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$1$

Short 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-by-step solution

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:

localid="1646218831507" $\frac{∆y}{∆x}=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Here, $a=0$and $b=2:$

localid="1646217782079" $$\begin{gathered} \frac{∆y}{∆x}=\frac{f\left(2\right)-f\left(0\right)}{2-0} \\ \frac{∆y}{∆x}=\frac{-{\left(2\right)}^3+1-\left(-{\left(0\right)}^3+1\right)}{2} \\ =-4 \end{gathered}$$

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:

localid="1646217897240" $\frac{∆y}{∆x}=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Here, $a=1$and $b=3$:

localid="1646217942520" $$\begin{gathered} \frac{∆y}{∆x}=\frac{f\left(3\right)-f\left(1\right)}{3-1} \\ \frac{∆y}{∆x}=\frac{-{\left(3\right)}^3+1-\left(-{\left(3\right)}^3+1\right)}{2} \\ =-13 \end{gathered}$$

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 $a=1$:

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