This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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“ ' ...
View
Full Document
 Fall '07
 ZHANG,TIAN

Click to edit the document details