Calculus Workbook: (X) The Mean Value Theorem

Section 3.2 (X) The Mean Value Theorem

Subsection 3.2.1 Before Class

We don’t cover this section, but feel free to read this material!

Subsubsection 3.2.1.1 Rolle’s Theorem

Theorem 3.2.1. Rolle’s Theorem.

Let \(f\) be a function that satisfies the following three properties:

\(f\) is continuous on the closed interval \([a,b]\text{.}\)
\(f\) is differentiable on the open interval \((a,b)\text{.}\)
\(f(a) = f(b)\text{.}\)

Then, there is some number \(c\) in \((a,b)\) such that \(f'(c) = 0\text{.}\)

Example 3.2.2.

Rolle’s Theorem has three hypotheses\(-\)conditions upon which the conclusion depends.

If we remove the first hypothesis, does Rolle’s Theorem still hold? Give an argument why or why not, and include a picture if possible.
If we remove the second hypothesis, does the theorem still hold? Give an argument why or why not, and include a picture if possible.
If the third hypothesis is removed, what then? Justify as before.

Subsubsection 3.2.1.2 Mean Value Theorem

Theorem 3.2.3. Mean Value Theorem.

Let \(f\) be a function that satisfies the two conditions:

\(f\) is continuous on the closed interval \([a,b]\)
\(f\) is differentiable on the open interval \((a,b)\)

Then, there is some number \(c\) in \((a,b)\) such that \(f'(c) = \dfrac{f(b)-f(a)}{b-a}\)

Example 3.2.4.

Restate the mean value theorem in words.

Example 3.2.5.

Let \(f(x) = x^2 + x\) on the interval \([0,5]\text{.}\) Find the value(s) of \(c\) which satisfy the Mean Value Theorem.

Subsection 3.2.2 Pre-Class Activities

Example 3.2.6.

Let \(f(x) = \tan x\text{.}\) Show that \(f(0) = f(\pi)\text{,}\) but that there is no number \(c\) in \((0,\pi)\) such that \(f'(c) = 0\text{.}\) Why does this not contradict Rolle’s Theorem?

Example 3.2.7.

Verify that \(f(x) = x^3-2x^2-4x+2\) satisfies the hypotheses of Rolle’s Theorem on \([-2,2]\text{,}\) then find all numbers \(c\) that satisfy the conclusion of Rolle’s Theorem.

Example 3.2.8.

Draw the graph of a function that is continuous on \([0,8]\text{,}\) with \(f(0) = 1\) and \(f(8) = 4\text{,}\) but that does not satisfy the conclusion of the Mean Value Theorem on \([0,8]\text{.}\)

Subsection 3.2.3 In Class

Example 3.2.9.

Use Rolle’s Theorem to prove that the equation \(x^3 + x -2 = 0\) has exactly one real root.

Example 3.2.10.

Show that the equation \(2x + \cos x = 0\) has exactly one real root.

Example 3.2.11.

Suppose \(f(0) = -3\) and \(f'(x)\leq 5\) for all values of \(x\text{.}\) How large can \(f(2)\) be?

Example 3.2.12.

Suppose that \(3\leq f'(x)\leq 5\) for all values of \(x\text{.}\) Show that \(18\leq f(8)-f(2)\leq 30\text{.}\)

Example 3.2.13.

At 1:00pm, a car’s speedometer reads 30 mi/h. At 1:15pm, it reads 50 mi/h. Show that at some time between 1:00 and 1:15, the acceleration is exactly 80 mi/h\(^2\text{.}\)

Example 3.2.14.

Find the number \(c\) that satisfies the conclusion of the Mean Value Theorem for the function \(f(x) = x^3-2x\) on the interval \([-2,2]\text{.}\)

Example 3.2.15.

Show that if \(f'(x) = 0\) on the interval \((a,b)\text{,}\) then \(f(x) = c\) on \((a,b)\) for some constant \(c\text{.}\)

Example 3.2.16.

Suppose that \(f\) is an odd function that is differentiable everywhere. Show that for every positive number \(b\text{,}\) there exists some number \(c\) in \((-b,b)\) such that \(f'(c) = \dfrac{f(b)}{b}\text{.}\)

Subsection 3.2.4 After Class Activities

Example 3.2.17.

Show that the equation \(2x - 1 - \sin x = 0\) has exactly one real root.

Example 3.2.18.

Show that the equation \(x^3 - 15x + C = 0\) has at most one root in the interval \([-2,2]\text{.}\)

Example 3.2.19.

Does there exist a function \(f\) such that \(f(0) = -1\text{,}\) \(f(2) = 4\text{,}\) and \(f'(x)\leq 2\) for all \(x\text{?}\) Why or why not? Justify your answer using a theorem from this section.

Example 3.2.20.

In the Mean Value Theorem, we assume that \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\text{.}\) Since differentiablity implies continuity, why do we have to assume continuity on \([a,b]\text{?}\)

Example 3.2.21.

Use the Mean Value Theorem to show that \(|\cos x - \cos y| \leq |x - y|\) for any choice of \(x\) and \(y\text{.}\)

Workbook for Stewart’s "Calculus"

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