1. 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

2. Review Graphs of f(x,y)

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

7. The Idea: Describe f(x,y,z) by finding the surfaces on which it takes constant values. Example:

9. Section 11.2

11. Summary: Section 11.2

13. Section 11.3: Partial Derivatives

16. New Notation

18. There is a similar interpretation of partial derivatives.

20. Higher derivatives:

22. 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.

23. Section 11.4: Tangent planes and linear approximations or On the differentiability of multivariable functions

24. The generalization of tangent line to a curve Is tangent plane to a surface

25. 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)).

27. 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.

28. 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

29. 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.

33. Section 11.5: THE CHAIN RULE

41. Example:

44. Section 11.6 Directional Derivatives and the Gradient Vector

45. Show pdf from Stewart’s textbook

47. We need a practical way to compute this!