Question

what is the average rate of change of the function ( h(x) = 8 - 3x^2 ) on the interval from ( x = 2 ) to ( x = 5 )?

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function \( f(x) \) over the interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\). Here, \( a = 2 \), \( b = 5 \), and \( f(x)=8 - 3x^{2} \).

Step2: Calculate \( f(5) \)

Substitute \( x = 5 \) into \( f(x) \):
\( f(5)=8 - 3(5)^{2}=8 - 3\times25 = 8 - 75=-67 \).

Step3: Calculate \( f(2) \)

Substitute \( x = 2 \) into \( f(x) \):
\( f(2)=8 - 3(2)^{2}=8 - 3\times4 = 8 - 12=-4 \).

Step4: Apply average rate of change formula

Using the formula \(\frac{f(5)-f(2)}{5 - 2}\), substitute the values:
\(\frac{-67-(-4)}{3}=\frac{-67 + 4}{3}=\frac{-63}{3}=-21\).

Answer:

\(-21\)