Question

What is the average rate of change of the function h(x) = -4x-7x² on the interval from x= 0 to x = 3?

Answer 1

To find the average rate of change of a function \( h(x) \) over a given interval \([a, b]\), we use the formula:

\[
\text{Average Rate of Change} = \frac{h(b) - h(a)}{b - a}
\]

For the function \( h(x) = -4x - 7x^2 \) and the interval from \( x = 0 \) to \( x = 3 \), we first need to evaluate \( h(0) \) and \( h(3) \).

1. Calculate \( h(0) \):
\[
h(0) = -4(0) - 7(0)^2 = 0
\]

2. Calculate \( h(3) \):
\[
h(3) = -4(3) - 7(3)^2 = -12 - 7 \times 9 = -12 - 63 = -75
\]

Now we will substitute \( h(0) \) and \( h(3) \) into the average rate of change formula:

\[
\text{Average Rate of Change} = \frac{h(3) - h(0)}{3 - 0} = \frac{-75 - 0}{3 - 0} = \frac{-75}{3} = -25
\]

Therefore, the average rate of change of the function \( h(x) = -4x - 7x^2 \) from \( x = 0 \) to \( x = 3 \) is

\[
\boxed{-25}
\]