Solution to Exam 3 for Math for Mgmt (summer 2009)

★★★★ Detailed Solution to Examination 3 (Int’l Business, WUCL) ★★★★
Summer 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
9:20 am ~ 10:30 am, July 15, 2009
Open books/notes exam. 100 points in 70 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。
Questions 1 to 10 are multiple choice questions. That is, choose the most appropriate one from four alternatives. 每題6分

1. Let f be a function mapping from A to B. To make f a real-valued function, we need
(A) the domain of f to be a subset of R;
(B) the range set of f to be a subset of R;
(C) the image set of f to be a subset of R;
(D) both A and B to be subsets of R.
Ans: C. [Recall Exam 2.]

2. Define f(x) := 2x3 + 3x2 - 3x + 8 for all x of R. The graph of f contains the point
(A) (0, 0);
(B) (1, 13);
(C) (2, 18);
(D) (3, 80).
Ans: D. [Obviously, f(0) = 8, f(1) = 10, f(2) = 30, and f(3) = 80]

3. Define f(x, y) := Min{x, y} for all (x, y) of R2+ (i.e., the first quadrant). The graph of f contains
(A) ((0, 0), 1) or (0, 0, 1);
(B) (0, (0, 0)) or (0, 0, 0);
(C) ((1, 3), 3) or (1, 3, 3);
(D) ((2, 0), 0) or (2, 0, 0).
Ans: D. [Trivial yet make sure you know why (B) is false.]

4. Consider the following constrained maximization problem. Choose (x, y) from the first quadrant to maximize 2x + y - 3 subject to (1) x + y ≦ 3, (2) 1 ≦ x ≦ 3, and (3) y ≧ 1. The slope of a typical level curve of the objective function is
(A) -2;
(B) -1;
(C) 1;
(D) 2.
Ans: A. [For any k > 0, the slope of 2x + y = k is -2.]

5. [continued from question 4] The constraint set (made by putting together those three weak inequalities) contains points
(A) (0, 0) and (1, 2);
(B) (2, 1) and (1, 2);
(C) (1, 2) and (2, 2);
(D) (1, 2) and (0, 3).
Ans: B. [The constraint set is the triangle with three vertex points, (1, 1), (2, 1), and (1, 2).]

6. [continued from question 4] The solution to this problem
(A) does not exist;
(B) is (2, 1);
(C) is (1, 2);
(D) is (3, 0).
Ans: B. [By visual inspection and recall that the slope of level curves is -2]

7. [continued from question 4] The maximized value of f is
(A) 2;
(B) 3;
(C) 5;
(D) None of the above is correct.
Ans: A. [2(2) + 1 - 3 = 2]

8. Consider the function f(x) := Min{x, 3} defined for all non-negative real numbers x. We know that
(A) f is differentiable;
(B) f has no local maximum;
(C) f has no local minimum;
(D) f can be maximized and minimized.
Ans: D. [Recall lectures]

9. [continued from question 8] The level curve of f at (level) 3 is
(A) a point;
(B) a ray;
(C) a line segment;
(D) an interval.
Ans: B. [Solutions to Min{x, 3} = 3 are those x satisfying x ≧ 3, which is a ray on the real line.]

10. [continued from question 8] The level curve of f at (level) 4 is
(A) a point;
(B) a line segment;
(C) an open interval;
(D) None of the above is correct.
Ans: D. [Solution to Min{x, 3} = 4 does not exist.]

以下每題 10 分

11. Choose x to maximize f(x) = (x)0.5 for all x ≧ 0 subject to Max{x, 3} ≦ 4. Solve it
with help from a diagram. What is the role of calculus here?
Ans: First notice that Max{x, 3} ≦ 4 is the same as x belonging to [0, 4]. Since f is strictly increasing, we see that the solution is x = 4. No calculus is needed here.

12. You are given a rope with length k > 0. You can use it to either make a square or
make a circle. If you wish to maximize the formed area, what will you do? A square or a circle? Justify your choice.
Ans: By [k/(2pi)]2 - (k/4)2 = k2/(4pi) - k2/(16) > 0 due to pi < 4, we know that a circle has larger area than a square does. 這是很多中學生也會的應用題。

13. Recall the movie “The Emperor’s Club” shown on July 14. Carefully choose a
scenario from that film and make it a decision tree having two players. Be precise and organized.
Ans: skipped.

14. 課程已過3/4，此題係測試大家對學習「管理數學」的正確認知。Have fun!

※ good job 這門課太酷了! ............根本不知道自己在幹嘛 = =

※ 感覺老師好像在賣弄學問，管理數學卻一直教經濟學，管理數學真正的內容到底是什麼???如果上學期是你教微積分，那也會把這堂課變成經濟學嗎?老師只重視經濟學而已嗎?

※ 老師你上課教好多經濟，管數好少，雖然上課題材很新鮮很有趣，希望能教一點感覺回家會有收穫的，還有就是能教仔細一點嗎，我還是老話一句希望少用ppt，數學類的這樣真的會很想睡覺

※ 字太醜 看不懂評語

The above (※) are written comments from the end-of-semester teaching evaluations for this course offered in Spring 2009 (i.e., the class of 2012). 在大學評鑑時，訪視委員與同學對談時，如果這些正好是這四位寶貝同學(天兵？)所表達的，訪視委員應如何接話？
Ans: Here are some samples.
(i) Ask students to recall what the course syllabus is; compare students’ recollection with the syllabus on e-learning. Also check if teacher issued typewritten solutions to all exams. [委員皆有電腦可上網]
(ii) Ask some simplest questions such as “Give me an example of constrained maximization”, “What is the use of second-order condition”, and “Tell me why prisoners’ dilemma game is well known outside the economics field.”
(iii) Ask students how, in their opinions, teachers should teach this course if a large number of students did fail calculus in the previous semester.
(iv) Ask students why the college strongly (and proudly) encourages teachers to use PPT in teaching; ask students whether they feel sleepy now.

[訪視委員是來評鑑打分數的，不是來輔導、調停、扮演包青天的。在我自己與友人的評鑑經驗裡，以上的(i)至(iii)皆偶有聽說，(iv)就不容易啦(應該不是文藻學生嗜睡或體弱，只是點名制度把所有人框在教室吧)。俗話說，一樣米養百種人，何況那一班59人(有16位遭期中預警、期末考平均成績退步近10分，學期成績不及格之比例高達42%)，背景迥異，也希望大家都能認真解析這一族群的想法。畢竟，我們不知道60~80%的同學的想法為何，委員也不會知道拔河活動與體育課，帶來的外部性。還好我懂一些訊息經濟學。]

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