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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![**Use Linear Approximation to Estimate \( \frac{1}{1.1} - 1 \):**
Linear approximation, or tangent line approximation, can be used to estimate the value of functions near a point. For a function \( f(x) \), the linear approximation at \( a \) is given by:
\[ L(x) = f(a) + f'(a)(x-a) \]
In this case, we want to estimate \( \frac{1}{1.1} - 1 \).
1. Define \( f(x) = \frac{1}{x} \).
2. Find \( f'(x) = -\frac{1}{x^2} \).
3. Choose \( a = 1 \), since it's close to 1.1.
4. Calculate \( f(1) = 1 \) and \( f'(1) = -1 \).
5. Use the linear approximation formula:
\[ L(x) = f(1) + f'(1)(x-1) = 1 - 1(x-1) = 1 - x + 1 = 2 - x \]
Now, apply this to estimate \( \frac{1}{1.1} - 1 \):
\[ L(1.1) = 2 - 1.1 = 0.9 \]
Thus, the linear approximation of \( \frac{1}{1.1} - 1 \) is approximately 0.9.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F81a3cd45-d601-42ca-9c9a-5a10cb20e3e5%2F9d452ecb-a03c-4439-8880-9f26a2f1b4c7%2Famuols_processed.png&w=3840&q=75)
Transcribed Image Text:**Use Linear Approximation to Estimate \( \frac{1}{1.1} - 1 \):**
Linear approximation, or tangent line approximation, can be used to estimate the value of functions near a point. For a function \( f(x) \), the linear approximation at \( a \) is given by:
\[ L(x) = f(a) + f'(a)(x-a) \]
In this case, we want to estimate \( \frac{1}{1.1} - 1 \).
1. Define \( f(x) = \frac{1}{x} \).
2. Find \( f'(x) = -\frac{1}{x^2} \).
3. Choose \( a = 1 \), since it's close to 1.1.
4. Calculate \( f(1) = 1 \) and \( f'(1) = -1 \).
5. Use the linear approximation formula:
\[ L(x) = f(1) + f'(1)(x-1) = 1 - 1(x-1) = 1 - x + 1 = 2 - x \]
Now, apply this to estimate \( \frac{1}{1.1} - 1 \):
\[ L(1.1) = 2 - 1.1 = 0.9 \]
Thus, the linear approximation of \( \frac{1}{1.1} - 1 \) is approximately 0.9.
![2. Let
\[ f(x) = \frac{1}{x} \]
Recall: \( L(x) = f(a) + f'(a)(x-a) \) and \( L(x) \approx f(x) \) when \( x \) is close to \( a \).
(a) Find the linearization \( L(x) \) of \( f \) at \( a = 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F81a3cd45-d601-42ca-9c9a-5a10cb20e3e5%2F9d452ecb-a03c-4439-8880-9f26a2f1b4c7%2Fxbhinos_processed.png&w=3840&q=75)
Transcribed Image Text:2. Let
\[ f(x) = \frac{1}{x} \]
Recall: \( L(x) = f(a) + f'(a)(x-a) \) and \( L(x) \approx f(x) \) when \( x \) is close to \( a \).
(a) Find the linearization \( L(x) \) of \( f \) at \( a = 1 \).
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