# Calculus Exercise Relative Rates of Growth & Exploring The Graphs Attached is a file with questions pertaining to calculus, no work is necessary unless ask

Calculus Exercise Relative Rates of Growth & Exploring The Graphs Attached is a file with questions pertaining to calculus, no work is necessary unless asked for.Exploring the Graphs of f, f Prime, and f Double Prime Relative Rates of Growth 1.

The graph of f “(x) is continuous and decreasing with an x-intercept at x = 0. Which of the

following statements is true? (4 points)

The graph of f has a relative maximum at x = 0.

The graph of f has a relative minimum at x = 0.

The graph of f has an inflection point at x = 0.

The graph of f has an x-intercept at x = 0.

2.

The graph below shows the graph of f (x), its derivative f “(x), and its second derivative f

“(x). Which of the following is the correct statement?

(4 points)

A is f “, B is f ‘. C is f.

A is f “, B is f, C is f ‘.

A is f ‘, B is f, C is f “.

A is f, B is f ‘, C is f “.

3.

Below is the graph of f ‘(x), the derivative of f(x), and has x-intercepts at x = -3, x = 1 and

x = 2. There are horizontal tangents at x = -1.5 and x = 1.5. Which of the following

statements is true?

(4 points)

f has an inflection point at x = -1.5.

f is increasing on the interval from x = -3.2 to x = -4.5.

f has a relative minimum at x = 1.5.

All of these are true.

4.

The graph of f ‘ (x), the derivative of f of x, is continuous for all x and consists of five line

segments as shown below. Given f (-3) = 6, find the absolute maximum value of f (x) over

the interval [-3, 0].

(4 points)

3

4.5

6

10.5

5.

The graph of y = f ‘(x), the derivative of f(x), is shown below. Given f(4) = 6, evaluate f(0).

(4 points)

-2

2

4

10

1.

Which of the following functions grows the fastest as x goes to infinity? (4 points)

2x

3x

ex

x20

2.

Compare the rates of growth of f(x) = x + sinx and g(x) = x as x approaches infinity. (4

points)

f(x) grows faster than g(x) as x goes to infinity.

g(x) grows faster than f(x) as x goes to infinity.

f(x) and g(x) grow at the same rate as x goes to infinity.

The rate of growth cannot be determined.

3.

What does

show? (4 points)

g(x) grows faster than f(x) as x goes to infinity.

f(x) and g(x) grow at the same rate as x goes to infinity.

f(x) grows faster than g(x) as x goes to infinity.

L’Hôpital’s Rule must be used to determine the true limit value.

4.

Which of the following functions grows at the same rate as

? (4 points)

x

x2

x3

x4

5.

Which of the following functions grows the slowest as x goes to infinity? (4 points)

0.01×3

0.001×3

.00001×3

They all grow at the same rate.

1.

The function f is continuous on the interval [3, 13] with selected values of x and f(x) given

in the table below. Find the average rate of change of f(x) over the interval [3, 13]. (4

points)

x 3 4 7 10 13

f(x) 2 8 10 12 22

2.

f is a differentiable function on the interval [0, 1] and g(x) = f(3x). The table below gives

values of f ‘(x). What is the value of g ‘(0.1)? (4 points)

x 0.1 0.2 0.3 0.4 0.5

f ‘(x) 1 2 3 -4 5

1

3

9

Cannot be determined

3.

f(x) and g(x) are a differentiable function for all reals and h(x) = g[f(5x)]. The table below

gives selected values for f(x), g(x), f ‘(x), and g ‘(x). Find the value of h'(1). (4 points)

x 123456

f(x) 0 3 2 1 2 0

g(x) 1 3 2 6 5 0

f ‘(x) 3 2 1 4 0 2

g ‘(x) 1 5 4 3 2 0

4.

The table of values below shows the rate of water consumption in gallons per hour at

selected time intervals from t = 0 to t = 12.

Using a left Riemann sum with 5 subintervals, estimate the total amount of water consumed

in that time interval. (4 points)

x 0 2 5 7 11 12

f(x) 5.7 5.0 2.0 1.2 0.6 0.4

22.1

34.8

35.8

None of these

5.

The function is continuous on the interval [10, 20] with some of its values given in the table

above. Estimate the average value of the function with a Trapezoidal Sum Approximation,

using the intervals between those given points. (4 points)

x 10 12 15 19 20

f(x) -2 -5 -9 -12 -16

-8.750

-7.000

-8.400

-5.500

Description

1.

Let

. Use your calculator to find F”(1). (4 points)

5.774

11.549

18.724

37.449

2.

Pumping stations deliver gasoline at the rate modeled by the function D, given

by

with t measure in hours and and R(t) measured in gallons per hour. How

much oil will the pumping stations deliver during the 3-hour period from t = 0 to t = 3? Give

3 decimal places. (4 points)

3.

A particle moves along the x-axis with velocity v(t) = sin(2t), with t measured in seconds

and v(t) measured in feet per second. Find the total distance travelled by the particle from t

= 0 to t = π seconds. (4 points)

2

1

0

4.

Find the range of the function

. (4 points)

[-4, 4]

[-4, 0]

[0, 4π]

[0, 8π]

5.

Use the graph of f(t) = 2t + 3 on the interval [-3, 6] to write the function F(x),

where

. (4 points)

F(x) = 2×2 + 6x

F(x) = 2x + 3

F(x) = x2 + 3x + 54

F(x) = x2 + 3x – 18

1.

