Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question
**Problem Statement:** Find \( f \). Given: \[ f'(t) = 8 \cos t + \sec^2 t, \] with the interval \(-\pi/2 < t < \pi/2\), and the condition \( f(\pi/3) = 2 \). --- **Explanation of Components:** 1. **Expression for Derivative:** - \( f'(t) = 8 \cos t + \sec^2 t \) - \( 8 \cos t \): This term involves the cosine function, which oscillates between -1 and 1. - \( \sec^2 t \): This term involves the secant function, which is the reciprocal of the cosine function. The square of the secant function is always non-negative and increases rapidly as \( t \) approaches the bounds where cosine is zero. 2. **Interval:** - \(-\pi/2 < t < \pi/2\) - This specifies the domain in which the function is defined, corresponding to one full cycle of the cosine function, but excluding the points where cosine is zero, preventing division by zero in the secant component. 3. **Initial Condition:** - \( f(\pi/3) = 2 \) - This provides a specific value of the function \( f \) at \( t = \pi/3 \), which is necessary for determining the constant of integration when solving for \( f(t) \).
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Transcribed Image Text:**Problem Statement:** Find \( f \). Given: \[ f'(t) = 8 \cos t + \sec^2 t, \] with the interval \(-\pi/2 < t < \pi/2\), and the condition \( f(\pi/3) = 2 \). --- **Explanation of Components:** 1. **Expression for Derivative:** - \( f'(t) = 8 \cos t + \sec^2 t \) - \( 8 \cos t \): This term involves the cosine function, which oscillates between -1 and 1. - \( \sec^2 t \): This term involves the secant function, which is the reciprocal of the cosine function. The square of the secant function is always non-negative and increases rapidly as \( t \) approaches the bounds where cosine is zero. 2. **Interval:** - \(-\pi/2 < t < \pi/2\) - This specifies the domain in which the function is defined, corresponding to one full cycle of the cosine function, but excluding the points where cosine is zero, preventing division by zero in the secant component. 3. **Initial Condition:** - \( f(\pi/3) = 2 \) - This provides a specific value of the function \( f \) at \( t = \pi/3 \), which is necessary for determining the constant of integration when solving for \( f(t) \).
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f apostrophe left parenthesis t right parenthesis equals 8 cos t plus s e c squared t comma space minus pi over 2 less than t less than straight pi over 2 f open parentheses straight pi over 3 close parentheses equals 2 F i n d space f.

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