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Workbook for Stewart’s "Calculus"

Cory Wilson

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Section 3.2 (X) The Mean Value Theorem

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Subsection 3.2.1 Before Class

We don’t cover this section, but feel free to read this material!
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Subsubsection 3.2.1.1 Rolle’s Theorem

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Example 3.2.2.
Rolle’s Theorem has three hypotheses−conditions upon which the conclusion depends.
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    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.
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    If we remove the second hypothesis, does the theorem still hold? Give an argument why or why not, and include a picture if possible.
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    If the third hypothesis is removed, what then? Justify as before.
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Subsubsection 3.2.1.2 Mean Value Theorem

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Example 3.2.4.
Restate the mean value theorem in words.
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Example 3.2.5.
Let f(x)=x2+x on the interval [0,5]. Find the value(s) of c which satisfy the Mean Value Theorem.
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Subsection 3.2.2 Pre-Class Activities

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Example 3.2.6.

Let f(x)=tan⁡x. Show that f(0)=f(π), but that there is no number c in (0,π) such that f′(c)=0. Why does this not contradict Rolle’s Theorem?
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Example 3.2.7.

Verify that f(x)=x3−2x2−4x+2 satisfies the hypotheses of Rolle’s Theorem on [−2,2], then find all numbers c that satisfy the conclusion of Rolle’s Theorem.
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Example 3.2.8.

Draw the graph of a function that is continuous on [0,8], with f(0)=1 and f(8)=4, but that does not satisfy the conclusion of the Mean Value Theorem on [0,8].
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Subsection 3.2.3 In Class

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Example 3.2.9.

Use Rolle’s Theorem to prove that the equation x3+x−2=0 has exactly one real root.
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Example 3.2.10.

Show that the equation 2x+cos⁡x=0 has exactly one real root.
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Example 3.2.11.

Suppose f(0)=−3 and f′(x)≤5 for all values of x. How large can f(2) be?
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Example 3.2.12.

Suppose that 3≤f′(x)≤5 for all values of x. Show that 18≤f(8)−f(2)≤30.
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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/h2.
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Example 3.2.14.

Find the number c that satisfies the conclusion of the Mean Value Theorem for the function f(x)=x3−2x on the interval [−2,2].
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Example 3.2.15.

Show that if f′(x)=0 on the interval (a,b), then f(x)=c on (a,b) for some constant c.
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Example 3.2.16.

Suppose that f is an odd function that is differentiable everywhere. Show that for every positive number b, there exists some number c in (−b,b) such that f′(c)=f(b)b.
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Subsection 3.2.4 After Class Activities

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Example 3.2.17.

Show that the equation 2x−1−sin⁡x=0 has exactly one real root.
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Example 3.2.18.

Show that the equation x3−15x+C=0 has at most one root in the interval [−2,2].
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Example 3.2.19.

Does there exist a function f such that f(0)=−1, f(2)=4, and f′(x)≤2 for all x? Why or why not? Justify your answer using a theorem from this section.
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Example 3.2.20.

In the Mean Value Theorem, we assume that f is continuous on [a,b] and differentiable on (a,b). Since differentiablity implies continuity, why do we have to assume continuity on [a,b]?
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Example 3.2.21.

Use the Mean Value Theorem to show that |cos⁡x−cos⁡y|≤|x−y| for any choice of x and y.
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