MT1_answerkey

MT1_answerkey - ECON 1088-100: Math Tools for Economists II...

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Unformatted text preview: ECON 1088-100: Math Tools for Economists II Summer 2007: Term A Midterm Examination 1 June 15th, 2007 Namez. ‘ AMWM Key * Student ID: Instructions: 0 There are 9 questions and 100 points total. 0 In order to get full credit, you should show all your work instead of putting down the answers. A correct answer with no work will receive no credit. Please make your final answer clear by writing legibly and place a box around it. 0 If you are unable to do a problem, you can get partial credit for explaining where you got stuck — doing so will help us to better understand what you know. 0 If you need a scratch paper, please raise your hand during the exam period. 0 The questions begin from the back of this page. Don’t turn this page until you are told to do so. ' 0 You have 95 minutes to complete the exam (12:45 — 2:20 PM). Good Luck. On my honor, as a University of Colorado at Boulder student; I have neither given nor received unauthorized assistance on this work. Signature: I Date: 06/15/07 \\ l. (10 points) Compute the following limits. 2 a. (2 points) limx x32— SUlDSlllulQ X = <3 lnlb HR QXWESS‘W‘A x—yS _ x 2/ Z 3&3 : (2W :i_-_4_ HEEL? ‘HTJ M ‘ ‘r ‘ x— b. (2 points) lily 2x2_4 = (Jill—[l _ if: : 9. “235 +x—2 (4330134 ' 4—2—1 0 m (“W/Ll : )m X~L (-2\~z ~4 4 X*>~7. (X/flXx—Q X—s-Z X‘l (-Z\~1 ~33 ‘ 3 c. (3points) limizzij— : gz‘g‘é 9‘3'6 .Q . -x—>3 x :51‘ +q 94% +9 0 Z . . I : l'wn Xl‘x‘é : \‘W ,k'bxflll _ x+z S I , “3 HM H3 “aw-ll ' x3“; ><~3 “ o ‘ l—x d. (3 points) lim 4“ (“l 9_ x—>11_\/; 1_ l\ O USe Jclne Comugale lo SKmKMQW Illa Whom) _ / (m 1:1 . [HR : Tm MCHJ—Xl } xal 1“ x “J? Am (V X\ 7 MW: lim/fl " *llfl 2. (10 points) Use the definition of the derivative, f '(x) = itingfli—f—gi-{Zto find the derivative off(x) = x4 + 3x2 — 7. (Hint: (x + h)4 = x4 + 4sz + 6le12 + 4x};3 + h“) WM z (“W + “my; 1. £(X 3:? “‘Q (A \ [X2 1* 6X1“; i-Kygf \44t + 3081 2A,] -1 __ [X‘u 3x1" qr) ‘ 4A : XJV Ana“ + 6%»). + (“Mi w + * (My flw" 7 M X ; M4X5+ 6%)] +4x\n”+\i’+ 6X +3” i “Wm' 4x41 M'Jr/fixihfl “73% z 4X: 6X I Z iim Mao 3. (10 points) Find f ’(x) of the following functions. a. (3 points) f (x)=x’2‘2 Yowev Wk ‘1 ~ -3>.2. 47x}: flax?“ : i—Mx ' . 1 4 1 3 "/ b. (4p01nts)f(x)=Zx ~§x +26 ; jlt‘xe‘q’axa 2X7. I \ m 3 % (4X1) -gw + 29 Xv; c. (4 points) f(x) =[g<x>h<x)13 iowu Me. [Chaim We) QNJVJ / . . ‘ . 4. (10 points) Find the equation of the line that is tangent to the graph of f(x)=2x2+8x atx=2. Vega) 3 £7,064) : 2x + (bx 759M : ML) (x4) I Sim: 4x” - 'Vzi -' 4W4 ; 46 : 2(27')+9(L) Y~24 =a6x~32 A c SZHQ: 2% 5. (10 points) Find the first and second order derivatives of the following functions. a. (5 points) f(x) = —(7 — 2x)5 €owet (fie, Rid C‘M‘m/ Kmla PM = «sh-MWJB : PM .