Question

True or false14. The average rate of change of the function f(x)=x2-1 as x changes from 1 to 7 is 7

Answer 1

Average rate of change

Initial explanation

We want to find the average rate of change as x changes from 1 to 7 is 7 of the function:

`f(x)=x^2-1`

This is:

A rate is always a division. The rate of change makes reference to the division between the change of the function in the y-axis, Δy, and its change in the x-axis, Δx:

`rate=(\Delta y)/(\Delta x)`

STEP 1: change Δy

In order to find Δy, we must use the values of the function at x=1 and x=7.

At x=1:

`\begin{gathered} f(x)=x^2-1 \\ \downarrow\text{replacing x by 1} \\ f(1)=1^2-1=1-1=0 \\ f(1)=0 \end{gathered}`

At x=7:

`\begin{gathered} f(x)=x^2-1 \\ \downarrow\text{replacing x by 7} \\ f(7)=7^2-1=49-1=48 \\ f(7)=48 \end{gathered}`

Then Δy=48 - 0 = 48

STEP 2: change Δx

from x=1 to x=7 the change of x is 6:

Δx = 7 - 1 = 6

STEP 3: rate of change

We have that, Δy = 48 and Δx=6

then,

`rate=(\Delta y)/(\Delta x)=(48)/(6)=8`

Then, the rate of change is 8 instead of 7.

ANSWER: False