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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Question
![**Problem:**
Find \( f \) if \( f''(t) = 2e^t + 3 \sin(t) \), \( f(0) = -4 \), \( f(\pi) = 9 \).
\[ f(t) = \text{[Input Box]} \]
**Solution:**
To find the function \( f(t) \) given the double derivative \( f''(t) = 2e^t + 3 \sin(t) \) and the initial conditions \( f(0) = -4 \), and \( f(\pi) = 9 \):
1. **Integrate** \( f''(t) \) to find the first derivative \( f'(t) \):
\[
f''(t) = 2e^t + 3 \sin(t)
\]
Integrate:
\[
f'(t) = 2e^t - 3 \cos(t) + C_1
\]
2. **Integrate** \( f'(t) \) to find \( f(t) \):
\[
f'(t) = 2e^t - 3 \cos(t) + C_1
\]
Integrate:
\[
f(t) = 2e^t - 3 \sin(t) + C_1 t + C_2
\]
3. **Determine** the constants \( C_1 \) and \( C_2 \) using the initial conditions \( f(0) = -4 \) and \( f(\pi) = 9 \):
For \( t = 0 \):
\[
f(0) = 2e^0 - 3 \sin(0) + C_1 \cdot 0 + C_2 = -4
\]
\[
2 + 0 + 0 + C_2 = -4
\]
\[
C_2 = -6
\]
For \( t = \pi \):
\[
f(\pi) = 2e^\pi - 3 \sin(\pi) + C_1 \cdot \pi + C_2 = 9
\]
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F276d7283-01d9-45ed-a762-7e45846115eb%2F5e5db47f-362c-4d94-ba08-6f199bdbaa4e%2F8tyqy96_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find \( f \) if \( f''(t) = 2e^t + 3 \sin(t) \), \( f(0) = -4 \), \( f(\pi) = 9 \).
\[ f(t) = \text{[Input Box]} \]
**Solution:**
To find the function \( f(t) \) given the double derivative \( f''(t) = 2e^t + 3 \sin(t) \) and the initial conditions \( f(0) = -4 \), and \( f(\pi) = 9 \):
1. **Integrate** \( f''(t) \) to find the first derivative \( f'(t) \):
\[
f''(t) = 2e^t + 3 \sin(t)
\]
Integrate:
\[
f'(t) = 2e^t - 3 \cos(t) + C_1
\]
2. **Integrate** \( f'(t) \) to find \( f(t) \):
\[
f'(t) = 2e^t - 3 \cos(t) + C_1
\]
Integrate:
\[
f(t) = 2e^t - 3 \sin(t) + C_1 t + C_2
\]
3. **Determine** the constants \( C_1 \) and \( C_2 \) using the initial conditions \( f(0) = -4 \) and \( f(\pi) = 9 \):
For \( t = 0 \):
\[
f(0) = 2e^0 - 3 \sin(0) + C_1 \cdot 0 + C_2 = -4
\]
\[
2 + 0 + 0 + C_2 = -4
\]
\[
C_2 = -6
\]
For \( t = \pi \):
\[
f(\pi) = 2e^\pi - 3 \sin(\pi) + C_1 \cdot \pi + C_2 = 9
\]
\
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