1. MA 242.003 Day 34- February 22, 2013 Review for test #2 Chapter 11: Differential Multivariable Calculus

4. Paraboloids

5. Ellipsoids

6. Cones

7. Planes

8. Cylinders

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

11. Example:

14. Section 11.2: Limits and Continuity

17. Types of functions we will study: 1. Polynomials: 2. Rational functions: 3. Compound functions:

18. Types of functions we will study: 1. Polynomials: Continuous everywhere 2. Rational functions: 3. Compound functions:

19. Types of functions we will study: 1. Polynomials: Continuous everywhere 2. Rational functions: Continuous where defined 3. Compound functions:

21. Summary: Section 11.2 In future work you will be required to be able to determine whether or not a function is continuous at a point.

22. Section 11.3: Partial Derivatives

29. Geometrical interpretation of Partial Derivatives

30. There is a similar interpretation of partial derivatives.

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

34. The generalization of tangent line to a curve

35. Is tangent plane to a surface

37. 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

38. at P = (1,2,4)

43. (continuation of example)

44. Section 11.5 THE CHAIN RULE

47. Example:

48. Section 11.6 Directional Derivatives and the Gradient Vector

50. We need a practical way to compute this!

54. Question: Can the right-hand-side be written as a DOT PRODUCT?

61. Tangent planes to level surfaces

67. Significance of the Gradient Vector

72. Section 11.7: OPTIMIZATION As an application of our work in chapter 11, we set up the theory of how to find the local maximum and minimum values of f(x,y)

73. Section 11.7: OPTIMIZATION As an application of our work in chapter 11, we set up the theory of how to find the local maximum and minimum values of f(x,y)

79. So to find the local maxima and minima of a differentiable f(x,y) do the following:

82. (Continuation of example)