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MAT 218 Final Practice Test
1. Let [pic], find
[pic] [pic]
2. Find the domain of the function in two variables
[pic]
3. Find the distance between the points[pic]and [pic]; the coordinates of the midpoint.
4. Find the equation of the sphere with the center at [pic]and radius 4
5. Show that [pic]is a level curve of [pic]
6. Find [pic]and [pic]for the following functions:
a. [pic] b. [pic]
7. Find [pic] for the function
[pic]
Multivariate Optimization.
8. Find the critical points of the function and determine if it is a max, min, or saddle point. [pic]
9. For the function [pic]
a. Find all the partial derivatives of the first and second order
b. Find the stationary point and classify it by using the second-derivative test.
10. Let [pic]and [pic].
a. Find all first order partial derivatives of[pic]and find the stationary point for f.
b. Find the Extreme points for f over S.
11. Find the max and min points for the following
[pic]subject to [pic]and [pic]
12. Which of the following sets are open, closed, bounded, compact?
a. [pic]
b. [pic]
13. Find the maximum and minimum values of [pic] under the constraint [pic]
Systems, Matrices.
14. Solve the system of equations and determine whether it is independent, dependent, or inconsistent.
[pic]
15. Perform the following operations for the given matrices. If it is impossible, explain why.
[pic] [pic] [pic]
a. [pic]
b. [pic]
c. [pic]
16. Multiply the given matrices. If it is impossible, explain why.
[pic] [pic] [pic]
a. [pic]
b. [pic]
c. [pic]
17. Show that the following matrices are inverses of each other:
a. [pic]
b. [pic]
18. Write the system of equations in a form of [pic], find [pic]and solve for X using [pic].
[pic]
Probability.
19. Draw a random card from a standard deck of 52 cards. The sample space is the collection of the 52 cards (outcomes of the draw). Probability of selecting any card is 1/52. Let
[pic]
[pic]
Find
a.[pic] b. [pic]
c. [pic] d. [pic]
20. Two fair six-sided dice are rolled and the outcomes are recorded.
a. Write a sample space for this event.
b. Find the probabilities of the following events:
[pic] [pic]
[pic] [pic]
21. In state lottery five numbers are drawn one at a time without replacement from 1 to 40. The six number – power ball is selected from the numbers from 1 to 100. Find the probability of buying a winning lottery ticket.
22. The following table classifies 100 people by their sex and their voting preference.
| |Male ([pic]) |Female ([pic]) |Totals |
|Democrat ([pic]) |15 |35 |50 |
|Republican ([pic]) |20 |30 |50 |
|Totals |35 |65 |100 |
Compute the following probabilities if one of these 100 students is selected randomly:
a. [pic]
b. [pic]
c. [pic]
23. Let A and B be two independent events with[pic]and[pic]Compute the probabilities of the following events:
a. [pic]
b. [pic]
c. [pic]
24. If [pic],[pic]and[pic] Are these events independent? Explain.
Discrete and Continuous Distributions.
25. 20 golden nuggets are randomly weighted in g (grams). Their weights are recorded below.
3 2 3 1 5
4 1 2 3 2
5 2 2 4 3
1 2 5 3 1
Create a histogram of their frequency distribution.
26. Find the standard deviation for the set of ungrouped sample data:
2,4,5,7,3,1,0,7,9,5
27. Find the standard deviation of the set of grouped sample data
|Interval |2-6 |6-10 |10-14 |14-18 |
|Frequency |2 |6 |4 |3 |
28. A play in the game costs $1.50. The play consists from tossing three coins and the payoff is equal to the number of tails showing up. For example, if there is one tail the net gain is -0.5$, if there are two tails the net gain is $0.5. Find the expected value of the game. Is it a fair game?
29. A payoff table for two courses of action[pic] is given below. Which of the two actions will produce the largest expected value? What is it?
|[pic] |[pic] |[pic] |
| |[pic] |[pic] |
|.4 |-$300 |-$150 |
|.1 |$200 |$300 |
|.2 |$400 |$300 |
|.3 |$50 |$40 |
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