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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![9. Answer the following questions about linearization.
a.
Find the linearization \( L(x) \) of the function at \( a \).
\[ f(x) = \cos x, \, a = \frac{3\pi}{2} \]
\[ L(x) = \boxed{} \]
b.
Find the linear approximation of the function \( f(x) = \sqrt{25-x} \) at \( x = 0 \) and use it to approximate the numbers \(\sqrt{24.9}\) and \(\sqrt{24.99}\).
\[ L(x) = \boxed{} \]
\[ \sqrt{24.9} \approx \boxed{} \]
\[ \sqrt{24.99} \approx \boxed{} \]
c.
Use a graphing calculator or computer to verify the given linear approximation at \( a = 0 \). Then determine the values of \( x \) for which the linear approximation is accurate to within 0.1.
\[ \sqrt{1-x} \approx 1 - \frac{1}{3}x^2 \]
\[ \left( \boxed{}, \boxed{} \right) \]
d.
Use a graphing calculator or computer to verify the given linear approximation at \( a = 0 \). Then determine the values of \( x \) for which the linear approximation is accurate to within 0.1.
\[ \frac{1}{(1+4x)^2} \approx 1 - 16x \]
\[ \left( \boxed{}, \boxed{} \right) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F966b76bc-b184-4898-af22-464e4af89029%2F555c1bee-ddf8-4a05-acb2-e68f354c6fcd%2F605ccli_processed.png&w=3840&q=75)
Transcribed Image Text:9. Answer the following questions about linearization.
a.
Find the linearization \( L(x) \) of the function at \( a \).
\[ f(x) = \cos x, \, a = \frac{3\pi}{2} \]
\[ L(x) = \boxed{} \]
b.
Find the linear approximation of the function \( f(x) = \sqrt{25-x} \) at \( x = 0 \) and use it to approximate the numbers \(\sqrt{24.9}\) and \(\sqrt{24.99}\).
\[ L(x) = \boxed{} \]
\[ \sqrt{24.9} \approx \boxed{} \]
\[ \sqrt{24.99} \approx \boxed{} \]
c.
Use a graphing calculator or computer to verify the given linear approximation at \( a = 0 \). Then determine the values of \( x \) for which the linear approximation is accurate to within 0.1.
\[ \sqrt{1-x} \approx 1 - \frac{1}{3}x^2 \]
\[ \left( \boxed{}, \boxed{} \right) \]
d.
Use a graphing calculator or computer to verify the given linear approximation at \( a = 0 \). Then determine the values of \( x \) for which the linear approximation is accurate to within 0.1.
\[ \frac{1}{(1+4x)^2} \approx 1 - 16x \]
\[ \left( \boxed{}, \boxed{} \right) \]
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