Calculus III Chapter 13

Partial Derivatives of f(x,y) z.z.z.z.

Tangent Plane and the Differential zThe tangent plane to the surface z = f(x,y): zThe tangent plane approximation: zThe differential: yFor a function z = f(x,y), the differential, df, at a point (a,b) is the linear function of dx and dy given by the formula:

The Gradient of z = f(x,y) zIf f is a differentiable function at the point (a,b) and  f (a,b)  0, then: yThe direction of  f (a,b) is xPerpendicular to the contour of f through (a,b) xIn the direction of increasing f yThe magnitude of the gradient vector, ||  f ||, is xThe maximum rate of change of at that point xLarge when the contours are close together and small when they are far apart.

The Gradient of w = f(x,y,z) Properties: yThe direction of the gradient vector is the direction in which f is increasing at the greatest rate, if it exists. yThe magnitude, ||grad f||, is the rate of change of f in that direction. yIf the directional derivative of f at (a,b,c) is zero in all directions then the grad f is defined to be 0.

Differentiability zFor a function f at a point (a,b), let E(x,y) be the error in the local linear approximation, that is the absolute value of the difference between the left and right hand sides, and let d(x,y) be the distance between (x,y) and (a,b). Then f is said to be locally linear, or differentiable at (a,b) if we can make the ratio E(x,y)/d(x,y) as small as we like by restricting (x,y) to a small enough non-zero distance from (a,b).