A differentiable function has the value y(2) = 4 and the derivative value y'(2) = -3. Approximate the value of y(2.1). O 3.7 O 3.75 O 3.8 O 3.85 17 18
A differentiable function has the value y(2) = 4 and the derivative value y'(2) = -3. Approximate the value of y(2.1). O 3.7 O 3.75 O 3.8 O 3.85 17 18
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 Statement
A differentiable function has the value \( y(2) = 4 \) and the derivative value \( y'(2) = -3 \). Approximate the value of \( y(2.1) \).
### Answer Options
- 3.7
- 3.75
- 3.8
- 3.85
### Explanation:
To approximate \( y(2.1) \), we can use the linear approximation formula:
\[ y(x) \approx y(a) + y'(a)(x - a) \]
Substitute \( a = 2 \) and \( x = 2.1 \):
\[ y(2.1) \approx y(2) + y'(2)(2.1 - 2) \]
Plug in the given values:
\[ y(2.1) \approx 4 + (-3)(0.1) \]
\[ y(2.1) \approx 4 - 0.3 \]
\[ y(2.1) \approx 3.7 \]
Thus, the approximate value of \( y(2.1) \) is 3.7.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d3da0a1-ed16-4e17-b0aa-3ed94d4e63aa%2F90447694-e55a-4499-9701-5f8bf3a2cb58%2F7xzbv6j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
A differentiable function has the value \( y(2) = 4 \) and the derivative value \( y'(2) = -3 \). Approximate the value of \( y(2.1) \).
### Answer Options
- 3.7
- 3.75
- 3.8
- 3.85
### Explanation:
To approximate \( y(2.1) \), we can use the linear approximation formula:
\[ y(x) \approx y(a) + y'(a)(x - a) \]
Substitute \( a = 2 \) and \( x = 2.1 \):
\[ y(2.1) \approx y(2) + y'(2)(2.1 - 2) \]
Plug in the given values:
\[ y(2.1) \approx 4 + (-3)(0.1) \]
\[ y(2.1) \approx 4 - 0.3 \]
\[ y(2.1) \approx 3.7 \]
Thus, the approximate value of \( y(2.1) \) is 3.7.
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