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?

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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( )= −7x+ , 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!

C. Correct! ( ) 7 5

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

D. Incorrect!

(E) Incorrect!

Solution

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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!

D. Incorrect!

E. Incorrect!

Solution

(3)

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

−

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Question

Question 4. Calculate the average rate of change of ( )f x = −ln(x+1) over [0,1] . (A) −lnx−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!

C. Correct!

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

D. Incorrect!

E. Incorrect!

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

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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

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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!

C. Incorrect!

D. Incorrect!

E. Correct!

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

Solution

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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!

D. Incorrect!

E. Incorrect!

One of the given answers is correct.

Solution

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Question

Question 8. Find a formula for the instantaneous rate of change of _{f x}_{( )}₌_x2_.

(A) _x2 (B) 2x (C) _x2₊₂_x (D) 2 (E) _x3

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!

D. Incorrect!

E. Incorrect!

Solution

The instantaneous rate of change is given by:

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Question

Question 9

.

If _{f x}_{( )}_{= −}₅_x2 _{+ , 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!

C. Incorrect!

D. Incorrect!

E. Incorrect!

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 → → → → → → + − ₌ − + + − − + − + + + + − = − − − + + − = − − = = − − = −

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Question

Question 10. An auto parts factory operates for x hours a day and produces _{f x}_{( ) 500}₌ _x₋₂_x2_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!

D. Incorrect!

E. Incorrect!

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 → → → → → + − ₌ + − + − − + − − − − + = − − = = − − = −