Question

What is the average rate of change of the function over the interval x = 0 to x = 8?

f(x)=3x+4
2x+7

Enter your answer, as a fraction, in the box.

Answer 1

Answer: 13/161

Explanation:

Answer 2

`\boxed{ARC=(13)/(161)}`

Explanation:

For a nonlinear graph whose slope changes at each point, the average rate of change between any two points
`(x_(1),f(x_(1)) \ and \ (x_(2),f(x_(2))` is defined as the slope of the line through the two points. We call the line through the two points the secant line and its slope is denoted as
`m_(sec)`, so:

`ARC=(f(x_(2))-f(x_(1)))/(x_(2)-x_(1)) =(Change \ in \ y)/(Change \ in \ x)=m_(sec)`

`If \ x_(1)=0 \ then: \\ \\ f(x_(1))=(3x_(1)+4)/(2x_(1)+7) \ \therefore f(x_(1))=(3(0)+4)/(2(0)+7) \ \therefore f(x_(1))=(4)/(7)`

`If \ x_(2)=8 \ then: \\ \\ f(x_(2))=(3x_(2)+4)/(2x_(2)+7) \ \therefore f(x_(2))=(3(8)+4)/(2(8)+7) \ \therefore f(x_(2))=(28)/(23)`

Then:

`ARC=((28)/(23)-(4)/(7))/(8-0) \\ \\ ARC=((104)/(161))/(8) \\ \\ ARC=(104)/(8* 161) \therefore \boxed{ARC=(13)/(161)}`