338 ACTIVITY 25: Review for Test

WHY:

Tests help you measure the level of understanding you have gained while learning the material. They can and should be opportunities for learning as well. Preparation for tests helps organize and synthesize the knowledge you have gained during the preceding part of the course.

LEARNING OBJECTIVES:

  1. Discover the general nature of a Math 338 test.
  2. Brainstorm and prioritize questions about topics which confuse one or more team members.
  3. Discover successful approaches to study for the test.

PERFORMANCE CRITERIA:

  1. Quality of the questions identified by the team.
  2. Success in finding an approach to use to study for the test.

RESOURCES:

  1. Sample Test
  2. List of test topics
  3. Activities, quizzes, homework, notes.
  4. 15 minutes

PLAN:

  1. Choose roles if you have not already done so.
  2. Brainstorm and come up with 5 topics which cause confusion in the material covered for the test.
  3. Identify 5 questions whose answers will help clarify the confusion.
  4. Prioritize the questions so the most pressing is asked first.

CRITICAL THINKING QUESTIONS:

  1. Did you have to omit areas of importance in coming up with the answer to Question 1? If so, what were they?

  2. How easy was it to come to consensus on the priority of each question? What were some of the obstacles to this process?

Math 338 Test Topics -- Chapters 4 and 5 (omit 4.5)

Note: You may bring in a 3-sided cheat sheet. There will be one proof (choose one out of two). 


Relative (local) max and min            Diagonalization test for definiteness
Absolute (global) max and min           Convex/concave functions
Stationary (critical) point             Strictly convex/concave functions
Point of inflection                     epigraph
Saddle point                            hypograph
Necessary conditions for max and min    Max and min of convex/concave functions
Sufficient conditions for max and min   Necessary/sufficient conditions if X >= 0 
Hessian matrix                          Equality constraints
Quadratic form                          Equality constraints with X >= 0
Positive/negative definite              Lagrange multipliers
Positive/negative semidefinite          Lagrangian function
Indefinite                              Kuhn Tucker conditions/ inequality constraints

Math 338 Sample Final Test

  1. (10 points) Prove ONE of the following two statements:

    1. If ac - b2 > 0 and a < 0, then prove that is negative definite.

    2. Prove that f is concave iff -f is convex.

  2. (20 points) Determine where the following function is convex, strictly convex, concave, and strictly concave over the reals. Also find any relative max or min points.

    f(x) = 3 - 4/x + 4/x2

  3. (20 points) Given the following problem: 
              minimize (x1 + 2)2 + 2(x2 - 1)2 + (x3)2
          subject to  x2 + x3 = 6
    1. Find the relative minima of this function.

    2. Find the minimum points of this function if we require that all the variables be nonnegative.

  4. (20 points) Find the stationary points, and determine whether each is a relative min, a relative max, or a saddle point for the following function:

    x4 - 2x2 + y2 + 2z2 - 16z + 33

  5. (30 points) Given the following nonlinear optimization problem:

    Suppose that the return on investment for a portfolio with x shares of stock A and y shares of stock B is given by the function:

    6x - 2x2 + 2xy - 2y2

    Market conditions dictate that the number of shares of stock A should never exceed 10 more than twice the number of shares of stock B. Also the total of 4 times the number of stock A shares plus the number of stock B shares should not exceed 76.

    1. Show that the return on investment function is concave everywhere.

    2. Plot the constraint region, verify that it is a convex polytope and find its extreme points.

    3. Write out the conditions of the Kuhn Tucker Problem for finding the maximum return on investment for this portfolio under the given market conditions. Verify that the portfolio should contain 2 shares of stock A and 1 share of stock B to maximize the return on investment. Also find the values of the Lagrange multipliers which yield this maximum value.

      Extra Credit: For tax purposes, the portfolio owner is interested in finding the number of shares of each stock in order to minimize her return on investment. Use the results of part b to solve this problem.

Math 338 Activity 25 -- Revised 12/4/98

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