So, we want to find the average rate of change of:
[tex]f(x)=-3x^2+5[/tex]
From 3 to 7.
First, remember that the average rate of change of a function f(x) from a to b, can be found using the following formula:
[tex]A=\frac{f(b)-f(a)}{b-a}[/tex]
Where:
f(b) is the value of the function at the point b.
f(a) is the value of the function at the point a.
b is the final point
a is the initial point.
In our problem, we're given that:
[tex]\begin{gathered} a=3 \\ b=7 \\ f(a)=-3(3)^2+5\to-3(9)+5\to-27+5=-22 \\ f(b)=-3(7)^2+5\to-147+5\to-142 \end{gathered}[/tex]
Now, let's just replace these values in the previous formula:
[tex]A=\frac{-142-(-22)}{7-3}=\frac{-142+22}{4}=\frac{-120}{4}=-30[/tex]
Therefore, the average rate of change is -30.
exercise:
Find the average rate of change of f(x):
[tex]f(x)=-2x^2+1[/tex]
From 4 to 6.
