Pages similar to: Introduction to differentiability in higher dimensions
- The multivariable linear approximation
Introduction to the linear approximation in multivariable calculus and why it might be useful. - 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. - 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. - 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. - Derivation of the directional derivative and the gradient
Derivation of the directional derivative and the gradient from the definition of differentiability of scalar-valued multivariable functions. - Introduction to Taylor's theorem for multivariable functions
Development of Taylor's polynomial for functions of many variables. - 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. - 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.