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 Description:**
Find the linear approximation of \( f(x) = \ln x \) at \( x = 1 \) and use it to estimate \( \ln(1.04) \).
**Solution:**
To find the linear approximation of a function \( f(x) \) at a point \( x = a \), we use the formula for linear approximation:
\[ L(x) = f(a) + f'(a)(x - a) \]
1. **Identify the function and the point:**
\( f(x) = \ln x \), \( a = 1 \)
2. **Calculate \( f(a) \):**
\( f(1) = \ln 1 = 0 \)
3. **Find the derivative \( f'(x) \):**
\( f'(x) = \frac{1}{x} \)
4. **Calculate \( f'(a) \):**
\( f'(1) = \frac{1}{1} = 1 \)
5. **Apply the linear approximation formula:**
\( L(x) = 0 + 1(x - 1) = x - 1 \)
Now, use the linear approximation to estimate \( \ln(1.04) \):
\[ L(1.04) = 1.04 - 1 = 0.04 \]
Therefore, the estimated value of \( \ln(1.04) \) is approximately 0.04.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb11ca76e-99cc-49cc-ab47-e18e3e84e4f7%2F8dff19e6-b334-47b7-b2d5-504f3cb8516b%2Fv1u67so_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
Find the linear approximation of \( f(x) = \ln x \) at \( x = 1 \) and use it to estimate \( \ln(1.04) \).
**Solution:**
To find the linear approximation of a function \( f(x) \) at a point \( x = a \), we use the formula for linear approximation:
\[ L(x) = f(a) + f'(a)(x - a) \]
1. **Identify the function and the point:**
\( f(x) = \ln x \), \( a = 1 \)
2. **Calculate \( f(a) \):**
\( f(1) = \ln 1 = 0 \)
3. **Find the derivative \( f'(x) \):**
\( f'(x) = \frac{1}{x} \)
4. **Calculate \( f'(a) \):**
\( f'(1) = \frac{1}{1} = 1 \)
5. **Apply the linear approximation formula:**
\( L(x) = 0 + 1(x - 1) = x - 1 \)
Now, use the linear approximation to estimate \( \ln(1.04) \):
\[ L(1.04) = 1.04 - 1 = 0.04 \]
Therefore, the estimated value of \( \ln(1.04) \) is approximately 0.04.
![Suppose that \( f(x) \) is a function with \( f(105) = 25 \) and \( f'(105) = 7 \). Estimate \( f(106) \).
= [Blank Input Field]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb11ca76e-99cc-49cc-ab47-e18e3e84e4f7%2F8dff19e6-b334-47b7-b2d5-504f3cb8516b%2Faa90ouo_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose that \( f(x) \) is a function with \( f(105) = 25 \) and \( f'(105) = 7 \). Estimate \( f(106) \).
= [Blank Input Field]
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