Q. 68 - Calculus

Question

For each function f and interval [a, b] given in Exercises 68–73, find a real number c ∈ (a, b) such that f(c) is the average value of f on [a, b]. Then use a graph of f to verify that your answer is reasonable.

$f\left(x\right)=3x+1, [a, b] = [2, 6]$

Step-by-Step Solution

Answer

The real number is $4,$ and the average value is $13$ and it is verified from the graph.

Step 1. Given Information.

The given function and the interval is $f\left(x\right)=3x+1, [a, b] = [2, 6].$

Step 2. Finding a real number c ∈ (a, b).

To find a real number c ∈ (a, b) such that f(c) is the average value of f on [a, b]. We will use the average value formula: $\frac{1}{b-a}{\int }_a^{b}f\left(x\right) dx.$

$$\begin{gathered} \frac{1}{b-a}{\int }_a^{b}f\left(x\right) dx \\ =\frac{1}{6-2}{\int }_2^6(3x+1) dx \\ =\frac{1}{4}{\left[\frac{3x^2}{2}+x\right]}_2^6 \\ =\frac{1}{4}\left[\frac{3{\left(6\right)}^2}{2}+6-\left(\frac{3{\left(2\right)}^2}{2}+2\right)\right] \\ =\frac{1}{4}\left[54+6-8\right] \\ =\frac{1}{4}\left(52\right) \\ =13 \end{gathered}$$

Thus, the average value is $13.$

Now,

$$\begin{gathered} f\left(c\right)=13 \\ 3\left(c\right)+1=13 \\ 3c=12 \\ c=4 \end{gathered}$$

The real number is $4.$

Step 3. Verification.

By using the graphing utility, the graph of f  is

From the graph, we can depict that $f\left(c\right) \mathrm{is} 13.$ Thus, the real number is $4, 4\in \left[2,6\right].$Hence, the answer is right.