Question

find the average rate of change of g(x) = 3x³ − 5x² between x = −0.2 and x = 0.2. enter the exact answer. the average rate of change of g(x) between x = −0.2 and x = 0.2 is

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( g(x) \) between \( x = a \) and \( x = b \) is given by \( \frac{g(b)-g(a)}{b - a} \). Here, \( a=- 0.2\) and \( b = 0.2\).

Step2: Calculate \( g(-0.2) \)

Substitute \( x=-0.2\) into \( g(x)=3x^{3}-5x^{2} \):
\[

$$\begin{align*} g(-0.2)&=3\times(-0.2)^{3}-5\times(-0.2)^{2}\\ &=3\times(-0.008)-5\times(0.04)\\ &=- 0.024 - 0.2\\ &=-0.224 \end{align*}$$

\]

Step3: Calculate \( g(0.2) \)

Substitute \( x = 0.2\) into \( g(x)=3x^{3}-5x^{2} \):
\[

$$\begin{align*} g(0.2)&=3\times(0.2)^{3}-5\times(0.2)^{2}\\ &=3\times(0.008)-5\times(0.04)\\ &=0.024-0.2\\ &=- 0.176 \end{align*}$$

\]

Step4: Calculate the average rate of change

Using the formula \( \frac{g(b)-g(a)}{b - a} \) with \( a=-0.2\), \( b = 0.2\), \( g(a)=-0.224\) and \( g(b)=-0.176\):
\[

$$\begin{align*} \frac{g(0.2)-g(-0.2)}{0.2-(-0.2)}&=\frac{-0.176-(-0.224)}{0.2 + 0.2}\\ &=\frac{-0.176 + 0.224}{0.4}\\ &=\frac{0.048}{0.4}\\ &=- 0.2 \end{align*}$$

\]

Answer:

-0.2