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

Transcribed Image Text:**Problem Statement:**
Find a function \( f(x) \) such that \( f'(x) = -3x^2 + 8 \) and \( f(0) = 1 \).
**Solution:**
To find \( f(x) \), integrate the derivative \( f'(x) = -3x^2 + 8 \).
1. Integrate \( -3x^2 \) to get \( -x^3 \).
2. Integrate \( 8 \) to get \( 8x \).
Thus, \( f(x) = -x^3 + 8x + C \), where \( C \) is the constant of integration.
**Apply the Initial Condition:**
Use the given condition \( f(0) = 1 \) to find \( C \):
1. Substitute \( x = 0 \) into \( f(x) = -x^3 + 8x + C \).
2. \( f(0) = 0 + 0 + C = 1 \).
Therefore, \( C = 1 \).
**Final Expression:**
The function is \( f(x) = -x^3 + 8x + 1 \).
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