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Q. 62

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Calculus

Found in: Page 249

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Last night at 6 p.m., Linda got up from her blue easy chair. She did not return to her easy chair until she sat down again at 8 p.m. Let s(t) be the distance between Linda and her easy chair t minutes after 6 p.m. last night.
(a) Sketch a possible graph of s(t), and describe what Linda did between 6 p.m. and 8 p.m. according to your graph. (Questions to think about: Will Linda necessarily move in a continuous and differentiable way? What are good ranges for t and s?
(b) Use Rolle’s Theorem to show that at some point between 6 p.m. and 8 p.m., Linda’s velocity v(t) with respect to the easy chair was zero. Find such a place on the graph of s(t).

(a) The graph of s(t) is,

(b) The graph for the tangent line is,

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Part (a). Step 1. Given Information.

s(t) is the distance between Linda and her chair t minutes after 6 p.m.

Part (a). Step 2. The graph of s(t)

The graph is,

Part (b). Step 1. Given Information.

s(t) is the distance between Linda and her chair t minutes after 6 p.m.

Part (b). Step 2. The Rolle's Theorem.

Rolle's Theorem states that if $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , and if $f (a) = f (b) = 0$ , then there exits atleast one value $c \in (a, b)$ for which $f' (c) = 0$ .

The tangent line to the graph at $x = c$ be a horizontal line.

The graph showing the tangent line is,

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