Solution to Midterm -- Introductory Calculus F2007

★★★★ Solution to Midterm Examination (International Business, WUCL) ★★★★
Fall 2007 Instructor: 衛忠欣 (Jong-Shin Wei)
微積分(Introductory Calculus) (07)342-6031 ext.6222
93001@mail.wtuc.edu.tw http://www.wtuc.edu.tw/ib

November 13, 2007
Closed books/notes exam. 104 points in 70 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。10:10 am ~ 11:20 am, Tuesday, November 13, 2007. Watch your time and good luck! 務必工整扼要、依序作答。Watch your time and have fun!
Part One: 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. I did encounter some less intelligent folks who confused it with “複選題”. 依慣例，答錯不倒扣！平均每題不該超過100秒。] 80 points

1. Let the market demand curve be given by P = -0.01Q2 - 0.2Q + 8; the market supply curve be given by P = 0.01Q2 + 0.1Q + 3, where P is the price and Q is the quantity. At market equilibrium, quantity demanded equals quantity supplied.
(A) The equilibrium price is 10;
(B) The equilibrium price is 5;
(C) The equilibrium quantity is 25;
(D) None of the above is correct.
Ans: B. [See p.83, text.]
2. [continued from question 1]
(A) The market demand function is P = f(Q) := -0.01Q2 - 0.2Q + 8;
(B) The market supply function is P = f(Q) := 0.01Q2 + 0.1Q + 3;
(C) Market demand curve and market supply curve only intersect once;
(D) The slope of demand curve is a negative constant.
Ans: C. [Straightforward]
3. If the graph of a function on the x-y place is simply the 45-degree ray starting at (0, 0), then
(A) the function must be f(x) := x for all non-negative reals;
(B) the function must be a constant function;
(C) the function must be f(x, y) := x + y;
(D) the function must be f(x) := 1/x for all positive reals.
Ans: A. [Trivial]
4. If the graph of a function on the x-y plane is the set {(0, 0), (1, 1), (2, 2), (3, 3)}, then
(A) the function must be f(x) := x for all non-negative reals;
(B) the range (set) of this function can be the set of all reals;
(C) the image set of this function must be the set of non-negative reals;
(D) the image of f at 4 is 4.
Ans: B. [Recall lectures]
5. The set {(1, 2), (3, 4), (5, 1)}
(A) can not be a graph;
(B) can not be a domain;
(C) can not be a range (set);
(D) is a subset of the plane.
Ans: D. [Recall lectures]
6. On the plane, consider two circles with different centers but with the same radius 3.
(A) they can only intersect once;
(B) they can only intersect twice;
(C) they can not intersect;
(D) the area inside each circle is 9.
Ans: D. [Trivial]
7. Connecting points (2, 0), (0, 2), and (2, 4) to get a triangle.
(A) the area within the triangle is 6;
(B) the area within the triangle is 8;
(C) the length of the boundary is 4(1 + 20.5);
(D) None of the above is correct.
Ans: C. [The area within the triangle is 4; the length of the boundary is 2(22 + 22)0.5 + 4.]
8. 4x(x + 1)0.5 – 2x2(0.5)(x + 1)-0.5 =
(A) (x + 1)0.5;
(B) x(x + 1)0.5(3x + 4) ;
(C) x(x + 1)-0.5(4x + 3) ;
(D) x(x + 1)-0.5(3x + 4).
Ans: D. [Straightforward ; p.13, text]
9. Let a and b be real numbers. The distance between a and the origin (i.e., 0) adds the distance between b and 0 must be no smaller than the distance between a + b and 0.
(A) This is called 畢氏定理；
(B) This is called 三角不等式；
(C) This is called 柯西定理；
(D) This is called 白努利不等式。
Ans: B. [p.5, text]
*10. Consider an equation ax + by = c, where a, b, and c are real numbers as parameters.
(A) This equation may have no solution;
(B) This equation may have (0, 0) as the unique solution;
(C) This equation can not have infinitely many solutions;
(D) This equation may have (1, 1) as the unique solution.
Ans: A. [Consider the case where a = b = 0 yet c ≠ 0.]
