Deck 2: Key Concept: the Derivative

Question 1

The length, x, of the side of a cube with volume V is given by .Is the average rate of change of x with respect to V increasing or decreasing as the volume V decreases?

Answer 1

increasing

Question 2

A runner planned her strategy for running a half marathon, a distance of 13.1 miles.She planned to run negative splits, faster speeds as time passed during the race.In the actual race, she ran the first 6 miles in 48 minutes, the second 4 miles in 28 minutes and the last 3.1 miles in 18 minutes.What was her average velocity over the first 6 miles? What was her average velocity over the entire race? Did she run negative splits?

A)7.50 mph for the first 5 miles, 8.36 mph for the race, No
B)8.36 mph for the first 5 miles, 7.50 mph for the race, No
C)8.12 mph for the first 5 miles, 7.35 mph for the race, Yes
D)7.35 mph for the first 5 miles, 8.12 mph for the race, No

Answer 2

7.50 mph for the first 5 miles, 8.36 mph for the race, No

Question 3

Estimate to 2 decimal places by substituting smaller and smaller values of h (use radians).

Answer 3

0

Question 4

Given the following data about a function f, estimate .

Question 5

The height of an object in feet above the ground is given in the following table.If heights of the object are cut in half, how does the average velocity change, over a given interval?

A)It is cut in half.
B)It is doubled.
C)It remains the same.
D)It depends on the interval.

Question 6

If is the length of the side of a cube in terms of its volume, V, calculate the average rate of change of x with respect to V over the interval to 2 decimal places.

Question 7

The height of an object in feet above the ground is given in the following table, .Make a graph of .On your graph , what does the average velocity over a the interval represent?

A)The average height between f(0)and f(3).
B)The slope of the line between the points (0, f(0)), and (3, f(3)).
C)The average of the slopes of the tangent lines to the points (0, f(0)), and (3, f(3)).
D)The distance between the points (0, f(0)), and (3, f(3)).

Question 8

Given the following table of values for a Bessel function, , estimate the derivative at x = 0.5.

Question 9

The data in the table report the average improvement in scores of six college freshmen who took a writing assessment before and again after they had x hours of tutoring by a tutor trained in a new method of instruction.When f(x)>0 the group showed improvement on average.
a)Find the average change in score from 6.5 to 9.5 hours of tutoring.
b)Estimate the instantaneous rate of change at 8 hours.
c)Approximate the equation of the tangent line at x = 8 hours.
d)Use the tangent line to estimate f(8.5).

Question 10

There is a function used by statisticians, called the error function, which is written y = erf (x).Suppose you have a statistical calculator, which has a button for this function.Playing with your calculator, you discover the following:
$\begin{array}{cc}x & \operatorname{erf}(x) \\1 & 0.29793972 \\0.1 & 0.03976165 \\0.01 & 0.00398929 \\0.001 & 0.000398942&\\0&0\end{array}$

Using this information alone, give an estimate for erf'(0), the derivative of erf at x = 0 to 4 decimal places.

Question 11

Given the following data about a function f(x), the equation of the tangent line at x = 5 is approximated by

A)
B)
C)
D)

Question 12

A horticulturist conducted an experiment to determine the effects of different amounts of fertilizer on the yield of a plot of green onions.He modeled his results with the function where Y is the yield in bushels and x is the amount of fertilizer in pounds.What are and ? Give your answers to two decimal places, specify units.

A)4.22 bushels, 1.25 bushels/pound, respectively
B)4.22 bushels, 6.25 bushels/pound, respectively
C)1.25 bushels, 1.56 bushels/pound, respectively
D)1.25 bushels, 4.22 bushels/pound, respectively

Question 13

For $f(x)=\log x$
, estimate f '(3)to 3 decimal places by finding the average slope over intervals containing the value x = 3.

Question 14

Let .Use a graph to decide which one of the following statements is true.

A)When x = -5, the derivative is negative; when x = 5, the derivative is positive; and as x approaches infinity, the derivative approaches 0.
B)When x = -6, the derivative is positive; when x = 6, the derivative is also positive, and as x approaches infinity, the derivative approaches 0.
C)When x = -7, the derivative is negative; when x = 7, the derivative is positive, and as x approaches infinity, the derivative approaches infinity.
D)The derivative is positive at at all values of x.

Question 15

For , estimate to 3 decimal places.

Question 16

Let f(x)= log(log(x)).Estimate to 3 decimal places using any method.

Question 17

Use the graph of at the point (0, 5.5)to estimate to three decimal places.

