Using the Mean Value Theorem

#5.1 Using the Mean Value Theorem

In the previous unit, we learned all about applying derivatives to different real-world contexts. What else are derivatives useful for? Turns out, we can also use derivatives to determine and analyze the behaviors of functions! 👀

#📈  Mean Value Theorem

The Mean Value Theorem states that if a function f is continuous over the interval $[a, b]$ and differentiable over the interval $(a, b)$, then there exists a point $c$ within that open interval $(a,b)$ where the instantaneous rate of change of the function at $c$ equals the average rate of change of the function over the interval $(a, b)$.

In other words, if a function f is continuous over the interval $[a, b]$ and differentiable over the interval $(a, b)$, there exists some $c$ on $(a,b)$ such that $f'(c)=\frac{f(b)-f(a)}{b-a}$.

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Visual representation of mean value theorem

Image Courtesy of Sumi Vora and Ethan Bilderbeek

Yet another way to phrase this theorem is that if the stated conditions of continuity and differentiability are satisfied, there is a point where the slope of the tangent line is equivalent to the slope of the secant line between $a$ and $b$.

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