11. Consider an equation ax + by = c, where a, b, and c are real numbers as parameters. If it is a line with slope -1, then we must have
(A) a = b ≠ 0 and c ≠ 0;
(B) a = b > 0 and c ≠ 0;
(C) a = b and abc ≠ 0;
(D) a = b ≠ 0.
Ans: D. [By -a/b = -1 we know a = b ≠ 0. It goes thru the origin if and only if c = 0.]
12. Consider an equation ax + by = c, where a, b, and c are real numbers as parameters. If it is the graph of a function f: R → R on the x-y plane, we must have
(A) a ≠ 0;
(B) b ≠ 0;
(C) ac ≠ 0;
(D) bc ≠ 0.
Ans: B. [f(x) := c/b - ax/b.]
13. The equation representing a line going thru points (1, 2) and (3, 4) is
(A) x + y = 3 ;
(B) 2x - y = 0 ;
(C) x - y = 1 ;
(D) x - y = -1.
Ans: D. [It follows immediately from (y - 2)/(x - 1) = (4 - 2)/(3 - 1), yielding x - y = -1.]
14. Consider an equation ax + by = c, where a, b, and c are real numbers as parameters. If it is associated with a function f(x, y), then at each solution (x*, y*),
(A) we say that the image of (x*, y*) under f is c;
(B) we say that the image of c under f is (x*, y*);
(C) we say that the image of x* under f is y*;
(D) None of the above is correct.
Ans: A. [Recall lectures]
15. [continued from question 14] Which of the following can not be a nice economic interpretation (or meaning) of f?
(A) x and y are amount of inputs used in production;
(B) x and y are amount of consumption goods;
(C) x is price while y is quantity;
(D) x and y are amount of goods produced in an economy.
Ans: C. [For A, think about the production function; for B, think about the utility function; for D, think about the production possibility frontier (or curve). The function f(P, Q) does not mean much in economics.]
16. Let a and b be real numbers. If ∣a + b∣ < ∣a ∣+ ∣b∣, then
(A) ab < 0;
(B) ab ≠ 0;
(C) a < 0 and b < 0;
(D) None of the above is correct.
*Ans: A. [∣a + b∣ < ∣a ∣+ ∣b∣holds only when a and b are of opposite signs.]
17. Which of the following piecewise-defined functions is actually a standard function (defined by a single rule)?
(A) f(x) := -x if x < 0 and f(x) := (x)0.5 if x ≧ 0;
(B) f(x) := x if x ≧ 0 and f(x) := -x-1 if x < 0;
(C) f(x) := x3 if x ≧ 0 and f(x) := -(-x)3 if x < 0;
(D) f(x) := x2 if x ≧ 0 and f(x) := -x2 if x < 0.
Ans: C. [It is simply f(x) := x3 for all real numbers x.]
18. Which of the following best serves as an illustration of the notion of function?
(A) marriage;
(B) dating;
(C) indexing room numbers;
(D) college admission application.
Ans: C. [To see why (A) fails, think of those who are in the domain but not married. Since a person may date more than one person of the opposite sex, (B) is incorrect.]
19. Let f and g be two functions defined by f(x) := x + 1 for all non-negative reals and g(x) := x2 for all non-negative reals. The composition of g and f is
(A) (g ○ f)(x) := (x + 1)2 defined for all non-negative reals;
(B) (g ○ f)(x) := x2 + 1 defined for all non-negative reals;
(C) (g ○ f)(x) = (f ○ g)(x) defined for all non-negative reals;
(D) (f ○ g)(x) := (x + 1)2 defined for all non-negative reals.
Ans: A. [(g ○ f)(x) := g(f(x)) = (x + 1)2 defined for all non-negative reals; (f ○ g)(x) := f(g(x)) = x2 + 1 defined for all non-negative reals.]
*20. Let f and g be two functions defined by f(x) := x2 for all reals and g(x) := x-2 for all non-zero reals. The domain of the composition of g and f is
(A) the set of real numbers;
(B) the set of non-zero real numbers;
(C) the set of positive real numbers;
(D) the set of non-negative real numbers.
Ans: B. [Recall slide no.12 of Lecture 3. Each element x in the domain of g ○ f must be in the domain of f such that f(x) lies in the domain of g. This is also in your text.]

