★★★ Detailed Solution to Midterm Examination (Int’l Business, WUCL) ★★★

Spring 2009 Instructor: 衛忠欣 (Jong-Shin Wei)

管理數學 (Mathematics for Management) (07)342-6031 ext.6222

93001@mail.wtuc.edu.tw jsw12011958@gmail.com

3:10 pm ~ 4:40 pm, April 22, 2009

Open books/notes exam. 110 points in 90 minutes; absolutely no talking nor borrowing items during exams. 可使用自己的資料、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。務必工整依序作答。平時上課頻尿的極少數同學，考試時，切勿改變習慣。理由一：有害健康；理由二：迫使教師合理懷疑平時是否謊報。只是題目很多，目的在於測試是否有平日閱讀英文資料之習慣。Watch your time and have fun!

Part One: 30 Multiple choice questions. [By multiple choice in an English-speaking academic environment, we mean that you must choose the most appropriate one from 4 alternatives. 依慣例，答錯不倒扣！] 90 points (別懷疑，看看誰是實力派！)

1. Once upon a time in Utopia a small kingdom was ruled by The Mighty King Richard (李察王). The actual weight of King Richard was way over 80 kg but he insisted that his weight was only 60 kg. One day, John, who was his prime minister, joked about it, “My Lord, I have no problem with your weight record. Yet, if your weight is 60 kg, some child might have zero in weight.” Since the King was also good in mathematics, he immediately found a flaw in John’s statement. Which of the following made the King detect the flaw?

(A)E = mc2;

(B)畢氏定理；

(C)兩條平行線永不相交；

(D)A:B = C:D.

Ans: D. [Recall lectures. Now we can see who actually paid attention to lectures.]

2. There are twenty basketball teams. Each team must play exactly one game with the rest 19 teams. How many games will be played in total?

(A) 200;

(B) 190;

(C) 400;

(D) None of the above is correct.

Ans: B. [Recall lectures. Note that (20)(20 - 1)/2 = 190.]

3. Let f be a function mapping from R to R, defined by f(x) := x(superscript 3). Find the incorrect statement.

(A) The graph is convex-shaped;

(B) It has no local minimum;

(C) It has no local maximum;

(D) df(x)/dx = 0 at x = 0 only;

Ans: A. [For non-negative x, it is convex-shaped; for non-positive x, it is concave-shaped.]

4. Let f be a function mapping from R to R, defined by f(x) := x(superscript 3).

(A) d(superscrpt 2)f(x)/dx(superscript 2) < 0 for all x;

(B) d(superscrpt 2)f(x)/dx(superscript 2) ≧ 0 for all x;

(C) d(superscrpt 2)f(x)/dx(superscript 2) = 0 only when x = 0;

(D) d(superscrpt 2)f(x)/dx(superscript 2) is not defined at x = 0.

Ans: C. [Obviously, d2f(x)/dx2 > 0 for all x > 0; d(superscrpt 2)f(x)/dx(superscript 2) < 0 for all x < 0; d(superscrpt 2)f(x)/dx(superscript 2) = 0 when x = 0.]

5. Let f be a function mapping from R to R, defined by f(x) := ∣x∣. The graph of f

(A) looks like the English letter V;

(B) is a straight line;

(C) looks like the putting the letter V upside down;

(D) is not in one piece.

Ans: A.

6. Let f be a function mapping from R to R, defined by f(x) := ∣x∣. We know that d2f(x)/dx2

(A) > 0 for all x > 0;

(B) < 0 for all x < 0;

(C) = 0 when x = 0;

(D) = 0 for all non-zero x.

Ans: D. [The function is not differentiable at x = 0.]

7. Let f be a function mapping from R to R, defined by f(x) := ∣x∣.

(A) f has a local minimum but no global minimum;

(B) f has a global minimum but no local minimum;

(C) f has neither local minimum nor global minimum;

(D) None of the above is correct.

Ans: B. [Clearly, f(0) < f(x) for all x ≠ 0.]

8. Recall that a relation can be a function. Which of the following is the graph of some function?

(A) {(a, 1), (b, 2), (a, 2), (c, 3)};

(B) {(1, (0, 0)), (2, (0, 1)), (3, (0, 2), …};

(C) {(x, y) of R (superscript 2): x (superscript 2) + y (superscript 2) = 1};

(D) {(x, y) of R (superscript 2): y = x or y = -x}.

Ans: B. [Recall lectures]

9. English teacher asks each student to pick his/her favorite English first name. This is a well-known way of illustrating the notion of functions. Here

(A) the set of all students is the domain of this function;

(B) the set of all English first names is the domain of this function;

(C) the graph of this function has more elements than the domain has;

(D) this function is a real-valued function.

