Pages similar to: The multivariable linear approximation
- Introduction to differentiability in higher dimensions
An introduction to the basic concept of the differentiability of a function of multiple variables. Discussion centers around the existence of a tangent plane to a function of two variables. - Examples of calculating the derivative
Examples showing how to calculate the derivative and linear approximation of multivariable functions. - The definition of differentiability in higher dimensions
The definition of differentiability for multivariable functions. Informal derivation designed to give intuition behind the condition for a function to be differentiable. - Introduction to Taylor's theorem for multivariable functions
Development of Taylor's polynomial for functions of many variables. - The multidimensional differentiability theorem
Discussion of theorem that gives conditions which guarantee that a multivariable function is differentiable. - A differentiable function with discontinuous partial derivatives
Illustration that discontinuous partial derivatives need not exclude a function from being differentiable. - The idea of the derivative of a function
The derivative of a function as the slope of the tangent line. - Tangent and normal lines
How to compute the tangent and normal lines to the graph of a function. - Linear approximations: approximation by differentials
Approximating the value of a function near a point by its tangent line formula. - Approximating a nonlinear function by a linear function
The secant line and tangent line are two ways to approximate a nonlinear function by a linear one. - Developing intuition about the derivative
An intuitive exploration into the properties of the derivative, illustrated by interactive graphics. - Elementary derivative problems
Sample problems illustrating the ordinary derivative. - Calculating the derivative of a linear function using the derivative formula
Exploring how the limit definition of the derivative gives the slope of a linear function. - Calculating the derivative of a quadratic function
Using the limit definition of the derivative to calculate the derivative of a quadratic. - The tangent line as a linear approximation
How the tangent line to the graph of a function is a linear approximation to the function close by the tangent point. - 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. - Introduction to the multivariable chain rule
Introduction to the multivariable chain rule. The basic concepts are illustrated through a simple example.