Question

find average rate of change from equation
question
given the function ( h(x) = x^2 + 2x - 2 ), determine the average rate of change of the function over the interval ( -5 leq x leq 4 ).

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( h(x) \) over the interval \([a, b]\) is given by \(\frac{h(b) - h(a)}{b - a}\). Here, \( a=-5 \) and \( b = 4 \).

Step2: Calculate \( h(-5) \)

Substitute \( x=-5 \) into \( h(x)=x^{2}+2x - 2 \):
\[

$$\begin{align*} h(-5)&=(-5)^{2}+2\times(-5)-2\\ &=25-10 - 2\\ &=13 \end{align*}$$

\]

Step3: Calculate \( h(4) \)

Substitute \( x = 4 \) into \( h(x)=x^{2}+2x - 2 \):
\[

$$\begin{align*} h(4)&=4^{2}+2\times4-2\\ &=16 + 8-2\\ &=22 \end{align*}$$

\]

Step4: Calculate the average rate of change

Using the formula \(\frac{h(4)-h(-5)}{4-(-5)}\), substitute \( h(4) = 22 \) and \( h(-5)=13 \):
\[

$$\begin{align*} \frac{h(4)-h(-5)}{4-(-5)}&=\frac{22 - 13}{4 + 5}\\ &=\frac{9}{9}\\ &=1 \end{align*}$$

\]

Answer:

\( 1 \)