Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.5 Derivatives of Trigonometric Functions - 3.5 Exercises: 14

Answer

\[\begin{gathered} = 0 \hfill \\ \hfill \\ \end{gathered} \]

Work Step by Step

\[\begin{gathered} \mathop {\lim }\limits_{\theta \, \to \,0} \,\,\,\,\,\frac{{\sec \,\,\,\theta - 1}}{\theta } \hfill \\ \hfill \\ identity\,\,\,\,\sec \,\,\theta = \frac{1}{{\cos \,\,\theta }}\,\, \hfill \\ then \hfill \\ \hfill \\ = \mathop {\lim }\limits_{\theta \, \to \,0} \,\,\,\frac{{\frac{1}{{\cos \theta }} - 1}}{\theta } = \mathop {\lim }\limits_{\theta \, \to \,0} \,\,\,\frac{{\frac{{1 - \cos \,\theta }}{{\cos \,\theta }}}}{\theta } \hfill \\ \hfill \\ simplify \hfill \\ \hfill \\ = \mathop {\lim }\limits_{\theta \, \to \,0} \,\,\,\frac{{1 - \cos \theta }}{{\theta \,\,\cos \,\,\theta }} \hfill \\ \hfill \\ = \,\left( {\mathop {\lim }\limits_{\theta \, \to \,0} \,\,\,\,\,\frac{1}{{\cos \,\,\theta }}} \right)\,\,\,\,\left( {\mathop {\lim }\limits_{\theta \, \to \,0} \,\,\,\frac{{1 - \cos \,\,\theta }}{\theta }} \right) \hfill \\ \hfill \\ from\,\,the\,\,theorem\,\,7.11\,\,\mathop {\lim }\limits_{\theta \, \to \,0} \,\,\,\frac{{1 - \cos \,\,\theta }}{\theta } = 0 \hfill \\ \hfill \\ = \,\left( 1 \right)\,\left( 0 \right) = 0 \hfill \\ \hfill \\ \end{gathered} \]
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