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Q. 11

Expert-verified

Calculus

Found in: Page 859

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Let $r (t) = (x (t), y (t), z (t)), t \in [a, b],$ be a vector-valued function, where a < b are real numbers and the functions x(t), y(t), and z(t) are continuous. Explain why the graph of r is contained in some sphere centered at the origin.

The graph of r is contained in a sphere $x^{2} + y^{2} + z^{2} = 3 p^{2}$ where $r (t) = (x (t), y (t), z (t)) .$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information.

The given vector-valued function is $r (t) = (x (t), y (t), z (t)), t \in [a, b],$ where a < b are real numbers and the functions x(t), y(t), and z(t) are continuous.

Step 2. Explanation.

As it is given that x(t), y(t), and z(t) are continuous functions, then by the extreme value theorem x(t), y(t), and z(t) have minimum and maximum values.

Let $P_{1}, P_{2} and P_{3}$ be the maximum values of x(t), y(t), and z(t). Now, let P be the highest value of these three values. Then the graph of r(t) on [a.b] lies within the sphere with center (0,0,0) and radius $\sqrt{3} p .$

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