A)16.500
B)36.823
C)3.500
D)146.507

Question 18

For any number r, let m(r)be the slope of the graph of the function at the point x = r.Estimate m(4)to 2 decimal places.

Question 19

The height of an object in feet above the ground is given in the following table.Compute the average velocity over the interval 1 $\le$ t $\le$ 3.
$\begin{array}{cccccccc}t \text { (sec) } & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\y \text { (feet) } & 10 & 45 & 70 & 85 & 90 & 85 & 70\end{array}$

Question 20

Estimate to 2 decimal places by substituting smaller and smaller values of h.

Question 21

Estimate a formula for for the function .Round constants to 3 decimal places.

Question 22

To find the derivative of at x = 8 algebraically, you evaluate the following expression.

A)
B)
C)
D)All of the above are correct.
E)None of the above is correct.

Question 23

Could the first graph, A be the derivative of the second graph, B? A B

Question 24

To study traffic flow along a major road, the city installs a device at the edge of the road at 1:00a.m.The device counts the cars driving past, and records the total periodically.The resulting data is plotted on a graph, with time (in hours since installation)on the horizontal axis and the number of cars on the vertical axis.The graph is shown below; it is the graph of the function C(t)= Total number of cars that have passed by after t hours.When is the traffic flow greatest?

A)2:00 am
B)3:00 am
C)4:00 am
D)5:00 am

Question 25

The graph below shows the velocity of a bug traveling along a straight line on the classroom floor. At what time(s)does the bug turn around?

A)At 3 seconds.
B)At 2 seconds and again at 7 seconds.
C)At 4 seconds and again at 7 seconds.
D)Never.

Question 26

Draw the graph of a continuous function y = g(x)that satisfies the following three conditions:
• g'(x)= 0 for x < 0
• g'(x)> 0 for 0 < x < 4
• g'(x)< 0 for x > 4

Question 27

Find the derivative of at x = 4 algebraically.

Question 28

he graph below shows the velocity of a bug traveling along a straight line on the classroom floor. Graph the bug's speed at time, t.How does it differ from the bug's velocity?

Question 29

Find the derivative of at x = 1 algebraically.

Question 30

Could the first graph, A be the derivative of the second graph, B? A B

Question 31

Use the limit of the difference quotient to find the derivative of at the point (1, 11/2).

A)
B)
C)-11
D)

Question 32

To study traffic flow along a major road, the city installs a device at the edge of the road at 3:00a.m.The device counts the cars driving past, and records the total periodically.The resulting data is plotted on a graph, with time (in hours since installation)on the horizontal axis and the number of cars on the vertical axis.The graph is shown below; it is the graph of the function C(t)= Total number of cars that have passed by after t hours.From the graph, estimate C'(6).

A)600
B)900
C)1200
D)1500

Question 33

A runner competed in a half marathon in Anaheim, a distance of 13.1 miles.She ran the first 7 miles at a steady pace in 48 minutes, the second 3 miles at a steady pace in 28 minutes and the last 3.1 miles at a steady pace in 18 minutes.
a)Sketch a well-labeled graph of her distance completed with respect to time.
b)Sketch a well-labeled graph of her velocity with respect to time.

Question 34

The definition of the derivative function is

Question 35

Suppose that f(T)is the cost to heat my house, in dollars per day, when the outside temperature is T ${ } ^ { \circ } F$ .If f(28)= 11.10 and f '(28)= -0.12, approximately what is the cost to heat my house when the outside temperature is 25 ${ } ^ { \circ } F$ ?

Question 36

The graph below shows the velocity of a bug traveling along a straight line on the classroom floor. When is the bug moving at a constant speed?

A)Between 4 and 7 seconds.
B)Whenever the velocity is linear with a positive slope.
C)Whenever the velocity is linear with a negative slope.
D)When the velocity is equal to zero.

Question 37

Use the definition of the derivative function to find a formula for the slope of the graph of .

Question 38

What is the equation of the tangent line to the graph of at the point (2, 8)?

A)
B)
C)
D)

Question 39

Could the first graph, A be the derivative of the second graph, B? A B

Question 40

Consider the function y = f(x)graphed below.At the point x = -3, is positive, negative, 0, or undefined?Note: f(x)is defined for -5 < x < 6, except x = 2.

Question 41

A function defined for all x has the following properties:
• f is increasing
• f is concave down
• f (3)= 2
• f '(3)= 1/2
How many zeros does f(x)have in the interval $1 \leq x \leq 3$
?

Question 42

The graph below represents the rate of change of a function f with respect to x; i.e., it is a graph of f '.You are told that f (0)= 0.What can you say about $f(x)$
at the point x = 1.3? Mark all that apply.