Ans: A. [Recall lectures]

10. Which of the following cannot serve as an application of equation x + y = 1?

(A) price-taking consumer’s budget line;

(B) production possibility frontier;

(C) demand curve;

(D) money supply curve.

Ans: D.

*11. Let f be a function mapping from R to R, defined by f(x) := 1 if x is rational (有理數); -1 if x is irrational (無理數).

(A) The function f is continuous but not differentiable;

(B) For any real number x, f(x) is either a global maximum or a global minimum (but not both);

(C) For any real number x, f(x) is either a local maximum or a local minimum (but not both);

(D) Neither global maximum nor global minimum exist for this function.

Ans: B.

12. Let f be a function defined as follows. f(x) := A if x belongs to [80, 100]; f(x) := B if x belongs to [70, 79]; f(x) := C if x belongs to [60, 69]; f(x) := F if x belongs to [0, 59]. Here,

(A) the domain of f is R;

(B) the range of f is R;

(C) f is not a real-valued function;

(D) f has a (global) maximum.

Ans: C.

13. On the x-y space, if we draw the graph of some function in a “continuous” way (一筆畫出來), then this function

(A) must be differentiable;

(B) must be continuous;

(C) must have at least one local maximum;

(D) must have the global maximum.

Ans: B. [Trivial]

14. The demand schedule is given as follows, P = 1, Q = 9; P = 2, Q = 8; P = 3, Q = 7; P = 4, Q = 6; P = 5, Q = 5; P = 6, Q = 4. We know nothing else. The price elasticity of demand

(A) is -1 at P = 5;

(B) is 1 at P = 5;

(C) is increasing as P goes up;

(D) is not defined here.

Ans: D. [Recall lectures]

15. Let f be a function mapping from an open interval in R to R such that d2f(x)/dx2 < 0 holds for all x. We know that

(A) df(x)/dx < 0 holds for all x;

(B) df(x)/dx > 0 holds for all x;

(C) df(x)/dx ≠ 0 holds for all x;

(D) df(x)/dx is a decreasing function of x.

Ans: D.

*16. Let f be a function mapping from an open interval in R to R such that

d(superscrpt 2)f(x)/dx(superscript 2) ≦ 0 holds for all x (in the domain). We know that

(A) if f has a local maximum at x*, then f(x*) is the global maximum;

(B) f must have at least one local maximum;

(C) f can be maximized;

(D) f cannot have any local minimum.

Ans: A. [Note that (B) and (C) are false in light of the case where df(x)/dx > 0. We eliminate (D) by considering the case where f is a constant function.]

*17. Let f be a function mapping from an open interval in R to R such that (1)

df(x)/dx = 0 at some x* in the domain, and (2) d(superscrpt 2)f(x)/dx(superscript 2)≦ 0 holds for all x. We know that

(A) f cannot have a local minimum at x*;

(B) f(x*) cannot be the global minimum;

(C) there exists no x** satisfying x** ≠ x* and df(x)/dx = 0 holds for x**;

(D) f has a local maximum at x* and that f(x*) is the global maximum.

Ans: D.

18. Let f be a function mapping from the set of non-negative real numbers to R, defined by f(x) := (x) (superscript 0.5) for all x. Then

(A) df(x)/dx > 0 for all x in the domain;

(B) df(x)/dx is defined everywhere but at x = 0;

(C) d2f(x)/dx2 < 0 for all x in the domain;

(D) f is not concave-shaped.

Ans: B. [是在幫忙複習微積分嗎？ Well, it is on the house.]

19. Let f be a function mapping from the set of non-negative real numbers to R, defined by f(x) := 2x + 3 for all x. Then

(A) df(x)/dx > 0 for all x in the domain;

(B) f cannot be minimized;

(C) d(superscrpt 2)f(x)/dx(superscript 2) = 0 for all x in the domain;

(D) f cannot be maximized.

Ans: D.

20. Let f be a function mapping from the set of non-negative real numbers to R, defined by f(x) := Min{x, 100} for all x. Then

(A) df(x)/dx > 0 for all x > 0;

(B) f cannot be minimized;

(C) d(superscrpt 2)f(x)/dx(superscript 2) = 0 for all x > 0 and x ≠ 100;

(D) f cannot be maximized.

Ans: C.

21. Let f be a function mapping from the set of non-negative real numbers to R, defined by f(x) := Max{x, 100} for all x. Then

(A) df(x)/dx = 1 for all x > 0;

(B) f cannot be minimized;

(C) d(superscrpt 2)f(x)/dx(superscript 2) is not defined for all x > 0;

(D) f cannot be maximized.

