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 Statement
Find the function \( f \) such that:
\[ f'(x) = \frac{7}{\sqrt{x}} \]
with the initial condition:
\[ f(16) = 70 \]
### Solution
Integrate \( f'(x) \) to find \( f(x) \):
Given:
\[ f'(x) = \frac{7}{\sqrt{x}} \]
First, rewrite the expression in a more familiar form for integration:
\[ f'(x) = 7x^{-\frac{1}{2}} \]
Integrate with respect to \( x \):
\[ f(x) = \int 7x^{-\frac{1}{2}} \, dx \]
Using the power rule for integration (\( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)), we get:
\[ f(x) = 7 \cdot \int x^{-\frac{1}{2}} \, dx \]
\[ f(x) = 7 \cdot \left( \frac{x^{\frac{1}{2}}}{\frac{1}{2}} \right) + C \]
\[ f(x) = 7 \cdot 2x^{\frac{1}{2}} + C \]
\[ f(x) = 14\sqrt{x} + C \]
Next, utilize the initial condition to solve for \( C \):
Given:
\[ f(16) = 70 \]
Substitute \( x = 16 \) and \( f(16) = 70 \):
\[ 70 = 14\sqrt{16} + C \]
\[ 70 = 14 \cdot 4 + C \]
\[ 70 = 56 + C \]
\[ C = 14 \]
Thus, the function \( f(x) \) is:
\[ f(x) = 14\sqrt{x} + 14 \]
### Final Answer
\[ f(x) = 14\sqrt{x} + 14 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Febbbc1cc-ba98-4671-a28a-276fb59e8560%2F641f76e4-52f0-4290-8cd0-d262a2d6f8a4%2F5o06j8p_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Find the function \( f \) such that:
\[ f'(x) = \frac{7}{\sqrt{x}} \]
with the initial condition:
\[ f(16) = 70 \]
### Solution
Integrate \( f'(x) \) to find \( f(x) \):
Given:
\[ f'(x) = \frac{7}{\sqrt{x}} \]
First, rewrite the expression in a more familiar form for integration:
\[ f'(x) = 7x^{-\frac{1}{2}} \]
Integrate with respect to \( x \):
\[ f(x) = \int 7x^{-\frac{1}{2}} \, dx \]
Using the power rule for integration (\( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)), we get:
\[ f(x) = 7 \cdot \int x^{-\frac{1}{2}} \, dx \]
\[ f(x) = 7 \cdot \left( \frac{x^{\frac{1}{2}}}{\frac{1}{2}} \right) + C \]
\[ f(x) = 7 \cdot 2x^{\frac{1}{2}} + C \]
\[ f(x) = 14\sqrt{x} + C \]
Next, utilize the initial condition to solve for \( C \):
Given:
\[ f(16) = 70 \]
Substitute \( x = 16 \) and \( f(16) = 70 \):
\[ 70 = 14\sqrt{16} + C \]
\[ 70 = 14 \cdot 4 + C \]
\[ 70 = 56 + C \]
\[ C = 14 \]
Thus, the function \( f(x) \) is:
\[ f(x) = 14\sqrt{x} + 14 \]
### Final Answer
\[ f(x) = 14\sqrt{x} + 14 \]
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