MATH 2153 - Calculus 3 (18324) Equations Midterm2

Multivariable Functions

Vertical Line Test for two variable functions

vertical line test: a relation F(x, y, z) = 0 is a function if every vertical line parallel to the z-axis intersects the graph of the relation at most once. If the relation fails the vertical line test, then this means that there is a point (x0, y0) that has at least two outputs under the relation.

Level Curves

Trace

A trace of a surface is the set of points at which the surface intersects a plane that is parallel to a coordinate plane.

Limit of a function w/ two variables

Two-path test

If f(x,y) approaches two different values as (x,y) approaches (a,b) along two different paths in the domain, then the lim (x,y) → (a,b) f(x,y) does not exist (DNE).

Boundary points and Interior Points

Interior → lies entirely in domain region

Boundary → lies on edge with only one point in R and one outside

Limits on boundary points

Continuous Function Rules

1. f is defined at (a,b)

2. limit at (a,b) exists

3. limit equals f(a,b)

Continuity Theorem

Partial Derivatives

Theorem on the fxy and fyx partial derivatives

Let f(x,y) be a function defined on an open set D in R². If fxy and fyx are continuous throughout D, then fxy = fyx for all points in D.

If a function is continuous at (a,b), then f is differentiable.

Chain Rule

Implicit Differentiation

Setting all values over to one side and putting it equal to 0 to then equal F(x,y)

For checking continuity with limits, you should what?

Not plug in the values. Instead alter the function by using conjugates or by using the two path method.

Directional derivatives

Gradient

The gradient meaning

It’s the direction of the max. rate of increase for f at (a,b)

The gradient meaning for it’s negative

The direction of the max. rate of decrease for f at (a,b)

The directional derivative for orthogonal directions to the gradient

Is zero.

Tangent line to z0 = f(x,y) at point (a,b) formula

Three variable circumstances for gradient

Tangent planes

Tangent Planes equation

Tangent planes (explicit)

Linear approximation

Linear approximation example

Differentials and change

Local max and min in relation to fx and fy

Critical points

Example on how to find critical points

Saddle points

Second derivative test

Abs max and min