Question

given the function f(x) = -x² + x + 3, determine the average rate of change of the function over the interval 0 ≤ x ≤ 7.

Explanation:

Step1: Recall the formula for average rate of change

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, \( a = 0 \) and \( b = 7 \).

Step2: Calculate \( f(0) \)

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

Step3: Calculate \( f(7) \)

Substitute \( x = 7 \) into \( f(x)=-x^{2}+x + 3 \):
\( f(7)=-(7)^{2}+7 + 3=-49 + 7+3=-39 \)

Step4: Calculate the average rate of change

Using the formula \(\frac{f(7)-f(0)}{7 - 0}\), substitute \( f(7)=-39 \) and \( f(0)=3 \):
\(\frac{-39 - 3}{7-0}=\frac{-42}{7}=-6\)

Answer:

\(-6\)