Chapter 3: Q. 37 (page 248)

For each graph of f in Exercises 37–40, explain why f satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. Then approximate any values c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem.
[a, b] = [−3, 1]

Short Answer

Expert verified

The hypotheses of Rolle's theorem is satisfied because the graph of f appears to be continuous on $[- 3, 1]$ and differentiable on $(- 3, 1)$ . The values of cthat satisfies the conclusion of Rolle's theorem are $c \approx - 2.3, c = - 1, c \approx 0.3$

Step by step solution

Step 1. Given information.

Consider the graph of f for the interval $[- 3, 1]$ .

Step 2. Satisfy hypothesis of Rolle's theorem.

It can be observed that the given graph has no break, hole or gap in the interval $[- 3, 1]$ . So, the graph of f appears to be continuous on $[- 3, 1]$ .

It can be observed that the graph has no corner, no vertical line or no discontinuous point in the interval $(- 3, 1)$ . So, the graph of f appears to be differentiable on $(- 3, 1)$ .

Thus, the hypothesis of Rolle's theorem is satisfied.

Step 3. Find values of c.

From the graph, $f (- 3) = 0$ and $f (1) = 0$ .

$\Rightarrow f (- 3) = f (1)$

So, Rolle's theorem applies then there exists some $c \in (- 3, 1)$ such that $f' (c) = 0$ or the graph has horizontal tangent line.

From the graph, such values of c where the graph has horizontal tangent line are $c \approx - 2.3, c = - 1, c \approx 0.3$ .

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For each graph of f in Exercises 37–40, explain why f satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. Then approximate any values c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem.
[a, b] = [−3, 1]

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Satisfy hypothesis of Rolle's theorem.

Step 3. Find values of c.

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For each graph of f in Exercises 37–40, explain why f satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. Then approximate any values c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem.[a, b] = [−3, 1]

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Satisfy hypothesis of Rolle's theorem.

Step 3. Find values of c.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Pure Maths

Calculus

Theoretical and Mathematical Physics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

For each graph of f in Exercises 37–40, explain why f satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. Then approximate any values c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem.
[a, b] = [−3, 1]