Question

what is the average rate of change for the function ( f(x) = 3x^2 - 5 ) on the interval ( -3 leq x leq 1 )? write the average rate of change in the box.

Explanation:

Step1: Recall the average rate of change formula

The average rate of change of a function \( f(x) \) on the interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\). Here, \( a=-3 \) and \( b = 1 \), and \( f(x)=3x^{2}-5 \).

Step2: Calculate \( f(-3) \)

Substitute \( x=-3 \) into \( f(x) \):
\( f(-3)=3(-3)^{2}-5=3\times9 - 5=27 - 5 = 22 \)

Step3: Calculate \( f(1) \)

Substitute \( x = 1 \) into \( f(x) \):
\( f(1)=3(1)^{2}-5=3 - 5=-2 \)

Step4: Apply the average rate of change formula

Now, use the formula \(\frac{f(b)-f(a)}{b - a}\) with \( a=-3 \), \( b = 1 \), \( f(-3)=22 \) and \( f(1)=-2 \):
\(\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2 - 22}{1 + 3}=\frac{-24}{4}=-6\)

Answer:

\(-6\)