To find the average rate of change of the function [tex]\( f(x) = 5x^2 - 7 \)[/tex] on the interval [tex]\([2, t]\)[/tex], we need to follow these steps:
1. Evaluate the function at the endpoints of the interval:
- At [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 5(2)^2 - 7 = 5 \cdot 4 - 7 = 20 - 7 = 13
\][/tex]
- At [tex]\( x = t \)[/tex]:
[tex]\[
f(t) = 5t^2 - 7
\][/tex]
2. Apply the formula for the average rate of change:
The average rate of change of a function [tex]\( f \)[/tex] over an interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[
\frac{f(b) - f(a)}{b - a}
\][/tex]
In this case, the interval is [tex]\([2, t]\)[/tex], so [tex]\( a = 2 \)[/tex] and [tex]\( b = t \)[/tex]. Hence, the average rate of change is:
[tex]\[
\frac{f(t) - f(2)}{t - 2}
\][/tex]
3. Substitute the function values into the formula:
We already have:
[tex]\[
f(2) = 13 \quad \text{and} \quad f(t) = 5t^2 - 7
\][/tex]
Therefore, the average rate of change becomes:
[tex]\[
\frac{5t^2 - 7 - 13}{t - 2}
\][/tex]
4. Simplify the expression:
Combine the constants in the numerator:
[tex]\[
5t^2 - 7 - 13 = 5t^2 - 20
\][/tex]
Thus, the formula for the average rate of change is:
[tex]\[
\frac{5t^2 - 20}{t - 2}
\][/tex]
So, the expression for the average rate of change of [tex]\( f(x) = 5x^2 - 7 \)[/tex] on the interval [tex]\([2, t]\)[/tex] is:
[tex]\[
\boxed{\frac{5t^2 - 20}{t - 2}}
\][/tex]
