Question

find the average rate of change for the function ( f(x) = \frac{1}{2}(4)^{x + 2}-1 ) on the interval ( -1 leq x leq 2 ).
a ( \frac{1}{42} )
b ( 42 )
c ( 126 )
d ( 128 )

Explanation:

Step1: Recall average rate formula

The average rate of change of $f(x)$ on $[a,b]$ is $\frac{f(b)-f(a)}{b-a}$.
Here, $a=-1$, $b=2$, $f(x)=\frac{1}{2}(4)^{x+2}-1$.

Step2: Calculate $f(2)$

Substitute $x=2$ into $f(x)$:

$$\begin{align*} f(2)&=\frac{1}{2}(4)^{2+2}-1\\ &=\frac{1}{2}(4^4)-1\\ &=\frac{1}{2}(256)-1\\ &=128-1\\ &=127 \end{align*}$$

Step3: Calculate $f(-1)$

Substitute $x=-1$ into $f(x)$:

$$\begin{align*} f(-1)&=\frac{1}{2}(4)^{-1+2}-1\\ &=\frac{1}{2}(4^1)-1\\ &=\frac{1}{2}(4)-1\\ &=2-1\\ &=1 \end{align*}$$

Step4: Compute average rate

Substitute values into the formula:

$$\begin{align*} \text{Average rate}&=\frac{f(2)-f(-1)}{2-(-1)}\\ &=\frac{127-1}{3}\\ &=\frac{126}{3}\\ &=42 \end{align*}$$

Answer:

B 42