Pages similar to: Linear approximations: approximation by differentials
- 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 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 multivariable linear approximation
Introduction to the linear approximation in multivariable calculus and why it might be useful. - 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. - Developing intuition about the derivative
An intuitive exploration into the properties of the derivative, illustrated by interactive graphics. - 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. - 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. - Tangent lines to parametrized curves
The tangent vector given by the derivative of a parametrized curve forms the basis for the equation of a line tangent to the curve. - Tangent line to parametrized curve examples
Examples showing how to calculate the tangent line to a parameterized curve from the derivative of the underlying vector-valued function. - Introduction to Taylor's theorem for multivariable functions
Development of Taylor's polynomial for functions of many variables. - Derivatives of polynomials
How to compute the derivative of a polynomial. - Derivatives of more general power functions
How to compute the derivative of power functions. - A refresher on the quotient rule
How to compute the derivative of a quotient. - A refresher on the product rule
How to compute the derivative of a product. - A refresher on the chain rule
How to compute the derivative of a composition of functions. - Minimization and maximization refresher
How to calculate the maxima and minima of a differentiable function. - Local minima and maxima (First Derivative Test)
How to calculate the local maxima and minima of a differentiable function. - An algebra trick for finding critical points
An old algebraic trick that can make finding critical points much easier.