**Question No. 1 of 10 **

**Instruction: (1) Read the problem statement and answer choices carefully (2) Work the problems on paper as needed (3) **
Pick the answer (4) Go back to review the core concept tutorial as needed.

**Question **

Question 1. If *f x*( )= −7*x*+ , find the average rate of change from 5 *x = to *1 3 *x = . *2 7

(A) 5
(B) 3
(C) −7
(D) 7
* (E) x *

**Feedback **

A. Incorrect!
What is the average rate of change of a linear function?

B. Incorrect!

What is the average rate of change of a linear function?

C. Correct! ( ) 7 5

*f x* = − *x*+ * is a linear function with slope m = *−7.

D. Incorrect!

What is the average rate of change of a linear function?

(E) Incorrect!

What is the average rate of change of a linear function?

**Solution **

RapidLearningCenter.com © Rapid Learning Inc. All Rights Reserved Pick the answer (4) Go back to review the core concept tutorial as needed.

**Question **

Question 2. The average rate of change of a constant function is ____.

(A) Always zero (B) Always positive (C) Always negative

(D) Sometimes negative and sometimes positive (E) Can't be determined

**Feedback **

A. Correct

A constant function always has a slope of zero.

B. Incorrect!

What is the slope of a constant function?

C. Incorrect!

What is the slope of a constant function?

D. Incorrect!

What is the slope of a constant function?

E. Incorrect!

What is the slope of a constant function?

**Solution **

Pick the answer (4) Go back to review the core concept tutorial as needed.

**Question **

Question 3. The average rate of change is the same thing as ______. (A) A linear function

(B) The slope of the tangent line (C) The instantaneous rate of change (D) The slope of the secant line (E) The position function

**Feedback **

A. Incorrect!

The average rate of change is represented by a slope.

B. Incorrect!

This is represents the instantaneous rate of change.

C. Incorrect!

The instantaneous rate of change requires only a single point. The average rate of change requires two points.

D. Correct!

This is correct; the average rate of change is the same thing as the slope of the secant line.

E. Incorrect!

The average rate of change is represented by a slope.

**Solution **

Recall that the average rate of change is given by *f b*( ) *f a*( )

*b a*

−

RapidLearningCenter.com © Rapid Learning Inc. All Rights Reserved Pick the answer (4) Go back to review the core concept tutorial as needed.

**Question **

Question 4. Calculate the average rate of change of ( )*f x* = −ln(*x*+1) over [0,1] .
(A) −ln*x*−ln1
(B) 0
(C) ln2−
(D) 1
(E) ln2

**Feedback **

A. Incorrect!
First find f(0) and then f(1). Now find the slope.

B. Incorrect!

First find f(0) and then f(1). Now find the slope.

C. Correct!

This is the correct answer. Find f(0) and then f(1). Then find the slope.

D. Incorrect!

First find f(0) and then f(1). Now find the slope.

E. Incorrect!

First find f(0) and then f(1). Now find the slope.

**Solution **

First, *f*(0)= −ln(0 1)+ = −ln1 0= and *f*(1)= −ln(1 1)+ = −ln2. Summarizing, we have *x =*1 0, *y = and *1 0
2 1

Pick the answer (4) Go back to review the core concept tutorial as needed.

**Question **

Question 5. Which of the following represents the slope of the line tangent to ( )*f x at (2,5) ? *

(A)
0
( ) (2)
lim
*h*
*f h* *f*
*h*
→
−
(B) *f*(2 *h*) *f*(2)
*h*
+ −
(C) *f*(2)
(D)
0
(2 ) (2)
lim
*h*
*f* *h* *f*
*h*
→
+ −
(E) (2,5)

**Feedback **

A. Incorrect!
*The numerator should be f(2 + h) - f(2) *

B. Incorrect!

Remember that the slope of the tangent line is a limit.

C. Incorrect!

f(2) is just a y-value for the function.

D. Correct!

This the correct answer. Use the tangent line formula.

E. Incorrect!

This is a point on the graph, not a slope.

**Solution **

Which of the following represents the slope of the line tangent to *f x*( ) at (2,5)?
Recall that the slope of the tangent line is

0
( ) ( )
lim
*h*
*f x* *h* *f x*
*h*
→

+ − _{. At the point (2,5), x is equal to 2 }

therefore the slope of the tangent line is

RapidLearningCenter.com © Rapid Learning Inc. All Rights Reserved Pick the answer (4) Go back to review the core concept tutorial as needed.

**Question **

Question 6. Suppose that the velocity of an object at a specific time it −74 feet/sec. What is the speed? (A) 0 feet/sec (B) 74− feet/sec (C) 16− feet/sec (D) Cannot be determined (E) 74 feet/sec

**Feedback **

A. Incorrect!
Remember what the relationship is between speed and velocity.

B. Incorrect!

Remember what the relationship is between speed and velocity.

