Q. 53 - Calculus

Question

For each function $f$ and interval [a, b] in Exercises 53–55, approximate the average value of $f$ from $x=a to x=b$, using a sample size of at least eight.

$f\left(x\right)=2^x,[-1,1]$ 

Step-by-Step Solution

Answer

The average value is $\approx 1.981$ .

Step 1. Given information.

The given function and interval is $f\left(x\right)=2^x,[-1,1]$.

Step 2. Value of Δ x .

The value of $\Delta x$ will be,

$$\begin{gathered} =\frac{1+1}{8} \\ =\frac{1}{4} \end{gathered}$$

Step 3. Average value.

The average value will be,

$$\begin{gathered} \left(f\left(x_0\right)+\dots f\left(x_n\right)\right)\Delta x \\ =\frac{2^{-1}+2^{-0.75}+2^{-0.5}+2^{-0.25}+2^0+2^{0.25}+2^{0.5}+2^{0.75}}{4} \\ \approx 1.981 \end{gathered}$$