f(xo+h, yo+h) = f(xo, yo) + [fx (xo, yo)h + fy(xo, yo)k] 1 + xo, Yo)h² +2fxy(xo, Yo)hk + fyy(xo, Yo)k²] + R. 2 [fxx (xo, 3 1. Find the second order approximation of the following functions near the given points using Taylor's Theorem. Your answer should be in terms of x and y. (a) f(x, y) = ln(x + cos y) near (0,0). (b) f(x, y) = x sin(x − y) near (π, 0). (c) f(x, y) = y√1 + x near (0, 1).
f(xo+h, yo+h) = f(xo, yo) + [fx (xo, yo)h + fy(xo, yo)k] 1 + xo, Yo)h² +2fxy(xo, Yo)hk + fyy(xo, Yo)k²] + R. 2 [fxx (xo, 3 1. Find the second order approximation of the following functions near the given points using Taylor's Theorem. Your answer should be in terms of x and y. (a) f(x, y) = ln(x + cos y) near (0,0). (b) f(x, y) = x sin(x − y) near (π, 0). (c) f(x, y) = y√1 + x near (0, 1).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Taylor's Theorem for Multivariable Functions**
For a function \( f(x, y) \), the second order Taylor approximation near a point \((x_0, y_0)\) is given by:
\[
f(x_0 + h, y_0 + k) = f(x_0, y_0) + [f_x(x_0, y_0)h + f_y(x_0, y_0)k]
+ \frac{1}{2} \left[ f_{xx}(x_0, y_0)h^2 + 2f_{xy}(x_0, y_0)hk + f_{yy}(x_0, y_0)k^2 \right] + R.
\]
**Task:**
1. Find the second order approximation of the following functions near the given points using Taylor’s Theorem. Your answer should be in terms of \( x \) and \( y \).
(a) \( f(x, y) = \ln(x + \cos y) \) near \((0, 0)\).
(b) \( f(x, y) = x \sin(x - y) \) near \((\pi, 0)\).
(c) \( f(x, y) = y \sqrt{1 + x} \) near \((0, 1)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcdb758fc-abaa-4ada-a842-cb31f023e11a%2F619bd38a-d710-4e01-9a31-a8250aac47a5%2Fm7uxte_processed.png&w=3840&q=75)
Transcribed Image Text:**Taylor's Theorem for Multivariable Functions**
For a function \( f(x, y) \), the second order Taylor approximation near a point \((x_0, y_0)\) is given by:
\[
f(x_0 + h, y_0 + k) = f(x_0, y_0) + [f_x(x_0, y_0)h + f_y(x_0, y_0)k]
+ \frac{1}{2} \left[ f_{xx}(x_0, y_0)h^2 + 2f_{xy}(x_0, y_0)hk + f_{yy}(x_0, y_0)k^2 \right] + R.
\]
**Task:**
1. Find the second order approximation of the following functions near the given points using Taylor’s Theorem. Your answer should be in terms of \( x \) and \( y \).
(a) \( f(x, y) = \ln(x + \cos y) \) near \((0, 0)\).
(b) \( f(x, y) = x \sin(x - y) \) near \((\pi, 0)\).
(c) \( f(x, y) = y \sqrt{1 + x} \) near \((0, 1)\).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
