Description

This class corresponds to MATH 251 at Texas A&M University

Prerequisites

Calculus III heavily requires knowledge from Calculus II and its prerequisites.

3 Dimensional Functions

Multivariate Derivatives

Partial Derivatives

Partial Derivative - taking the derivative of a multivariable function. They are taken with respect to one variable while treating the other variables as constants.

For a function of 2 variables (z= f(x, y)), this is the slope of a surface along either the x or y axis.

• fx(x, y) is the partial derivative with respect b x (df/dx)
• fy(x, y) is the partial derivative with respect to y (df/dy)
• fxx(x, y) is the partial derivative of fx(x, y) with respect to x (d2f/dx2)
• fyy(x, y) is the partial derivative of fy(x, y) with respet to y (d2f/dy2)
• fxy(x, y) = fyx(x, y) are the mixed second order partial derivatives (equal by Clairaut's Theorem) (d2f/dxdy or d2f/dydx)

Approximating Multivariable Functions

A surface in space has planes tangent to points on the surface. The equation for a tangent plane of the surface z=f(x,y) at the point (x0,y0,f(x0,y0)) is found by:

Be aware that we will present another version of this formula using vectors later on.

To approximate z = f(x,y), note that a change in x and a change in y will result in a change in z. This can be found in exact form:

But also differential form (an approximation):