Question

given the function ( f(x) = -x^2 - 8x + 21 ), determine the average rate of change of the function over the interval ( -10 leq x leq 0 ).

Explanation:

Step1: Recall the 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, \( a=-10 \) and \( b = 0 \).

Step2: Calculate \( f(0) \)

Substitute \( x = 0 \) into \( f(x)=-x^{2}-8x + 21 \):
\( f(0)=-(0)^{2}-8(0)+21=21 \)

Step3: Calculate \( f(-10) \)

Substitute \( x=-10 \) into \( f(x)=-x^{2}-8x + 21 \):
\( f(-10)=-(-10)^{2}-8(-10)+21=-100 + 80+21=-19 \)

Step4: Apply the average rate of change formula

Now, use the formula \(\frac{f(b)-f(a)}{b - a}\) with \( a=-10 \), \( b = 0 \), \( f(a)=-19 \), and \( f(b)=21 \):
\(\frac{f(0)-f(-10)}{0-(-10)}=\frac{21-(-19)}{0 + 10}=\frac{21 + 19}{10}=\frac{40}{10}=4\)

Answer:

4