The graph of f ‘(x) is continuous, positive, and has a relative maximum at x = 0. Which of

the following statements must be true? (5 points)

The graph of f is always concave down.

The graph of f is always increasing.

The graph of f has a relative maximum at x = 0.

The graph of f has a relative minimum at x = 0.

2.

Below is the graph of f ‘(x), the derivative of f(x), and has x-intercepts at x = -3, x = 1, and

x = 2 and a relative maximum at x = -1.5 and a relative minimum at x = 1.5. Which of the

following statement is false?

(5 points)

f is concave up from x = -1.5 to x = 1.5.

f has an inflection point at x = 1.5.

f has a relative minimum at x = 2.

All of these are false.

3.

The graph of y = f ‘(x), the derivative of f(x), is shown below. List the intervals where the

graph of f is concave down.

(5 points)

(-4, -2) U (2, 4)

(-2, 2)

(-4, 0)

(0, 4)

4.

Which of the following functions grows the fastest as x grows without bound? (5 points)

f(x) = ex

g(x) = ecosx

h(x) =

They all grow at the same rate.

5.

Compare the growth rate of the functions f(x) = 4x and g(x) =

. (5 points)

f(x) grows faster than g(x).

g(x) grows faster than f(x).

f(x) and g(x) grow at the same rate.

It cannot be determined.

6.

f is a function that is differentiable for all reals. The value of f ‘(x) is given for several values

of x in the table below.

x -8 -3 0 3 8

f ‘(x) -4 -2 0 4 5

If f ‘(x) is always increasing, which statement about f(x) must be true? (5 points)

f(x) passes through the origin.

f(x) is concave downwards for all x.

f(x) has a relative minimum at x = 0.

f(x) has a point of inflection at x = 0.

7.

f is a differentiable function on the interval [0, 1] and g(x) = f(2x). The table below gives

values of f ‘(x). What is the value of g ‘(0.1)? (5 points)

x 0.1 0.2 0.3 0.4 0.5

f ‘(x) 1 2 3 -4 5

1

2

4

Cannot be determined

8.

Use the graph of f(t) = 2t + 2 on the interval [-1, 4] to write the function F(x),

where

. (5 points)

F(x) = x2 + 3x

F(x) = x2 + 2x – 12

F(x) = x2 + 2x – 3

F(x) = x2 + 4x – 8

9.

The velocity of a particle moving along the x-axis is v(t) = t2 + 2t + 1, with t measured in

minutes and v(t) measured in feet per minute. To the nearest foot find the total distance

travelled by the particle from t = 0 to t = 2 minutes. (5 points)

10.

Find the range of the function

. (5 points)

[0, 4π]

[0, π]

[-4, 0]

[0, 4]

Must Show work

The figure below shows the graph of f ‘, the derivative of the function f, on the closed

interval from x = -2 to x = 6. The graph of the derivative has horizontal tangent lines at x =

2 and x = 4.

Find the x-coordinate of each of the points of inflection of the graph of f. Justify your

answer. (10 points)

2.

A car travels along a straight road for 30 seconds starting at time t = 0. Its acceleration in

ft/sec2 is given by the linear graph below for the time interval [0, 30]. At t = 0, the velocity

of the car is 0 and its position is 10.

What is the total distance the car travels in this 30 second interval? Your must show your

work but you may use your calculator to evaluate. Give 3 decimal places in your answer and

include units.

(10 points)

3.

Show that f(x) = 2000×4 and g(x) = 200×4 grow at the same rate. (10 points)

4.

A radar gun was used to record the speed of a runner (in meters per second) during

selected times in the first 2 seconds of a race. Use a trapezoidal sum with 4 intervals to

estimate the distance the runner covered during those 2 seconds. Give a 2 decimal place

answer and include units. (10 points)

t 0 0.5 1.2 1.5 2

v(t) 0 4.5 7.8 8.3 9.0

5.

Oil flows into a tank according to the rate

, and at the same time empties out at

the rate

, with both F(t) and E(t) measured in gallons per minute. How much

oil, to the nearest gallon, is in the tank at time t = 12 minutes. You must show your setup

but can use your calculator for all evaluations. (10 points)

Must show work

A calculator is permitted.

Use the following information for Questions 1–3:

The rate at which water flows into a tank, in gallons per hour, is given by a differentiable function R

of time t. The table below gives the rate as measured at various times in an 8-hour time period.

t (hours)

R(t) (gallons per hour)

0

2

3

7

8

1.95

2.5

2.8

4

4.26

1. Use a trapezoidal sum with the four sub-intervals indicated by the data in the table to

estimate . Using correct units, explain the meaning of your answer in terms of water flow. Give 3

decimal places in your answer.

2. Is there some time t, 0 < t < 8, such that R′(t) = 0? Justify your answer.
3. The rate of water flow R(t) can be estimated by W(t) = ln(t + 7). Use W(t) to approximate the
average rate of water flow during the 8-hour time period. Indicate units of measure.
2
Use the following information for Questions 4 and 5.
f is a continuous function with a domain [−3, 9] such that
and let .
4. On what interval is g increasing? Justify your answer.
5. For 0 ≤ x ≤ 6, express g(x) in terms of x. Do not include +C in your final answer
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