Question

3 formula 21 is find the average rate of change for ( f(x) = x^2 - x^3 ) from -3 to 1. round your answer to one decimal place.

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) from \( a \) to \( b \) is given by \( \frac{f(b) - f(a)}{b - a} \). Here, \( a=-3 \) and \( b = 1 \), and \( f(x)=x^{2}-x^{3} \).

Step2: Calculate \( f(1) \)

Substitute \( x = 1 \) into \( f(x) \):
\( f(1)=1^{2}-1^{3}=1 - 1=0 \)

Step3: Calculate \( f(-3) \)

Substitute \( x=-3 \) into \( f(x) \):
\( f(-3)=(-3)^{2}-(-3)^{3}=9-(-27)=9 + 27 = 36 \)

Step4: Calculate the average rate of change

Using the formula \( \frac{f(b)-f(a)}{b - a} \), substitute \( a=-3 \), \( b = 1 \), \( f(1)=0 \) and \( f(-3)=36 \):
\( \frac{f(1)-f(-3)}{1-(-3)}=\frac{0 - 36}{1 + 3}=\frac{-36}{4}=-9.0 \) (already to one decimal place)

Answer:

\(-9.0\)