Question

find average rate of change from equation
question
given the function $f(x)=-x^{2}+x+3$, determine the average rate of change of the function over the interval $0leq xleq7$.
answer
attempt 1 out of 2

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 0\) and \(b=7\).

Step2: Find \(f(0)\)

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

Step3: Find \(f(7)\)

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 \(\frac{f(b)-f(a)}{b - a}\), substitute \(a = 0\), \(b = 7\), \(f(0)=3\), and \(f(7)=-39\).
\(\frac{f(7)-f(0)}{7-0}=\frac{-39 - 3}{7}=\frac{-42}{7}=-6\)

Answer:

\(-6\)