Calculating errors in Functions of two variables Taylor Series

By substituting x = x_{0} + h and y = y_{0} + k into the expansion, we can simplify to:\Delta f(x, y) \approx f(x_{0}, y_{0}) + f_x(x_0, y_0)h + f_y(x_0, y_0)k - f(x_{0},y_{0})Using the definition of partial derivatives, we can rewrite this as:\Delta f(x, y) \approx f(x_{0}, y_{0}) + \frac{\partial f}{\partial x}\Delta x + \frac{\partial f}{\partial y}\Delta y - f(x_{0},y_{0})Recognizing
  • #1
thomas49th
655
0

Homework Statement


From the taylor series we can replace [tex]x =x_{0} + h[/tex]
but how does
[tex]\delta f = f(x_{0} + h, y_{0} + k) - f(x_{0},y_{0})[/tex]
become
[tex]\delta f = hf(x_{0}, y_{0}) + kf(x_{0}, y_{0})[/tex]
I can see the first step, but how do you get it to the second?

Homework Equations


The Attempt at a Solution

 
Physics news on Phys.org
  • #2
thomas49th said:

Homework Statement


From the taylor series we can replace [tex]x =x_{0} + h[/tex]
but how does
[tex]\delta f = f(x_{0} + h, y_{0} + k) - f(x_{0},y_{0})[/tex]
become
[tex]\delta f = hf(x_{0}, y_{0}) + kf(x_{0}, y_{0})[/tex]
I can see the first step, but how do you get it to the second?


Homework Equations





The Attempt at a Solution


[tex]\Delta f(x, y) = f(x_{0} + h, y_{0} + k) - f(x_{0},y_{0})[/tex]
[tex]\approx f(x_{0}, y_{0}) + f_x(x_0, y_0)\Delta x + f_y(x_0, y_0)\Delta y - f(x_{0},y_{0})[/tex]

In the Taylor expansion above, terms of order 2 and higher are omitted.
 

1. What is a Taylor Series?

A Taylor Series is a mathematical representation of a function as an infinite sum of terms, using the function's derivatives at a single point. It is used to approximate the value of a function at a given point.

2. How do you calculate the Taylor Series for a function of two variables?

To calculate the Taylor Series for a function of two variables, you need to take the partial derivatives of the function with respect to each variable at a given point. These derivatives are then used to create the terms in the Taylor Series.

3. What is the error in a Taylor Series?

The error in a Taylor Series is the difference between the value of the function at a given point and the value calculated using the Taylor Series approximation. It is also known as the remainder term.

4. How do you calculate the error in a Taylor Series for a function of two variables?

To calculate the error in a Taylor Series for a function of two variables, you need to use the remainder formula, which involves taking the maximum value of the partial derivatives of the function at a given point and finding the product of those values with the appropriate powers of the difference between the given point and the center point.

5. When is it necessary to use a Taylor Series to calculate the value of a function of two variables?

A Taylor Series is necessary to use when the function cannot be easily evaluated at a given point, but the derivatives of the function can be calculated. It is also useful for approximating the value of a function at points close to the given point.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
667
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
  • Calculus and Beyond Homework Help
Replies
27
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
851
  • Calculus and Beyond Homework Help
Replies
3
Views
972
  • Calculus and Beyond Homework Help
Replies
18
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
944
Back
Top