Please solve the screenshot and explain each step clearly. Thanks! Consider a function f : R² → R where f(x, y) = (1 + x²)e²v for all (æ, y) E R². Apply thesecond-order Taylor approximation around (0,0) to estimate f(-0.1,0.2). (You do not have

Answered: Consider a function f : R² → R where…

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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Please solve the screenshot and explain each step clearly. Thanks!

Consider a function f : R² → R where f(x, y) = (1 + x²)e²v for all (æ, y) E R². Apply the
second-order Taylor approximation around (0,0) to estimate f(-0.1,0.2). (You do not have

Expert Solution

Step 1

For a function $f : ℝ^{2} \to ℝ$ i.e. $f (x, y)$ , the second-order Taylor approximation at a point $(a, b)$ is defined as

$\begin{array}{rcl} f (x, y) & = & f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b) \\ + \frac{1}{2} f_{x x} (a, b) {(x - a)}^{2} + f_{x y} (a, b) (x - a) (y - b) + \frac{1}{2} f_{y y} (a - b) {(y - b)}^{2} \end{array}$

where $f_{x} (a, b)$ is the partial derivative of the function at the point $(a, b)$ .

Step 2

Given: $f (x, y) = (1 + x^{2}) e^{2 y}$ , $(a, b) = (0, 0)$

On differentiating $f (x, y)$ partially with respect to $x$ we get

$f_{x} (x, y) = 2 x e^{2 y}$

On differentiating this partially with respect to $y$ we get

$f_{x y} (x, y) = 4 x e^{2 y}$

On differentiating this again partially with respect to $x$ we get

$f_{x x} (x, y) = 2 e^{2 y}$

On substituting $(x, y) = (a, b)$ we get $x$ we get

$f_{x} (0, 0) = 0$

$f_{x y} (0, 0) = 0$

$f_{x x} (0, 0) = 2$

On differentiating $f (x, y)$ partially with respect to $y$ we get

$f_{y} (x, y) = 2 (1 + x^{2}) e^{2 y}$

On differentiating this again partially with respect to $y$ we get

$f_{y} (x, y) = 4 (1 + x^{2}) e^{2 y}$

On substituting $(x, y) = (a, b)$ we get

$f_{y} (0, 0) = 2$

$f_{y y} (0, 0) = 4$

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