# The average rate of change of function f between x = a and x = b is slope of which line between ( a , f ( a ) ) and ( b , f ( b ) ) .

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 2.4, Problem 4E

(a)

To determine

## To fill: The average rate of change of function f between x=a and x=b is slope of which line between (a,f(a)) and (b,f(b)) .

Expert Solution

The average rate of change of a function is the slope of the secant line between (a,f(a)) and (b,f(b)) .

### Explanation of Solution

Given:

Average rate of change of function y=f(x) between x=a and x=b .

Averagerateofchange=ChangeinyChangeinx=f(b)f(a)ba

So, the average rate of change of a function is the slope of the secant line between (a,f(a)) and (b,f(b)) .

(b)

To determine

### To fill: The average rate of change of linear function f(x)=3x+5 between any two point.

Expert Solution

The average rate of change of linear function f(x)=3x+5 between any two point is 3.

### Explanation of Solution

Formula used:

Average rate of change of function y=f(x) between x=a and x=b .

Averagerateofchange=ChangeinyChangeinx=f(b)f(a)ba (1)

Given:

The given function is,

f(x)=3x+5 . (2)

Calculation:

Suppose the two point between which the average rate of change of the linear function is to be calculated is x=x1 and x=x2 .

Substitute x1 for a and x2 for b in equation (1)

Averagerateofchange=f(x1)f(x2)x1x2 (3)

For, f(x1) substitute x1 for x in equation (1)

f(x1)=3x1+5

For, f(x2) substitute x2 for x in equation (1)

f(x2)=3x2+5

Substitute 3x1+5 for f(x1) and 3x2+5 for f(x2) in equation (3), to calculate the average rate of change.

Averagerateofchange=3x1+5(3x2+5)(x1x2)=3(x1x2)+55(x1x2)=3(x1x2)(x1x2)=3

Therefore, the average rate of change of linear function f(x)=3x+5 between any two point is 3.

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