Q. 19 - Calculus

Question

State the Mean Value Theorem for Integrals, and explain what this theorem means. Include a picture with your explanation. What does the Mean Value Theorem for Integrals have to do with average values? 

Step-by-Step Solution

Answer

The height of the function is equal to its absolute value at $x=c$.

 

Step 1. Given Information

Mean Value Theorem for Integrals.

Step 2. Explanation

The Mean Value Theorem states that, 

If $f$ is a continuous function on $\left[a,b\right]$, then there exists some $c\in \left(a,b\right)$ such that

$f\left(c\right)=\frac{1}{b-a}{\int }_a^{b}f\left(x\right)dx$

At this point$c$, the slope of the function $f$is equal to the average rate of change of $f$ on$\left[a,b\right]$.

It is actually at $x=c$ where the height of the function is equal to its absolute value.