Q. 20 - Calculus

Question

Explain what the Mean Value Theorem for Integrals has to do with the Intermediate Value Theorem.

Step-by-Step Solution

Answer

Value between $f\left(a\right)$and $f\left(b\right)$at some point within the interval in Intermediate Value Theorem is the point $c$of the Mean Value Theorem.

Step 1. Given Information

Mean Value Theorem for Integrals and Intermediate Value Theorem. 

Step 2. Explanation

The objective is to explain what the Mean Value Theorem for Integrals has to do with the intermediate Value Theorem.

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.

The Intermediate Theorem states that,

If $f$ is a continuous function on $\left[a,b\right]$, then the function takes the values $f\left(a\right)$ and $f\left(b\right)$ at each end of interval, then it may also take any value between $f\left(a\right)$ and $f\left(b\right)$ at some point within the interval.

And that point is the point $c$ of the Mean Value Theorem.