Q. 57 - Calculus

Question

For each function f and interval [a, b] in Exercises 56–67, use definite integrals and the Fundamental Theorem of Calculus to find the exact average value of f from x = a to x = b. Then use a graph of f to verify that your answer is reasonable. 

$f\left(x\right)=3x+1, [0, 4]$

Step-by-Step Solution

Answer

The exact average value of f is $7$ and it is verified from the graph of f.

The graph is

Step 1. Given Information.

The given function and interval is $f\left(x\right)=3x+1, [0, 4].$

Step 2. Finding the exact average value.

To find the exact average value of f from $x=a to x=b,$ we will use the formula: $\frac{1}{b-a}{\int }_a^{b}f\left(x\right) dx.$

Thus,

$$\begin{gathered} \frac{1}{b-a}{\int }_a^{b}f\left(x\right) dx \\ =\frac{1}{4-0}{\int }_0^4(3x+1) dx \\ =\frac{1}{4}{\left[\frac{3x^2}{2}+x\right]}_0^4 \\ =\frac{1}{4}\left[\frac{3{\left(4\right)}^2}{2}+4-\left(\frac{3{\left(0\right)}^2}{2}+0\right)\right] \\ =\frac{1}{4}\left[\frac{48}{2}+4-0\right] \\ =\frac{1}{4}\left[\frac{56}{2}\right] \\ =7 \end{gathered}$$

Step 3. Verification.

By using the graphing utility the graph of f is 

 

From the graph, we can depict that the average value is $7.$ Thus, the answer is right.