Pages similar to: Introduction to local extrema of functions of two variables
- Two variable local extrema examples
Examples of calculating the critical points and local extrema of two variable functions. - Local minima and maxima (First Derivative Test)
How to calculate the local maxima and minima of a differentiable function. - Minimization and maximization refresher
How to calculate the maxima and minima of a differentiable function. - Introduction to Taylor's theorem for multivariable functions
Development of Taylor's polynomial for functions of many variables. - Solutions to minimization and maximization problems
Solutions to sample problems illustrating minimization and maximization. - 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. - 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. - Approximating functions by quadratic polynomials
Determining how to use information from the second derivative to form a quadratic approximation to a function. - Developing an initial model to describe bacteria growth
By analyzing some data and hypothesizing rules for cell division, we develop a discrete dynamical system for the growth of a population of bacteria. - Spruce budworm outbreak model
A graphical analysis of a differential equation model reveals how the spruce budworm could devastate forests every 40 years. - 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. - Using the Forward Euler algorithm to solve pure-time differential equations
By pretending that the slope of a function is constant over small intervals, we following tangent lines to estimate the solution to pure-time differential equations. - Determinants and linear transformations
A description of how a determinant describes the geometric properties of a linear transformation. - 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 conservative vector fields
An introduction to the concept of path-independent or conservative vector fields, illustrated by interactive graphics. - Introducing rabbit predators
Exploring the interactions between fox and rabbit populations when foxes are introduced to an exploding population of rabbits. - Initial dynamical systems exploration
Exploring models of anteater population growth and lead decay in the bloodstream using interactive applets. - How to determine if a vector field is conservative
A discussion of the ways to determine whether or not a vector field is conservative or path-independent. - An algebra trick for finding critical points
An old algebraic trick that can make finding critical points much easier. - The idea of a dynamical system
The basic concept of a dynamical system as the evolution of something over time. The fundamental ideas of the state space and temporal evolution rules are illustrated with examples featuring interactive graphics.