Midterm Exam -- Math for Mgmt (Spring 2008)

Midterm Examination (Int'l Business, WUCL)
Spring 2008 Instructor: 衛忠欣 (Jong-Shin Wei)
管理數學(Mathematics for Management) (07)342-6031 ext.6222
93001@mail.wtuc.edu.tw http://www.wtuc.edu.tw/ib

10:30 am ~ 11:30 am, Tuesday, April 22, 2008. 另卷作答。三日後公佈解答於e-learning，也歡迎以email函索
Closed books/notes exam. 110 points in 60 minutes; absolutely no talking nor borrowing items during exams. 可使用自己的字典或翻譯機。行動電話若響起，該生扣十分，以強調基本禮貌。務必工整扼要、依序作答。違反規定，視同放棄調分機會。Watch your time and have fun!
Part One: Multiple choice questions. [Choose the most appropriate one from 4 alternatives. 依慣例，答錯不倒扣。] 80 points 每五題成一列，由左至右，逐題標明題號作答。
1. The graph of function f(x) := 5 if x is in [0, 5]; f(x) := x if x > 5 is the same as the graph of
(A) g(x) := Max{x, 5} defined for all non-negative reals x;
(B) g(x) := Min{x, 5} defined for all non-negative reals x;
(C) g(x) := Min{x + 5, x – 5} defined for all non-negative reals x;
(D) g(x) := 5 + Min{x, 5}.
2. How many elements can we find in the set {x is a real: x ≧ 1 and x ≦ 1}?
(A) zero;
(B) one;
(C) two;
(D) None of the above is correct.
3. The incorrect statement about (0, 1) is
(A) it is a point on the first quadrant;
(B) {x is a real: 0 < x < 1};
(C) {(x, y) of R(square): x = 0 and y = 1};
(D) it is a closed interval with endpoints 0 and 1.
4. The set {(x, y) of R2: the slope of the line going thru (x, y) and (1, 2) is 4} is
(A) 4x + y = 6;
(B) 4x - y = 2;
(C) x - 4y = 7;
(D) x + 4y = 9.
5. Choose (x, y) to maximize 2x + y subject to x + y ≦ 5, x ≧ 0, and y ≧ 0.
(A) The solution is (5, 0) only;
(B) The solution is (0, 5) only;
(C) Both (0, 5) and (5, 0) are solutions;
(D) None of the above is correct.
6. [continued from question 5] Which of the following cannot serve as an economic application?
(A) Prices of two consumption goods are 2 and 1 respectively;
(B) Utility function is U(x, y) := 2x + y;
(C) Utility function is U(x, y) := (2x + y)2 + 1;
(D) Disposable income is 10.
7. The set {(x, y) of R2: xy = 1, x > 0, and y > 0} can be
(A) the graph of function f(x, y) := xy for all x > 0 and y > 0;
(B) the graph of function f(x) := 1/x for all x > 0;
(C) the graph of function f(x) := 1/y for all y > 0;
(D) None of the above is correct.
8. 「我對你百分之兩百的支持。」In English, it is better put as
(A) You have my 110% support;
(B) You have my 200% support;
(C) I give you 200% support;
(D) You have my 200 degrees in support.
9. Let f(x) be twice-differentiable (二次微分存在) and be defined for all non-negative reals. If we know that f(x*) is a local maximum, then
(A) x* = 0 is possible;
(B) we can find an interval (a, b) containing x* such that f(x) ≧ f(x*) for all x in (a, b);
(C) The equation df(x)/dx = 0 has only one solution, which is x*;
(D) d2(f(x))/dx2 ≦ 0 for all x > 0.
10. Consider the function f(x) := x4 defined for all real numbers x.
(A) f(0) is the unique local maximum and unique global maximum;
(B) f can be neither maximized nor minimized;
(C) d2(f(x))/dx2 > 0 for all x;
(D) The local minimum is also the global minimum.
11. Recall the kissing problem (and experiment). If there are 50 kids, then there will be
(A) 100 kissing activities to be performed;
(B) 1,225 kissing activities to be performed;
(C) 1,250 kissing activities to be performed;
(D) None of the above is correct.
*12. [continued from question 11] Recall the paper “Having Fun with Organized Kissing” which you can find on e-learning. By using the proposed method (or scheme) and assuming that each kiss takes 10 seconds, how long does it take to complete the process?
(A) 1,850 seconds;
