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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COMPLETLY SOLVE PLEASE!!!!
![**Problem Statement:**
Find the function \( f \).
Given:
\[ f''(x) = x^{-2}, \, x > 0, \, f(1) = 0, \, f(3) = 0 \]
**Solution:**
To find \( f(x) \), we first need to integrate the second derivative \( f''(x) = x^{-2} \) twice to obtain \( f(x) \).
1. **First Integration:**
Integrate \( f''(x) = x^{-2} \) to find \( f'(x) \).
\[
f'(x) = \int x^{-2} \, dx = -x^{-1} + C_1 = -\frac{1}{x} + C_1
\]
where \( C_1 \) is the constant of integration.
2. **Second Integration:**
Integrate \( f'(x) = -\frac{1}{x} + C_1 \) to find \( f(x) \).
\[
f(x) = \int \left(-\frac{1}{x} + C_1\right) dx = -\ln|x| + C_1 x + C_2
\]
where \( C_2 \) is another constant of integration.
3. **Applying Initial Conditions:**
Use the conditions \( f(1) = 0 \) and \( f(3) = 0 \) to solve for \( C_1 \) and \( C_2 \).
- For \( f(1) = 0 \):
\[
-\ln|1| + C_1 \cdot 1 + C_2 = 0 \quad \Rightarrow \quad 0 + C_1 + C_2 = 0 \quad \Rightarrow \quad C_1 + C_2 = 0
\]
- For \( f(3) = 0 \):
\[
-\ln|3| + C_1 \cdot 3 + C_2 = 0 \quad \Rightarrow \quad -\ln 3 + 3C_1 + C_2 = 0
\]
Solving the system of linear equations:
\[
\begin{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70fe4f99-9e69-4bef-afe6-80d54e8903ee%2F97ca057a-8247-4837-b982-3342f52a59aa%2Fc5lsmnn_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the function \( f \).
Given:
\[ f''(x) = x^{-2}, \, x > 0, \, f(1) = 0, \, f(3) = 0 \]
**Solution:**
To find \( f(x) \), we first need to integrate the second derivative \( f''(x) = x^{-2} \) twice to obtain \( f(x) \).
1. **First Integration:**
Integrate \( f''(x) = x^{-2} \) to find \( f'(x) \).
\[
f'(x) = \int x^{-2} \, dx = -x^{-1} + C_1 = -\frac{1}{x} + C_1
\]
where \( C_1 \) is the constant of integration.
2. **Second Integration:**
Integrate \( f'(x) = -\frac{1}{x} + C_1 \) to find \( f(x) \).
\[
f(x) = \int \left(-\frac{1}{x} + C_1\right) dx = -\ln|x| + C_1 x + C_2
\]
where \( C_2 \) is another constant of integration.
3. **Applying Initial Conditions:**
Use the conditions \( f(1) = 0 \) and \( f(3) = 0 \) to solve for \( C_1 \) and \( C_2 \).
- For \( f(1) = 0 \):
\[
-\ln|1| + C_1 \cdot 1 + C_2 = 0 \quad \Rightarrow \quad 0 + C_1 + C_2 = 0 \quad \Rightarrow \quad C_1 + C_2 = 0
\]
- For \( f(3) = 0 \):
\[
-\ln|3| + C_1 \cdot 3 + C_2 = 0 \quad \Rightarrow \quad -\ln 3 + 3C_1 + C_2 = 0
\]
Solving the system of linear equations:
\[
\begin{
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