Question

find the average rate of change of $f(x) = 4x^2 - 3$ on the interval $3, b$. your answer will be an expression involving $b$

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) on the interval \([a, b]\) is given by \(\frac{f(b)-f(a)}{b - a}\). Here, \( a = 3 \) and the function is \( f(x)=4x^{2}-3 \).

Step2: Calculate \( f(3) \)

Substitute \( x = 3 \) into \( f(x) \):
\( f(3)=4(3)^{2}-3=4\times9 - 3=36 - 3 = 33 \)

Step3: Calculate \( f(b) \)

Substitute \( x = b \) into \( f(x) \):
\( f(b)=4b^{2}-3 \)

Step4: Substitute into the average rate of change formula

Using the formula \(\frac{f(b)-f(3)}{b - 3}\), substitute \( f(b)=4b^{2}-3 \) and \( f(3) = 33 \):
\[

$$\begin{align*} \frac{(4b^{2}-3)-33}{b - 3}&=\frac{4b^{2}-3 - 33}{b - 3}\\ &=\frac{4b^{2}-36}{b - 3}\\ &=\frac{4(b^{2}-9)}{b - 3}\\ &=\frac{4(b - 3)(b + 3)}{b - 3}\\ \end{align*}$$

\]
Since \( b
eq3 \) (otherwise the denominator is zero), we can cancel out \( b - 3 \) from the numerator and the denominator:
\( = 4(b + 3)=4b+12 \)

Answer:

\( 4b + 12 \)