(B) 1,470 seconds;
(C) 1,180 seconds;
(D) None of the above is correct.
13. [Jump to question 21] In this scenario,
(A) the disutility of hard-working is w - 10;
(B) the rate of unemployment is 1/4;
(C) the rate of unemployment is 0.05;
(D) the unemployment benefit is 6.
14. [continued from question 13] Which of the following will increase the efficiency wage?
(A) A decrease in the disutility of hard-working;
(B) An increase in the rate of unemployment;
(C) An increase in the probability of getting caught while shirking;
(D) An increase in the wage at the next available job.
15. Recall lecture 1. There are three workers, indexed by A, B, and C, ready to accomplish a labor-intensive job of which the amount is normalized to 1, that is, k = 1. Assume fixed productivity, perfect divisibility, and the absence of externalities. Suppose that it takes 10 days if A and B work together (while C does not work at all); if takes 10 days if A and C work together (while B does not work at all); if takes 8 days if B and C work together (while A does not work at all). Denote by x the amount of work done by A working for one day. Similarly y and z represent the amount of work done by B and C working for one day respectively. Obviously,
(A) A is the most efficient or productive;
(B) C is the most efficient or productive;
(C) B is the least efficient or productive;
(D) None of the above is correct.
16. [continued from question 15] How many days does A need if A works alone?
(A) 20 days;
(B) 70/3 days;
(C) 80/3 days;
(D) None of the above is correct.
17. [continued from question 15] How many days does B need if B works alone?
(A) 16 days;
(B) 15 days;
(C) 14 days;
(D) None of the above is correct.
18. [continued from question 15] If three workers work together, then how many days do they need to finish this job?
(A) 4 days;
(B) 70/13 days;
(C) 80/11 days;
(D) None of the above is correct.
19. 本棟大樓的每間教室，皆被分配一個號碼(如506)。此一real-valued function
(A) does not have any local maximum;
(B) does not have any global maximum;
(C) cannot be minimized;
(D) is continuous but not differentiable.
20. (Now it is time to find out who have paid adequate attention to lecturing.) 瀏覽97.03.24網路新聞「張正傑新竹彈三角戀鬧美猴王」之用意是
(A) 修習管理數學可以訓練作戰策略；
(B) 數學訓練可以培養精確性；
(C) 示範沒有備課的教師，如何打發時間；
(D) 誘使大家指出誰是班上最調皮的同學。

Part Two: Analytical questions. [注意字體工整、邏輯順暢與不要出現錯別字或注音(與火星文)！如果字跡難以辨認，視同未答。畫蛇添足、答非所問、不知所云、自曝其短，將不利得分。如昔，教師有權公告具有特色之作答。] 30 points
21. [20 points] Recall lecture 2. This is about the determination of efficiency wage that provides an incentive for a worker to work hard, not to shirk. A worker is paid at wage w > 0. If he works hard (as expected), his utility is w - 10. If he shirks, then with probability 1/4, he will be caught. Once being caught, he will be fired at once (and getting no pay). After being fired, with probability 0.05, he will remain unemployed and will be entitled to receiving unemployment benefit, which is 2. And with probability 0.95, he will be getting a new job at wage 6. If he shirks, then with probability 3/4, he will not be caught and his utility will be w.
(i) Use a decision tree to illustrate this worker’s decision-making.
(ii) Find the minimal wage providing an incentive for him to work hard. In other words, find the efficiency wage w.

22. [10 points] Suppose that you and I are playing the scissors-rock-paper game. We both show our choices simultaneously. The loser gives $1 to the winner. Nobody pays if there is a tie. Carefully draw a game tree to describe it.

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