Math 242, Spring 2019, Assignment 8

Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning.

- John Locke, Second Reply to the Bishop of Worcester

Read:

1. Section 14.7.
2. Section 14.8.

Carefully define the following terms, then give one example and one non-example of each:

1. Local maximum.
2. Local maximum value.
3. Local minimum.
4. Local minimum value.
5. Local extremum.
6. Local extreme value.
7. Absolute maximum.
8. Absolute maximum value.
9. Absolute minimum.
10. Absolute minimum value.
11. Absolute extremum.
12. Absolute extreme value.
13. Critical point.
14. Closed set.
15. Bounded set.

Carefully state the following theorems (you need not prove them):

1. Interpretation of the direction of the gradient vector (in terms of rates of change).
2. Interpretation of the magnitude of the gradient vector (in terms of rates of change).
3. Relationship between the gradient of a function and its level curves (or level surfaces).
4. Relationship between local extrema and critical points (Theorem 2 on page 960).
5. Second derivative test.
6. Extreme value theorem.
7. Higher-dimensional analog of the Closed Interval Method (at the top of page 966).
8. Method of Lagrange multipliers (on page 972).

Solve the following problems:

1. Section 14.7, problems 1, 3, 5, 7, 17, 31, 35, 37, 41, and 48.
2. Section 14.8, problems 1 and 3.

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