To find the average rate of change of the function [tex]\( f(x) = 9x^2 - 3 \)[/tex] on the interval [tex]\([4, t]\)[/tex], we can follow these steps:
1. Calculate [tex]\( f(t) \)[/tex]:
The function [tex]\( f(x) \)[/tex] is given as [tex]\( f(x) = 9x^2 - 3 \)[/tex]. For [tex]\( x = t \)[/tex], we substitute [tex]\( t \)[/tex] into the function:
[tex]\[
f(t) = 9t^2 - 3
\][/tex]
2. Calculate [tex]\( f(4) \)[/tex]:
Next, we substitute [tex]\( x = 4 \)[/tex] into the function:
[tex]\[
f(4) = 9(4)^2 - 3 = 9 \times 16 - 3 = 144 - 3 = 141
\][/tex]
3. Determine the average rate of change:
The average rate of change of a function on the interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[
\frac{f(b) - f(a)}{b - a}
\][/tex]
For our interval [tex]\([4, t]\)[/tex], [tex]\([a, b]\)[/tex] translates to [tex]\([4, t]\)[/tex]. Therefore, we substitute [tex]\( a = 4 \)[/tex] and [tex]\( b = t \)[/tex]:
[tex]\[
\frac{f(t) - f(4)}{t - 4}
\][/tex]
4. Substitute the values of [tex]\( f(t) \)[/tex] and [tex]\( f(4) \)[/tex] into the expression:
We already calculated [tex]\( f(t) \)[/tex] as [tex]\( 9t^2 - 3 \)[/tex] and [tex]\( f(4) \)[/tex] as 141. Substituting these into the average rate of change formula, we get:
[tex]\[
\frac{f(t) - f(4)}{t - 4} = \frac{9t^2 - 3 - 141}{t - 4}
\][/tex]
5. Simplify the expression:
Simplify the numerator of the fraction:
[tex]\[
9t^2 - 3 - 141 = 9t^2 - 144
\][/tex]
Therefore, the average rate of change becomes:
[tex]\[
\frac{9t^2 - 144}{t - 4}
\][/tex]
Hence, the average rate of change of the function [tex]\( f(x) = 9x^2 - 3 \)[/tex] on the interval [tex]\([4, t]\)[/tex] is:
[tex]\[
\boxed{\frac{9t^2 - 144}{t - 4}}
\][/tex]