A) $f(x)$ is decreasing.
B) $f(x)$
is increasing.
C) $f(x)$
is concave up.
D) $f(x)$
is concave down.

Question 43

Let L(r)be the amount of board-feet of lumber produced from a tree of radius r (measured in inches).What does L(16)mean in practical terms?

A)The amount of board-feet of lumber produced from a tree with a radius of 16 inches.
B)The radius of a tree that will produce 16 board-feet of lumber.
C)The rate of change of the amount of lumber with respect to radius when the radius is 16 inches (in board-feet per inch).
D)The rate of change of the radius with respect to the amount of lumber produced when the amount is 16 board-feet (in inches per board-foot).

Question 44

The graph of is given in the following figure.What happens to at the point ?

A) has an inflection point.
B) has a local minimum or maximum.
C) changes sign.
D)none of the above

Question 45

The graph below gives the position of a spider moving along a straight line on the forest floor for 10 seconds.On the same axes, sketch a graph of the spider's velocity over the 10 seconds.Then write a description of the spider's movement for the 10 second period.

Question 46

Esther is a swimmer who prides herself in having a smooth backstroke.Let s(t)be her position in an Olympic size (50-meter)pool, as a function of time (s(t)is measured in meters, t is seconds).Below we list some values of s(t), for a recent swim.Based on the data, was Esther's instantaneous speed ever greater than 3 meters/second?

Question 47

Given the following data about a function f, estimate the rate of change of the derivative at x = 4.5.
.

Question 48

Let t(h)be the temperature in degrees Celsius at a height h (in meters)above the surface of the earth.What does t '(1200)mean in practical terms?

A)The temperature in degrees Celsius at a height 1200 meters above the surface of the earth.
B)The height above the surface of the earth at which the temperature is 1200 degrees Celsius.
C)The rate of change of temperature with respect to height at 1200 meters above the surface of the earth (in degrees per meter).
D)The rate of change of height with respect to temperature when the temperature is 1200 degrees Celsius (in meters per degree).

Question 49

On the axes below, sketch a smooth, continuous curve , and which clearly satisfies the following conditions:
• Concave up to the left of P
• Concave down to the right of P
• Increasing for x > 0
• Decreasing for x < 0
• Does not pass through the origin.

Question 50

Every day the Office of Undergraduate Admissions receives inquiries from eager high school students.They keep a running account 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.
Based on the table determine a formula that approximates the total number of inquiries received by a given week.Use your formula to estimate how many inquiries the admissions office will have received by November 24.

Question 51

The cost in dollars to produce q bottles of a prescription skin treatment is given by the function .The manufacturing process is difficult and costly when large quantities are produced.The marginal cost of producing one additional bottle when q bottles have been produced is the derivative .
a)Find the marginal cost function.
b)Compute C(50)and explain what the number means in terms of cost and production.
c)Compute C'(50)and explain what the number means in terms of cost and production.

Question 52

Let t(h)be the temperature in degrees Celsius at a height of h meters above the surface of the earth.What does t(h)+ 15 mean in practical terms?

A)The temperature in degrees Celsius at a height h meters above the surface of the earth plus an additional 15 degrees
B)The height above the surface of the earth at which the temperature is h degrees Celsius plus an additional 15 meters.
C)The rate of change of temperature with respect to height at 15 additional meters above the surface of the earth (in degrees per meter).
D)The rate of change of height with respect to temperature when the temperature is 15 additional degrees Celsius (in meters per degree).

Question 53

The graph below represents the rate of change of a function f with respect to x; i.e., it is a graph of f'.You are told that f(0)= -2.For approximately what value of x other than x = 0 in the interval 0 $\le$ x $\le$ 2 does $f(x)$
= -2?

A)0.6
B)1
C)1.4
D)2
E)None of the above

Question 54

A concert promoter estimates that the cost of printing p full color posters for a major concert is given by a function Cost = c(p)where p is the number of posters produced.
a)Interpret the meaning of the statement c(450)= 5400.
b)Interpret the meaning of the statement c'(450)= 11.

Question 55

Esther is a swimmer who prides herself in having a smooth backstroke.Let s(t)be her position in an Olympic size (50-meter)pool, as a function of time (s(t)is measured in meters, t is seconds).Below we list some values of s(t)for a recent swim.Find Esther's average speed over the entire swim in meters per second.Round to 2 decimal places.

Question 56

One of the following graphs is of , and the other is of .Is the first graph or the second graph?

Question 57

Let t(h)be the temperature in degrees Celsius at a height of h meters above the surface of the earth.What does h such that t(h)= 8 mean in practical terms?

