Math 242, Fall 2013, Assignment 8

Do not imagine that mathematics is hard and crabbed and repulsive to common sense. It is merely the etherealization of common sense.

- Lord Kelvin

Carefully define the following terms:

1. Directional derivative.
2. Gradient vector.
3. Local maximum.
4. Local maximum value.
5. Critical point.
6. Saddle point.
7. Absolute maximum.
8. Absolute maximum value.
9. Closed set.
10. Bounded set.

Carefully state the following theorems:

1. Theorem relating directional derivatives to the gradient (equation 14 on page 794).
2. Interpretation of the gradient vector (Theorem 15 on page 795).
3. Theorem relating level surfaces to gradients (paragraph following equation 18 on page 798).
4. Fermat's Theorem for functions of two variables.
5. Second Derivatives Test.
6. Extreme Value Theorem for functions fo two variables.
7. Extension of the Closed Interval Method to two variables (box 9 on page 808).
8. Method of Lagrange Multipliers.
9. Method of Lagrange Multipliers with two contraints.

Solve the following problems:

1. Section 11.6, problems 1, 5, 7, 11, 21, 36, and 41.
2. Section 11.7, problems 1, 3, 5, 7, 11, 27, and 46.
3. Section 11.8, problems 3, 11, 19, and 21.

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