MA 242.003

• Day 25- February 11, 2013• Review of last week’s material• Section 11.5: The Chain Rule• Section 11.6: The Directional Derivative

ReviewGraphs of f(x,y)

Graphs of f(x,y)

Graphs of f(x,y)

Graphs of f(x,y)

This leads to the idea of level surfaces of f(x,y,z).

The Idea: Describe f(x,y,z) by finding the surfaces

on which it takes constant values.

Example:

Section 11.2

Summary: Section 11.2

Section 11.3: Partial Derivatives

New Notation

There is a similar interpretation of partial derivatives.

Higher derivatives:

Corollary: The corresponding mixed second partial derivatives of polynomials are always equal.

Corollary: The corresponding mixed second partial derivatives of a rational function f are equal at each point of the domain of f.

Section 11.4: Tangent planes and linear approximations

or On the differentiability of multivariable functions

The generalization of tangent line to a curve

Is tangent plane to a surface

So we will say that a function f(x,y) is differentiable at a point (a,b) if its graph has a tangent plane at (a,b,f(a,b)).

We are going to show that if f(x,y) has continuous first partial derivatives at (a,b) then we can write down an equation for the tangent plane at (a,b,f(a,b)).

DEF: Let f(x,y) have continuous first partial derivatives at (a,b). The tangent plane to z = f(x,y) is the plane that contains the two tangent lines to the curves of intersection of the graph and the plane x = a and y = b.

DEF: Let f(x,y) have continuous first partial derivatives at (a,b). The tangent plane to z = f(x,y) is the plane that contains the two tangent lines to the curves of intersection of the graph and the plane x = a and y = b.

Theorem: When f(x,y) has continuous partial derivatives at (a,b) then the equation for the tangent plane to the graph z = f(x,y) is

Def: When f(x,y) has continuous partial derivatives at (a,b) then the linear approximation of f(x,y) near (a,b) is:

Let us next formulate the definition of differentiability for f(x,y) based on the linear approximation idea.

Section 11.5: THE CHAIN RULE

Example:

Example:

Section 11.6Directional Derivatives

andthe Gradient Vector

Show pdf from Stewart’s textbook

We need a practical way to compute this!