A)The temperature in degrees Celsius at a height 8 meters above the surface of the earth.
B)The height above the surface of the earth at which the temperature is 8 degrees Celsius.
C)The rate of change of temperature with respect to height at 8 meters above the surface of the earth (in degrees per meter).
D)The rate of change of height with respect to temperature when the temperature is 8 degrees Celsius (in meters per degree).

Question 58

A typhoon is a tropical cyclone, like a hurricane, that forms in the northwestern Pacific Ocean.The wind speed of a typhoon is given by a function W = w(r)where W is measured in meters/sec., and r is measured in kilometers from the center of the typhoon.What does the statement that w'(15)> 0 tell you about the typhoon?

A)At a distance of 15 kilometers from the center of the typhoon, the wind speed is increasing.
B)At a distance of 15 kilometers from the center of the typhoon, the wind speed is positive.
C)The wind speed of the typhoon is 15 meters per second at any distance from the center of the typhoon.

Question 59

A function defined for all x has the following properties:
• f is increasing
• f is concave down
• f (4)= 2
• Is it possible that ?

Question 60

Every day the Office of Undergraduate Admissions receives inquiries from eager high school students.They keep a running account 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)the first
B)the second
C)the third

Question 61

Assume that f and g are differentiable functions defined on all of the real line.Is it possible that f '(x)> g'(x)for all x and f(x)< g(x)for all x?

Question 62

Given the function .Is differentiable at r = -1?

Question 63

Assume that f and g are differentiable functions defined on all of the real line.If f'(x)= g'(x)for all x and if

Question 64

A golf ball thrown directly upwards from the surface of the moon with an initial velocity of 17.00 meters per second and will attain a height of meters in t seconds.Find a formula for the velocity of the golf ball at time t.

A) meters/sec
B) meters/sec
C) -1.36 meters/sec
D) meters/sec

Question 65

A golf ball thrown directly upwards from the surface of the moon with an initial velocity of 14.00 meters per second and will attain a height of meters in t seconds.What is the acceleration of the golf ball at time t?

A) meters/sec/sec
B) meters/sec²
C) meters/sec/sec
D)14.00 meters/sec²

Question 66

Suppose a function is given by a table of values as follows:
Give your best estimate of .

Question 67

Let S(t)represent the number of students enrolled in school in the year t.If the number of students enrolling is increasing faster and faster, then is S '(t)positive, negative, or 0?

Question 68

Assume that f and g are differentiable functions defined on all of the real line.If f ' > 0 everywhere and f > 0 everywhere then must ?

Question 69

A company graphs C'(t), the derivative of the number of pints of ice cream sold over the past ten years.At approximately what year was C ''(t)greatest?

Question 70

A golf ball thrown directly upwards from the surface of the moon with an initial velocity of 20 meters per second and will attain a height of meters in t seconds.On Earth, its height would be given by .Compare the velocity and acceleration of the golf ball on the moon after two seconds with its velocity and acceleration on Earth after two seconds.

Question 71

Describe two ways that a continuous function can fail to have a derivative at a point, x = a.Illustrate your description with graphs.

Question 72

A husband and wife purchase life insurance policies.Over the next 40 years, one policy pays out when the husband dies, and the other pays out when both husband and wife die.Their life expectancy is 20 years, and the probability that both die before year t is given by the function .How fast is the probability that both are dead increasing in 25 years?

A)0.0313
B)0.3906
C)0.0500
D)50.0006

Question 73

Is the graph of continuous at x = -9?

Question 74

The cost of mining a ton of coal is rising faster every year.Suppose C(t)is the cost of mining a ton of coal at time t.Must C ''(t)be concave up?

Question 75

If the Figure 1 is , could Figure 2 be ? Figure 1 Figure 2

Question 76

Assume that f is a differentiable function defined on all of the real line.Is it possible that f > 0 everywhere, f ' > 0 everywhere, and f ' < 0 everywhere?

Question 77

A golf ball thrown directly upwards from the surface of the moon with an initial velocity of 20 meters per second and will attain a height of meters in t seconds.How fast is the golf ball going at its high point?

A)0 meters/sec
B)-0.8 meters/sec
C)20 meters/sec
D)-20 meters/sec

Question 78

Sketch a graph y = f(x)that is continuous everywhere on -6 < x < 6 but not differentiable at x = -3 or x = 3.

Question 79

Is the graph of continuous at x = -3?

Question 80

Sketch a graph of a continuous function f(x)with the following properties:
• f'(x)< 0 for x < 4
• f'(x)> 0 for x > 4
• f'(4)is undefined