B)The function in graph B is the derivative of the function in graph A.
C)Neither function is the derivative of the other.

Question 34

Consider the two functions shown below. A. B.

A)The function in graph A is the derivative of the function in graph B.
B)The function in graph B is the derivative of the function in graph A.
C)Neither function is the derivative of the other.

Question 35

Let be the temperature in degrees Celsius at a height h (in meters) above the surface of the earth. Which of the following gives the rate of change of temperature with respect to a height at 70 meters above the surface of the earth, in degrees per meter?

A)
B)
C)h such that
D)h such that

Question 36

Suppose that is the cost to heat my house, in dollars per day, when the outside temperature is Fahrenheit. If and , approximately what is the cost to heat my house when the temperature is ?

Question 37

Let be the time, in minutes, that it takes for an oven to heat up to F. What is the sign of ?

A)positive
B)negative

Question 38

The graph below is the graph of , the derivative of . At 2 is the original function M(x) increasing, decreasing, constant or undefined?

Question 39

Let be the fuel efficiency of a car moving at v miles per hour. with efficiency measured in miles per gallon. Suppose and . What would you expect to be?

Question 40

Consider the two functions shown below. A. B.

A)The function in graph A is the derivative of the function in graph B.
B)The function in graph B is the derivative of the function in graph A.
C)Neither function is the derivative of the other.

Question 41

The following table gives the wind chill factor ( ) as a function of the wind speed (miles/hour) when the air temperature is 20 . What is the derivative of wind chill with respect to wind speed when the air temperature is 20 and the wind speed is 20 miles per hour? Use the nearest right-hand value to make your estimate.

Question 42

Let be the temperature in degrees Celsius at a height h (in meters) above the surface of the earth. Which of the following gives the temperature in degrees Celsius at a height of 1000 meters?

A)
B)
C)h such that
D)h such that

Question 43

Let represent the dollar amount charged per hour by a computer consultant to a client when they sign a contract t hours of work. The consultant gives a discount to the client if the contract is increased by 10 hours. Estimate the amount charged per hour when the client orders 130 hours of work.

Question 44

The following table gives the wind chill factor ( ) as a function of the wind speed (miles/hour) when the air temperature is 20 . What are the units for the derivative of wind chill with respect to wind speed when the air temperature is 20 ?

A)mph/
B) /mph

Question 45

Let represent the dollar amount charged per hour by a computer consultant to a client when they sign a contract t hours of work. The consultant gives a discount to the client if the contract is increased by 10 hours. Interpret the following statements.

A) .
B) .

Question 46

To study traffic flow along a major road, the city installs a device at the edge of the road at 4:00 am. The device counts the cars driving past, and records the total periodically. The resulting data is plotted on a graph, with time (in hours) on the horizontal axis and the number of cars on the vertical axis. The graph is shown below. It is a graph of the function = Total number of cars that have passed by after t hours. When is the traffic flow the greatest?

A)At t=6 hours.
B)At t=3 hours.
C)At t=4 hours.
D)At t=5 hours.

Question 47

Let be the temperature in degrees Celsius at a height h (in meters) above the surface of the earth. Which of the following gives the height, in meters, at which the rate of change of temperature with respect to height is 25 degrees per meter?

A)
B)
C)h such that
D)h such that

Question 48

Let be the amount of lumber, in board-feet, produced from a tree of radius r (measured in inches). Which of the following gives the rate of change in the amount of lumber, in board-feet per inch, with respect to the radius when the radius is 21 inches?

A)
B)
C)r such that
D)r such that

Question 49

To study traffic flow along a major road, the city installs a device at the edge of the road at 4:00 am. The device counts the cars driving past, and records the total periodically. The resulting data is plotted on a graph, with time (in hours) on the horizontal axis and the number of cars on the vertical axis. The graph is shown below. It is a graph of the function = Total number of cars that have passed by after t hours. Estimate .

A)1000
B)1300
C)1600
D)1900

Question 50

Every day the Undergraduate Office of Admissions receives inquiries from eager high school students (e.g. "Please send me an application",etc.) They keep a running count of the number of inquiries received each day, along with the total number received until that point. Below is a table of weekly figures from about the end of August to about the end of October of a recent year. One of these columns can be interpreted as a rate of change. Which one is it?

A)"Week of"
B)"Inquiries That Week"
C)"Total for Year"

Question 51

Let be the amount of lumber, in board-feet, produced from a tree of radius r (measured in inches). Which of the following gives the radius (in inches) of a tree that produces 150 board-feet of lumber ?