C. Incorrect!

Remember what the relationship is between speed and velocity.

D. Incorrect!

Remember what the relationship is between speed and velocity.

E. Correct!

This is correct. Speed is the absolute value of velocity.

**Solution **

Pick the answer (4) Go back to review the core concept tutorial as needed.

**Question **

Question 7. Suppose that an object is moving horizontally in such a way that its position is given by ( ) 4 8

*s t* = *t+ at time t. What is the velocity of the object at time t? *

* (A) 4t *
(B) 4
(C) 8
(D) 0

(E) None of these

**Feedback **

A. Incorrect! 7

Note that s(t) is a linear function. What is the instantaneous rate of change of a linear function?

B. Correct!

This is correct. Since s(t) is a linear function, the rate of change is the same as the slope.

C. Incorrect!

Note that s(t) is a linear function. What is the instantaneous rate of change of a linear function?

D. Incorrect!

Note that s(t) is a linear function. What is the instantaneous rate of change of a linear function?

E. Incorrect!

One of the given answers is correct.

**Solution **

**Question **

Question 8. Find a formula for the instantaneous rate of change of _{f x}_{( )}_{=}* _{x}*2

_{. }

(A) * _{x }*2
(B) 2

*x*(C)

*2*

_{x}_{+}

_{2}

*(D) 2 (E)*

_{x}*3*

_{x }**Feedback **

A. Incorrect!
Take the limit of the average rate of change as h goes to 0.

B. Correct!

By using the definition of instantaneous rate of change, this is the correct answer.

C. Incorrect!

Take the limit of the average rate of change as h goes to 0.

D. Incorrect!

Take the limit of the average rate of change as h goes to 0.

E. Incorrect!

Take the limit of the average rate of change as h goes to 0.

**Solution **

The instantaneous rate of change is given by:

Pick the answer (4) Go back to review the core concept tutorial as needed.

**Question **

Question 9

## .

If

_{f x}_{( )}

_{= −}

_{5}

*2*

_{x}_{+ , find the instantaneous rate of change at }

_{2}

_{x}_{= . }

_{1}

(A) 10−
(B) 5−
(C) 2
(D) 1
(E) 10− *x*

**Feedback **

A. Correct!
This is the correct answer. Take the limit of the average rate of change as *h* goes to 0.

B. Incorrect!

Take the limit of the average rate of change as *h* goes to 0.

C. Incorrect!

Take the limit of the average rate of change as *h* goes to 0.

D. Incorrect!

Take the limit of the average rate of change as *h* goes to 0.

E. Incorrect!

Take the limit of the average rate of change as *h* goes to 0.

**Solution **

The instantaneous rate of change is given by:

2 2
0 0
2 2 2
0
2 2 2
0
2
0
0
( ) ( ) 5( ) 2 ( 5 2)
lim lim
5( 2 ) 2 5 2
lim
5 10 5 2 5 2
lim
10 5
lim
lim( 10 5 )
10
*h* *h*
*h*
*h*
*h*
*h*
*f x h* *f x* *x* *h* *x*
*h* *h*
*x* *xh h* *x*
*h*
*x* *xh* *h* *x*
*h*
*xh* *h*
*h*
*x* *h*
*x*
→ →
→
→
→
→
+ − _{=} − + + − − +
− + + + + −
=
− − − + + −
=
− −
=
= − −
= −

**Question **

Question 10. An auto parts factory operates for *x *hours a day and produces _{f x}_{( ) 500}_{=} _{x}_{−}_{2}* _{x}*2

_{ parts. }

When during the day will the factory be producing parts at the rate of 448 parts per hour? (A) 4 hours into the day

(B) 13 hours into the day (C) 5 hours into the day (D) 10 hours into the day

(E) it will never reach 448 parts/hour

**Feedback **

A. Incorrect!

Find the instantaneous rate of change and set it equal to 448.

B. Correct!

Find the instantaneous rate of change and set it equal to 448. Solve for *x*.

C. Incorrect!

Find the instantaneous rate of change and set it equal to 448.

D. Incorrect!

Find the instantaneous rate of change and set it equal to 448.

E. Incorrect!

Find the instantaneous rate of change and set it equal to 448.

**Solution **

The instantaneous rate of change is given by

2 2
0 0
2 2 2
0
2
0
0
( ) ( ) 500( ) 2( ) (500 2 )
lim lim
500 500 2 4 2 500 2
lim
500 4 2
lim
lim(500 4 2 )
500 4
*h* *h*
*h*
*h*
*h*
*f x* *h* *f x* *x* *h* *x* *h* *x* *x*
*h* *h*
*x* *h* *x* *xh* *h* *x* *x*
*h*
*h* *xh* *h*
*h*
*x* *h*
*x*
→ →
→
→
→
+ − _{=} + − + − −
+ − − − − +
=
− −
=
= − −
= −