Sure, let's find the average rate of change of the function [tex]\( g(t) = -(t-1)^2 + 5 \)[/tex] over the interval [tex]\([-4, 5]\)[/tex].
### Step 1: Evaluate [tex]\( g(t) \)[/tex] at [tex]\( t = -4 \)[/tex]
We need to substitute [tex]\( t = -4 \)[/tex] into the function:
[tex]\[
g(-4) = -(-4 - 1)^2 + 5
\][/tex]
First, calculate [tex]\((-4 - 1)\)[/tex]:
[tex]\[
-4 - 1 = -5
\][/tex]
Then, square this result:
[tex]\[
(-5)^2 = 25
\][/tex]
Now, negate that result:
[tex]\[
-25
\][/tex]
Finally, add 5:
[tex]\[
-25 + 5 = -20
\][/tex]
So, [tex]\( g(-4) = -20 \)[/tex].
### Step 2: Evaluate [tex]\( g(t) \)[/tex] at [tex]\( t = 5 \)[/tex]
Next, substitute [tex]\( t = 5 \)[/tex] into the function:
[tex]\[
g(5) = -(5 - 1)^2 + 5
\][/tex]
First, calculate [tex]\(5 - 1\)[/tex]:
[tex]\[
5 - 1 = 4
\][/tex]
Then, square this result:
[tex]\[
4^2 = 16
\][/tex]
Now, negate that result:
[tex]\[
-16
\][/tex]
Finally, add 5:
[tex]\[
-16 + 5 = -11
\][/tex]
So, [tex]\( g(5) = -11 \)[/tex].
### Step 3: Calculate the Average Rate of Change
The average rate of change of the function over the interval [tex]\([-4, 5]\)[/tex] is given by the formula:
[tex]\[
\frac{g(t_2) - g(t_1)}{t_2 - t_1}
\][/tex]
Substitute [tex]\(t_1 = -4\)[/tex] and [tex]\(t_2 = 5\)[/tex]:
[tex]\[
\frac{g(5) - g(-4)}{5 - (-4)}
\][/tex]
We already found [tex]\(g(5) = -11\)[/tex] and [tex]\(g(-4) = -20\)[/tex]:
[tex]\[
\frac{-11 - (-20)}{5 - (-4)}
\][/tex]
Simplify the numerator [tex]\(-11 - (-20)\)[/tex]:
[tex]\[
-11 + 20 = 9
\][/tex]
Simplify the denominator [tex]\(5 - (-4)\)[/tex]:
[tex]\[
5 + 4 = 9
\][/tex]
Thus, the fraction becomes:
[tex]\[
\frac{9}{9} = 1
\][/tex]
So, the average rate of change of [tex]\( g(t) \)[/tex] over the interval [tex]\([-4, 5]\)[/tex] is [tex]\( 1.0 \)[/tex].
### Final Answer
[tex]\[
\boxed{1.0}
\][/tex]
