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. 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. 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. Examples of calculating the derivative Examples showing how to calculate the derivative and linear approximation of multivariable functions. The multidimensional differentiability theorem Discussion of theorem that gives conditions which guarantee that a multivariable function is 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. Exploring the derivative of the exponential function A guided tour into the reasons that the derivative of the exponential function with base e is the function itself. The idea of the chain rule An illustration of the basic concept of the chain rule using interactive graphics to diagram the relevant points on the graphs and the corresponding slopes. 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 partial derivatives The concept of partial derivatives is introduced with an illustration of heating costs. Interactive graphics demonstrate the properties of partial derivatives. Introduction to the multivariable chain rule Introduction to the multivariable chain rule. The basic concepts are illustrated through a simple example. Introduction to Taylor's theorem for multivariable functions Development of Taylor's polynomial for functions of many variables. 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. Newton's Method A method to approximate the roots to an equation. 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.