Math 119 - Week 1

Contour plot =

Take multiple level curves to construct the full function

Always 1 dimension below actual function dimension

Level curve =

A "slice" of a 3D multi-variable function at a specific constant/height to get a 2D image

Clairaut's Theorem =

Let f_x and f_y be partial derivatives

if f_x and f_y and f_xy exists near (a, b) AND f_xy is continuous near (a, b) then f_yx must also exist

also: f_xy(a, b) = f_yx(a, b)

Tangent plane equation =

z = z_0 + [f_x(x_0, y_0)](x - x_0) + [f_y(x_0, y_0)])(y - y_0)

Where (x_0, y_0, z_0) is the point of tangency

Define mini-max/saddle point =

point that can be local max or min depending on the angle

partial derivative =

∂

Derivative of one variable in a multi-variable function, rest of the variables are constant