Q. 62 - Precalculus Enhanced with Graphing Utilities

Question

Find the average rate of change of $f\left(x\right)=-x^3+1$

(a) From $0$ to $2$

(b) From $1$ to $3$

(c) From $-1$ to $1$

Step-by-Step Solution

Answer

(a) The average rate of change is $-4$.

(b) The average rate of change is $-13.$

(c) The average rate of change is $-1.$

Step 1. Given information.

We have:

$f\left(x\right)=-x^3+1$

Part a. Step 1. Find the average rate of change from 0 to 2 .

The function is $f\left(x\right)=-x^3+1$.

Average rate of change is given by: 

$\frac{∆y}{∆x}=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Here, $a=0$ and $b=2:$

$$\begin{gathered} \frac{∆y}{∆x}=\frac{f\left(2\right)-f\left(0\right)}{2-0} \\ \frac{∆y}{∆x}=\frac{-{\left(2\right)}^3+1-\left(-{\left(0\right)}^3+1\right)}{2} \\ =-4 \end{gathered}$$

Part b. Step 1. Find the average rate of change from 1 to 3 .

The function is $f\left(x\right)=-x^3+1$.

Average rate of change is given by: 

$\frac{∆y}{∆x}=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Here, $a=1$ and $b=3$:

$$\begin{gathered} \frac{∆y}{∆x}=\frac{f\left(3\right)-f\left(1\right)}{3-1} \\ \frac{∆y}{∆x}=\frac{-{\left(3\right)}^3+1-\left(-{\left(3\right)}^3+1\right)}{2} \\ =-13 \end{gathered}$$

Part c. Step 1. Find the average rate of change from - 1 to 1 .

The function is $f\left(x\right)=-x^3+1$.

Average rate of change is given by: 

$\frac{∆y}{∆x}=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Here, $a=-1$ and $a=1$:

$$\begin{gathered} \frac{∆y}{∆x}=\frac{f\left(3\right)-f\left(1\right)}{-1-1} \\ \frac{∆y}{∆x}=\frac{-{\left(1\right)}^3+1-\left(-{\left(-1\right)}^3+1\right)}{-2} \\ =-1 \end{gathered}$$