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Essential Calculus: Early Transcendentals
Found in: Page 209
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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Short Answer

Find the critical numbers of the function.

34. \(g(\theta ) = 4\theta - \tan \theta \).

The critical numbers of the function \(g(\theta ) = 4\theta - \tan \theta \) are \(\theta = 2n\pi \pm \frac{\pi }{3},\theta = 2n\pi \pm \frac{{2\pi }}{3}\), for all \(n = 0,1,2,3 \ldots ..\)

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Given data

The function is \(g(\theta ) = 4\theta - \tan \theta \).

Step 2: Concept of critical number

A critical number of a function \(f\) is a number \(c\), if it satisfies either of the below conditions:

(1) \({f^\prime }(c) = 0\)

(2) \({f^\prime }(c)\), Does not exist.

Step 3: Obtain the first derivative of the given function

Obtain the first derivative of the given function.

\(\begin{aligned}{c}{g^\prime }(\theta ) &= \frac{d}{{d\theta }}(4\theta - \tan \theta )\\ &= \frac{d}{{d\theta }}(4\theta ) - \frac{d}{{dt}}(\tan \theta )\\ &= 4 - {\sec ^2}\theta \end{aligned}\)

Take \({g^\prime }(\theta ) = 0\)and obtain the critical numbers.

\(\begin{aligned}{c}4 - {\sec ^2}\theta &= 0\\{\sec ^2}\theta &= 4\\\sec \theta &= \pm 2\\\cos \theta &= \pm \frac{1}{2}\end{aligned}\)

Therefore, \(\cos \theta = \frac{1}{2}\) and \(\cos \theta = \frac{{ - 1}}{2}\).

Step 4: Obtain the critical numbers of the function \(g(\theta ) = 4\theta  - \tan \theta \)

Compute the value of \(\theta \) for the two possible cases.

Recall the fact that the general solution of the equation \(\cos \theta = \cos \beta \):

\(\theta = 2n\pi \pm \beta \). ……. (1)

Case (1):

Consider, \(\cos \theta = \frac{1}{2}\). ……. (2)

Express equation (2).

\(\cos \theta = \cos \left( {\frac{\pi }{3}} \right)\) ……. (3)

Apply the general solution as shown in equation (1) and obtain the solution for equation (3).

\(\theta = 2n\pi \pm \frac{\pi }{3}\)

Therefore, the general solution of equation (3) is \(\theta = 2n\pi \pm \frac{\pi }{3}\).

Case (2):

Consider, \(\cos \theta = \frac{{ - 1}}{2}\). ……. (4)

Express the equation (3).

\(\cos \theta = \cos \left( {\frac{{2\pi }}{3}} \right)\) ……. (5)

Apply the general solution as shown in equation (1) and obtain the solution for equation (5).

\(\theta = 2n\pi \pm \frac{{2\pi }}{3}\).

The critical numbers of the function \(g(\theta ) = 4\theta - \tan \theta \) are \(\theta = 2n\pi \pm \frac{\pi }{3},\;\theta = 2n\pi \pm \frac{{2\pi }}{3}\), for all \(n = 0,1,2,3 \ldots ..\)

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