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Unformatted text preview: ECON 1088100: 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—sZ 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 “awll ' 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“; iKygf \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 = «shMWJB : 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—312“): Q~6=3
(9/60 — Zxz :7 00%;) ; 2w ‘2 if value of v'(3) ? SlYICQ. V(x) = H“)
%) [email protected] ,( 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) ElxWmél 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 proﬁtmaximizing outputlevel and price? liqu— mmxmxz‘mg ,' M = MC SM 9 >130 ~ a
Mao—2Q : QWO WHO30
. «we 6. (2 points) Calculate the maximal proﬁt.
SW11 1E . g @LJr ‘3on 42,29 V
WW  ghcﬂwoao) 4526
: ~l%So+z?oo~[%zg : l $24
f. (5 points) Find the breakeven 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/<mo4(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 ‘(ohllbxad—L é ’lol
‘0“ ' ...
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