~ 40 (21—fo (-2) 7 _ \ J, J b. (5 points) f(x) = i + _1_ + L .4 4 4 ' 1 L 9 x «[9; x U ><( + ><§ ,3- 6. v (5 points) Define v(x) = f(x) andg(x) = x2 — 2x. Iff(3) = 9,f'(3) = 3 , What is the g(x) War—31-2“): Q~6=3 (9/60 —- Zx-z :7 00%;) ; 2w ‘2 if value of v'(3) ? SlYICQ. V(x) = H“) %) vhhl9@-WM ,( WW (3r \1 a) : m MLHMM : 9—24, —2a 3 } [Wye T = ‘77” r i 7. (15 points) Let f(x) = x3 —%x2 —36x+3 a. (6 points) Determine the intervals Where f is increasing / decreasing. . / Z We: 2x ‘ ax—ae ’L = ’37 (x — x - 47A < '5 (X 4+)(X T3) Elx-Wmél 7/ O M w" 1510M [4 0e. CYNCR’Q' W:th I‘K "‘$ 0le Li I see b. (5 points) Determine the intervals Where f is convex / concave. Qz/(A : 6X‘3 ‘5 «meme \l- 6X -’5 4 0 6X 4 a x49; g0 is cowex ill X 7/ 4/2. 8? c. (4 points) According to your results in (a) and (b), which one of the following figures is most likely to represent f (x) for1 S x S 4 ? Why? J L: k l; (1) (2) @ <4)" FOY %éxé‘+7 H0 {5 (iQMASWZr owl Convex lyllnich i5 5 969m 8. (10 points) Find the derivative of the following functions and do not simplify‘ your ‘ answers. a. (5 points) f(x) 2 (4x3)(2x +1)4 llde elf Rm l6, . 1C KO .—-\ (lull {2X HTr i+ (2X Milli) b. (SpointS)f(x)=”“3_2—x2 audit/ml We. 9. (20 points) The price P per unit obtained by a publisher in producing and selling Q units of “Economics with Calculus” textbook is P =130—Q and the cost of producing and selling Q units is C(Q) : éQz + 4OQ +1326 a. .(3 points) Write down the total revenue function and marginal revenue fimction. Tuna two—old glw Mir TR) ; @ b. (3 points) Write down the average cost function and marginal cost function. AC = 3(9). : EQZMOQHM Q o (“C : Cleo : @ c. (3 points) Write down the profit function and marginal profit function. 7g: Te m r @zo uni—L izal+4oa+iszel : -% 621+9’06t 4326 l margiMQ \omgl 2 TC : -%<26Q Argo d. (4 points) What are the profit-maximizing outputlevel and price? liqu— mmxmxz‘mg ,' M = MC SM 9 >130 ~ a Mao—2Q : QWO WHO-30 . «we 6. (2 points) Calculate the maximal profit. SW11 1E -. g @L-Jr ‘3on 42,29 V WW - ghcflwoao) 4526 : ~l%So+z?oo~[%zg : l $24 f. (5 points) Find the break-even points. ' _ + 1/ 2 _ Hint: the quadratic formula is ax2 + bx + c = 0 (—> x = w Email raven oivx‘R ' T '- 0 M V / - ‘ ' : #40 i M ~73 ‘22 (limo 6 432.6 s 0 ~‘ ~Qo’r12 a.- bwo 04326 “ ’7‘) <9 - —9o:i/<mo-4(-w.mé) & is ~93ng M 4042 ’3 2 (3m : Jioim ' ‘3 Extra Credit (5 points): 10. If u( y) denotes an individual’s utility function of having income _ u "()0 u '(y) ComputeR for the following utility function. is the coefficient of relative risk aversion. (or consumption) y , then R = — y u( y) = Ay“ , A and a are positive constants, y 2 Oand 0 < a < 1 (AZ/val r A ova4 Huh/0i : A (Akin) \Udig‘ V 2“ MW \afigg4 -— “(ebb 3 ‘(ohllb-xad—L é ’l-ol ‘0“ ' ...
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This note was uploaded on 03/06/2008 for the course ECON 1088 taught by Professor Zhang,tian during the Summer '07 term at Colorado.

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MT1_answerkey - ECON 1088-100: Math Tools for Economists II...

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