Review for Test

338 ACTIVITY 9: 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:

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

PERFORMANCE CRITERIA:

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

RESOURCES:

Sample Test
List of test topics
Activities, quizzes, homework, notes.
15 minutes

PLAN:

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

CRITICAL THINKING QUESTIONS:

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

Note: You may bring in a 1 page cheat sheet. There will be one proof (choose one out of two).

Finding equation, graph, slope, y-intercept, direction vector of a line
Dot product                             Standard Form of an LP problem
Matrix operations                       Slack variables
Row Echelon Form                        Surplus variables
Elementary matrices                     Line segment
Rank                                    Convex set
Inverse of a matrix                     Hyperplane
Column space and its basis              Upper & Lower halfspaces
Solution space and its basis            Linear Combination & Hull               
Linear combination of vectors           Affine Combination & Hull
Linear independence                     Convex Combination & Hull
Linear dependence                       Conical Combination & Hull
Dimension of a space or subspace        Convex cone with vertex at the origin
Spanning set                            Convex Polytope
Math model diagram                      Convex Polyhedron
Math programming problem definition     Extreme Points of a Convex Polyhedron
Linear vs. Nonlinear problem            Direction Vectors of a Convex Polyhedron
Unconstrained vs. constrained problem   Finite Basis Theorem
Problem Solving Methodology             Extreme Point Theorem
   Graphical solution of Linear Programming Problems of two variables

Math 338 Sample Test 1

(10 points) Prove one of the following:
1. If S and T are convex sets then S intersect T is convex.
2. If matrix B is the inverse of A then B = En*En-1 ... *E1 where E1, E2, ... En are the elementary matrices formed while reducing A to row echelon form.
(5 points) What do we mean when we say that a set {V1, ..., Vn} of vectors in Rm spans Rm? How can we tell from the row echelon form of an mxn matrix if its columns span Rm?
(3 points) If the feasible region for a math programming problem is the whole space, what is this type of problem called?
(3 points) What are the key differences between a polytope and a polyhedron? Which of these figures can possess direction vectors?
(4 points) Describe or sketch the linear, affine, convex, and conical hulls of {[0 0]T, [1 2]T, [4 8]T}
(20 points) Set up the following problem as a Math Programming problem; (i.e. identify the following: problem definition, issues, data, assumptions, variables, objective function, constraints).

Karlin Enterprises manufactures two games. Standing orders require that at least 24,000 monopoly and 5,000 scrabble games be produced each month. The company has two factories: the Gainesville plant can produce 600 monopoly and 100 scrabble games per day; the Sacramento plant can produce 300 monopoly and 100 scrabble games per day. If the Gainesville plant costs $20,000 per day to operate and the Sacramento plant costs $15,000 per day, find the number of days per month that each factory should operate to minimize the cost.
(10 points) Plot the feasible region for the following problem and find the extreme points and direction vectors of the region. Find the convex hull of the extreme points. Solve the problem.
```
          minimize 4x + 3y
          subject to:
                  2x +  y >= 80
                   x +  y >= 50
                        x >= 0
                        y >= 0
```
(15 points) Given the system of equations
```
 
      x + 2y + 2z -  s + 3t = 0
      x + 2y + 3z +  s +  t = 0
     3x + 6y + 8z +  s + 5t = 0
```
1. Find a basis and dimension of the solution space for this system.
2. Is the vector [10 1 -4 1 -1]T in the solution space? Give a reason for your answer.
3. Find a basis and dimension for the column space of the coefficient matrix of this system.
4. Determine whether the columns of the coefficient matrix are linearly independent. Do they span R3?

(10 points) Given the following L.P. problem, write it in standard form.

     Maximize  x1 + x2 + x3 + x4
   Subject to  x1 + x2        >=  4
               x2 + x3         =  8
                    x3 + x4   >= -1 
          x2 >= 0,   0 <= x3 <= 6,    x4 <= 0

(14 points) Given the function f(x) = x2ln x3 - 3x2
1. Find its domain and derivative.
2. Find all the critical points for this function
3. Find all relative max and min points for this function.
4. Find the absolute max and min over the interval: [ 1, e]
(6 points) Find the gradient of the function x3y2z - xz2

Extra Credit: Solve the math programming problem given in problem 6.

Math 338 Activity 9 -- Revised 9/15/98