Pages similar to: Multivariable Taylor polynomial example
- Introduction to Taylor's theorem for multivariable functions
Development of Taylor's polynomial for functions of many variables. - Taylor polynomials: formulas
Different ways of writing Taylor's formula with remainder term. - Introduction to partial derivatives
The concept of partial derivatives is introduced with an illustration of heating costs. Interactive graphics demonstrate the properties of partial derivatives. - Partial derivative examples
Examples of how to calculate partial derivatives. - Partial derivative by limit definition
Description with example of how to calculate the partial derivative from its limit definition. - Subtleties of differentiability in higher dimensions
A description of some of the tricky ways where a function of multiple variables can fail to be differentiable. Example two variable functions are illustrated with interactive graphics. - The derivative matrix
The derivative matrix is presented as a natural generalization of the single variable derivative to multivariable functions. - An introduction to the directional derivative and the gradient
Interactive graphics about a mountain range illustrate the concepts behind the directional derivative and the gradient of scalar-valued functions of two variables. - Introduction to local extrema of functions of two variables
Introduction to the idea of critical points and local extrema of two variable functions. - Two variable local extrema examples
Examples of calculating the critical points and local extrema of two variable functions. - The multidimensional differentiability theorem
Discussion of theorem that gives conditions which guarantee that a multivariable function is differentiable. - Non-differentiable functions must have discontinuous partial derivatives
A visual tour demonstrating discontinuous partial derivatives of a non-differentiable function, as required by the differentiability theorem. - A differentiable function with discontinuous partial derivatives
Illustration that discontinuous partial derivatives need not exclude a function from being differentiable. - Historical and theoretical comments: Mean Value Theorem
Comments on the traditional way that Taylor polynomials are taught as inextricably linked with issues about infinite series. - Classic examples of Taylor polynomials
Some of the most famous (and important) examples of Taylor expansions. - Computational tricks regarding Taylor polynomials
A short cut to calculating Taylor polynomials in terms of Taylor polynomials of simple functions. - Prototypes: More serious questions about Taylor polynomials
Questions regarding using Taylor polynomials to approximate a function over an interval. - Determining tolerance/error in Taylor polynomials.
Calculating with what tolerance a Taylor polynomial approximates a function on an interval. - How large an interval with given tolerance for a Taylor polynomial?
Given a Taylor polynomial expanded around a point, on how large an interval around the point does it achieve a required tolerance? - Achieving desired tolerance of a Taylor polynomial on desired interval
For a given interval around a point, how many terms of a Taylor polynomial expanded around that point must be used to achieve a required tolerance?