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Find ƒ if f"(t) = 2e* + 3 sin(t), f(0) = -2, f(T) = -4. %3D %3D f(t) =

Find ƒ if f"(t) = 2e* + 3 sin(t), f(0) = -2, f(T) = -4. %3D %3D f(t) =

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...
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**Problem Statement**: Find \( f \) if \( f''(t) = 2e^t + 3\sin(t) \), \( f(0) = -2 \), \( f(\pi) = -4 \). **Solution**: The objective is to determine the function \( f(t) \) that satisfies the given second derivative \( f''(t) \) and the initial conditions \( f(0) = -2 \) and \( f(\pi) = -4 \). Given: - \( f''(t) = 2e^t + 3\sin(t) \) - \( f(0) = -2 \) - \( f(\pi) = -4 \) **Steps**: 1. To find \( f(t) \), first integrate \( f''(t) \) with respect to \( t \) to obtain \( f'(t) \). 2. Integrate \( f'(t) \) with respect to \( t \) to find \( f(t) \). 3. Apply the initial conditions to solve for the constants of integration. 4. Substitute the constants back into the general expression for \( f(t) \). This setup allows students to explore the process of integrating to find a function from its derivatives, using initial conditions to determine specific solutions, and verifying results by substituting back into the original conditions.
Transcribed Image Text:**Problem Statement**: Find \( f \) if \( f''(t) = 2e^t + 3\sin(t) \), \( f(0) = -2 \), \( f(\pi) = -4 \). **Solution**: The objective is to determine the function \( f(t) \) that satisfies the given second derivative \( f''(t) \) and the initial conditions \( f(0) = -2 \) and \( f(\pi) = -4 \). Given: - \( f''(t) = 2e^t + 3\sin(t) \) - \( f(0) = -2 \) - \( f(\pi) = -4 \) **Steps**: 1. To find \( f(t) \), first integrate \( f''(t) \) with respect to \( t \) to obtain \( f'(t) \). 2. Integrate \( f'(t) \) with respect to \( t \) to find \( f(t) \). 3. Apply the initial conditions to solve for the constants of integration. 4. Substitute the constants back into the general expression for \( f(t) \). This setup allows students to explore the process of integrating to find a function from its derivatives, using initial conditions to determine specific solutions, and verifying results by substituting back into the original conditions.
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