f(x, y) = x³ex, (1, 0) The partial derivatives are f(x, y) = Find the linearization L(x, y) of the function at (1, 0). L(x, y) = and fy(x, y) = , so fx(1, 0) = and f,(1, 0) = [ . Both fx and fy are continuous functions, so f is differentiable at (1, 0).
f(x, y) = x³ex, (1, 0) The partial derivatives are f(x, y) = Find the linearization L(x, y) of the function at (1, 0). L(x, y) = and fy(x, y) = , so fx(1, 0) = and f,(1, 0) = [ . Both fx and fy are continuous functions, so f is differentiable at (1, 0).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Explain Why the Function is Differentiable at the Given Point
Given the function:
\[ f(x, y) = x^5 e^y \]
at the point \((1, 0)\),
**1. Calculate the Partial Derivatives:**
The partial derivatives are:
\[ f_x(x, y) = \]
\[ f_y(x, y) = \]
**2. Evaluate at the Point \((1, 0)\):**
\[ f_x(1, 0) = \]
\[ f_y(1, 0) = \]
Both \( f_x \) and \( f_y \) are continuous functions, so \( f \) is differentiable at \((1, 0)\).
**3. Find the Linearization \( L(x, y) \) of the Function at \((1, 0)\):**
\[ L(x, y) = \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F51323dd8-f785-4fee-9caa-f5ed8457fa8f%2F1166406c-a38c-471d-8f26-0b94de50178d%2Foj50a5i_processed.png&w=3840&q=75)
Transcribed Image Text:### Explain Why the Function is Differentiable at the Given Point
Given the function:
\[ f(x, y) = x^5 e^y \]
at the point \((1, 0)\),
**1. Calculate the Partial Derivatives:**
The partial derivatives are:
\[ f_x(x, y) = \]
\[ f_y(x, y) = \]
**2. Evaluate at the Point \((1, 0)\):**
\[ f_x(1, 0) = \]
\[ f_y(1, 0) = \]
Both \( f_x \) and \( f_y \) are continuous functions, so \( f \) is differentiable at \((1, 0)\).
**3. Find the Linearization \( L(x, y) \) of the Function at \((1, 0)\):**
\[ L(x, y) = \]
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