Ans: D.

22. Let the graph of some function f be {(x, y): x can be either man or woman; y = 1 if x is a man; y = 0 if x is a woman}. We know that

(A) the domain of f is {0, 1};

(B) f can be both maximized and minimized;

(C) f has a local maximum;

(D) f is continuous.

Ans: B. [The maximum of f is 1; the minimum of f is 0. See slide no.9 of Lecture 1.]

23. The derivation of delta TR/TR = n*(delta P/Po)(square) + (1 + n*)(delta P/Po) is mainly for

(A) estimating the price elasticity of demand;

(B) maximizing total revenue;

(C) using the rate of change in price to find the rate of change in total revenue;

(D) using the change of price to find the change of total revenue.

Ans: C. [Recall handout]

24. Let the demand curve be P(square)Q = 9 defined for all P > 0.

(A) The price elasticity of demand is -1 when P = 5;

(B) The price elasticity of demand is -1 when P = 3;

(C) The price elasticity of demand is -1 when P = 1;

(D) None of the above is correct.

Ans: D. [Straightforward]

25. Let the demand curve be downward-sloping, continuous (in shape) but not linear. Suppose that the demand function is not in the form of f(P) := P (superscript -0.5) for all P > 0.

(A) Total revenue can be maximized;

(B) Total revenue will rise when we increase the price;

(C) Total revenue will rise when we decrease the price;

(D) None of the above is correct.

Ans: D. [See Proposition 3]

26. Let the demand curve be given by Q = P(superscript -1) for all P > 0.

(A) Total revenue can be minimized;

(B) Total revenue will rise when we increase the price;

(C) Total revenue will rise when we decrease the price;

(D) None of the above is correct.

Ans: A. [Obvious]

27. If the demand curve is linear and downward-sloping. Currently n* = -0.5 and total revenue is 1,200. If we double the price, the total revenue will be

(A) 1,000;

(B) 2,400;

(C) 1,200;

(D) None of the above is correct.

Ans: C. [By TR/TR = n*(delta P/Po)(square) + (1 + n*)(delta P/Po) we know that delta TR/TR = 0.]

28. If the demand curve is linear and downward-sloping. Currently n* = -2 and total revenue is 1,000. If we cut the price by one-half, the total revenue will be

(A) 525;

(B) 625;

(C) 1,000;

(D) None of the above is correct.

Ans: C. [Note TR/TR = n*(delta P/Po)(square) + (1 + n*)(delta P/Po) = (-2)(-0.5)(square) + (1 - 2)(superscript -0.5) = 0.]

29. If the demand curve is linear and downward-sloping. At any point with n* < -1,

(A) any increase in price will increase total revenue;

(B) any increase in price will decrease total revenue;

(C) any reduction in price will increase total revenue;

(D) total revenue remains constant regardless of the price change.

Ans: B. [See Proposition 1 and related discussion]

30. 以上29題選擇題的正確答案中，出現最頻繁的是

(A) A;

(B) B;

(C) C;

(D) D.

Ans: D. [There are 5 As, 8 Bs, 6 Cs, and 10 Ds. Have you ever seen a weird question like this one? – I can picture that many of you are looking up the word weird at the moment. I doubt but I am shamelessly proud of my design here. Now think strategically why I made this question.]

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。教師也有權公告具有特色之作答。] 20 points

31. Let f be a continuous function mapping from R to R. It has a local maximum at x = 1. It has another local maximum at x = 3. It has the global maximum at x = 1 only. It has no local minimum. Furthermore, it is differentiable everywhere but at x = 2. Now you are asked to carefully sketch the graph of f to satisfy all these conditions listed above.

Ans: (to be given on the chalkboard at our next meeting)

32. By now you should have read, with great care, my handout on price elasticity, which is also on e-learning. 請扼要陳述此份講義的特色與貢獻。

Ans: (trivial and skipped here)

Extra credits question (bonus range: 0 ~ 6 points)

red vs. green -- 選紅球呢？還是選綠球？規則如下：

如果在你(妳)以外的相對多數選的是紅球，你(妳)正好也選紅球，你(妳)會被加 5分。

如果在你(妳)以外的相對多數選的是綠球，你(妳)正好也選綠球，你(妳)會被加 2分。

如果在你(妳)以外的相對多數選的是紅球，而你(妳)卻選綠球，你(妳)會被加6 分。

如果在你(妳)以外的相對多數選的是綠球，而你(妳)卻選紅球，你(妳)會被加0 分。

Now make your pick.

(歡迎全貌下載流傳。)