A)
B)
C)r such that
D)r such that

Question 52

The population of a certain town is given by the function where t is the number of years since the town was incorporated. If is constant for t >185, what will be if and ?

Question 53

The following table shows the number of oranges sold in one month, , against the price per bag, p (in cents). Find an approximation for . Use the nearest right-hand value to make your estimate.

Question 54

The noise level, N, in decibels, of a rock concert is given by , where d is the distance in meters from the concert speakers. Which of the following gives the rate of change, in decibels per meter, of noise 600 meters away from the speakers?

A)
B)
C)d such that
D)d such that

Question 55

The graph of is shown in the following figure. Is positive, negative, or zero?

A)zero
B)positive
C)negative

Question 56

The noise level, N, in decibels, of a rock concert is given by , where d is the distance in meters from the concert speakers. Which of the following gives the distance, in meters, away from the speakers at which the noise is 80 decibels?

A)
B)
C)d such that
D)d such that

Question 57

Let be the amount of lumber, in board-feet, produced from a tree of radius r (measured in inches). Which of the following gives the number of board-feet obtained from a tree of radius 8 inches?

A)
B)
C)r such that
D)r such that

Question 58

The cost of extracting T tons of ore from a copper mine is dollars. Would you expect to be positive or negative?

A)positive
B)negative

Question 59

The graph of is shown in the following figure. Give an estimate for

A)-10
B)10
C)-20
D)20

Question 60

The cost of extracting T tons of ore from a copper mine is dollars. What are the units for ?

A)dollars/ton
B)tons/dollar

Question 61

A company graphs , the derivative of the number of pints of ice cream sold over the past ten years. Out of t=1,2,4,8, and 10, in what year was C ''(t) greatest?

Question 62

Suppose the graph of f is in the figure below. Is positive, negative, or zero?

Question 63

The cost of mining a ton of coal is rising faster every year. Suppose is the cost of mining a ton of coal at time t. Which of the following must be concave up? Select all that apply.

A)
B)
C)

Question 64

Consider the function f sketched in the following figure. Do you expect to be positive, negative, or zero?

Question 65

Cost and revenue functions for a certain chemical manufacturer are given in the following figure. How much does it cost to produce 10 tons?

A)$4,500
B)$3,200
C)$4.50
D)$3.20

Question 66

Consider the function f sketched in the following figure. Do you expect to be positive, negative, or zero?

Question 67

Consider the following graph. In region III, is ________ (positive/negative) and is ________ (positive/negative).

Question 68

Suppose a function is given by the following table of values. Estimate the instantaneous rate of change of f at x=1.7, and use this estimate to find the equation for the tangent line to f at x=1.7. The line is y = _____x+_____. Use the nearest right-hand value to make your estimate.

Question 69

A driver obeys the speed limit as she travels past different towns in the order A, B, C. In town A, the speed limit is 35 mph. In town B, the speed limit is 60 mph, and in town C the speed limit is 55 mph. It always takes her two minutes to reach the new speed limit when she passes by a new town. If represents the driver's position at time t, then is for the first two minutes she is passing town B positive or negative?

Question 70

The cost of mining a ton of coal is rising faster every year. Suppose is the cost of mining a ton of coal at time t. Which of the following must be positive? Select all that apply.

A)
B)
C)

Question 71

Suppose a function is given by the following table of values. Is most likely positive or negative at x=1.9?

A)positive
B)negative

Question 72

Using the graph of , at x=D is positive, negative or zero?

Question 73

Write the Leibniz notation for the first and second derivatives of the given function and include units.
"The amount of paint, G, in gallons, is a function of time m, in minutes"

Question 74

Suppose the graph of f is in the figure below. Is positive, negative, or zero?

Question 75

A function f satisfies the following conditions: , , , , and for . Which of the following are possible values for ? Select all that apply.

A)4
B)8
C)14

Question 76

Consider the function f sketched in the following figure. Do you expect to be positive, negative, or zero?

Question 77

Let represent the number of students enrolled in school in the year t. If enrollment is decreasing steadily, then _____0 and _____0. (Enter "<",">", or "=")

Question 78

Could the function on the right be the second derivative of the function on the left?

Question 79

The cost of mining a ton of coal is rising faster every year. Suppose is the cost of mining a ton of coal at time t. Which of the following must be increasing? Select all that apply.

A)
B)
C)

Question 80

Suppose the graph of f is in the figure below. Is f (B) positive, negative, or zero?