Part Two: Analytical questions. [皆是數學應用題。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。] 24 points
21. [10 points] Recall the problem with organizing kissing raised by Professor Harbaugh at the Univ. of Oregon (Lecture 1).
(i) For a class of 35 kids, exactly how many kissing must be done?
Ans: C(35, 2) = (35)(35 - 1)/2 = 595.
*(ii) Suggest a way to efficiently make all these kissing take place in a short period of time.
Ans: [I posed this entertaining yet educational puzzle on the lecture notes for you to think it over for several weeks. Now let’s see who did it and who failed to do so.] What follows is what I got after thinking for 10 minutes or so. 將學生(little girls at some French school)編號，由1到35。依據我的教學經驗，行動緩慢、興趣缺缺、聽力較差與習慣性觀望者，當然會得到較大的號碼。小女孩依號碼圍成一圓圈坐下。No.1起身，由 No.2、No.3、… 依序逐一親吻臉頰。大約在No.1走向No.4之時，No.2應起身，由 No.3、No.4、… 逐一親吻臉頰。大約在No.2走向No.5之時，No.3起身，由 No.4、No.5、… 逐一親吻臉頰。以此類推，很快地，35位小女孩將井然有序地完成此任務。當然，No.1親吻No.35的臉頰後，應該迅速由邊緣離開該圈，去喝水或洗臉之類。No.2親吻No.35的臉頰後，應該撤出該圈，去喝水或洗臉。… 此設計還有兩項特色：(1)團體的任務，不會被行動緩慢、興趣缺缺、聽力較差與習慣性觀望者所延宕；(2)積極主動的小女孩，編號較小，將可以較早離席，不必乾等。Of course, I am always open to any better solution. [During the semester break, I will mail my solution (in English) to Prof. Harbaugh and see how he thinks about it.]
Remarks:
(1) If each kiss takes 10 seconds, then my proposed procedure will take 670 seconds (slightly over 11 minutes) to complete. Can you prove it?
*(2) Any other applications or extensions? 假設有35支球隊，每兩隊要比賽一場，賽事可以同時進行。如果採用上述流程，大會需提供多少座球場？Hint: Figure out the peak load.

22. [7 points]
給你一條繩子，長度為k > 0. 你可以圍成一個正方形，或是圓形，目標是面積極大化。假設極大後的面積是a，當然，a將是k 的函數。接著，在面積是a的新限制下，你可以圍成一個正方形，或是圓形，這一回目標是週邊長度極大化。請算出極大後的長度。Hint: Of course, your final answer will be expressed as a function of k only.
Ans: From slide no.16 in lecture 1, we know that a circle has more area than the square does. So, the maximized area is a = [k/(2pi)]2 = k2/(4pi). Next, from slide no.18, we know that the maximized length is 4[k2/(4pi)]0.5 = 2k(pi)-0.5 > k.

23. [7 points]
你需要桌面的面積是a > 0. 你可以製作一個方桌，或是圓桌，目標是週邊長度極大化。假設極大後的週邊長度是k，當然，k將是a 的函數。接著，在週邊長度是k的新限制下，你可以圍成一個正方形，或是圓形，這一回目標是面積極大化。請算出極大後的面積。Hint: Of course, your final answer will be expressed as a function of a only.
Ans: From slide no.18 in lecture 1, we know that a square is better than a circle. So, the maximized boundary is k = 4(a)0.5Next, from slide no.16, we know that the maximized area is [4(a)0.5/(2pi)]2 =[2(a)0.5/pi]2 = 4a(pi)-1 > a.

**N.B. Compare questions 22 with 23, what can you possibly learn? This is a real challenge for brave soul.

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