Homework Help is Here – Start Your Trial Now!
Math
Calculus
Find f. f'(t) = 8 cos t + sec²t, -π/2 < t < π/2, f(π/3) = 2

Find f. f'(t) = 8 cos t + sec²t, -π/2 < t < π/2, f(π/3) = 2

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
See similar textbooks
icon
Related questions
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 \( f(t) \).
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 \( f(t) \).
Expert Solution
Step 1: Requirement of the question

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.

steps

Step by step

Solved in 3 steps with 5 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS