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Unformatted text preview: We: u 1. Evaluate the following limits. (1 point each). l, sin3t9 km {l" 36 KIQ COS Y9 ' 2 1 ;
game940 39 sme y L/ a,
x \‘x k l/
\ ( l
. LI
332—1 . hm (iv/00ml) r, hm ‘
3:qu x—l x31 x~ x— t < ) 2 _ ac — 1 l'
11m = “"1.
z—r1+ (a: — 1)2 >0“ . 1.
11m —smz=
z——+oo.’,L‘ \fx+h———\/—z—1_ mm W ,
h ‘ 2.
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LY SiMPEZ—e (g Q i X 5}” K W44]?!
—”l_?0 (’7 \lxrAl +J7Q
»<+A~( —(s<'!) a um i r “ hat.) Uxmq'ﬂ/KT? a lim
h—>0 a (ll/V) f M
L‘—)0 L,(.l)¢h~l +t/K’!) am 2. Use the deﬁnition of the derivative to ﬁnd f’ if f = 32:2 + 2x. (5 points). Hm gymH'iEL) 2 lino 3(X1—hr1—2(xrh) ~3yZ—2x ln‘m in H90 k (1m CXk+351+2A [m
logo A :L,(~>o (6x1'3L‘f2) : éxfz. 3x+1 a3<1 3. Find a, I) such that f = (1102 + b 1 S a: S 2 is continuous. (5 points).
6 — x3 2 < a: "M snow, if} wab = F“) Y“?! Ifm X~> ' 9'64): 4. \ ; 2 “a 5 ’93), 5:3; m: —2_ 7?) 5% cpnfil ﬂue/yﬂu‘ﬂ) on each h‘ne abut/e WW5" be eciuqu
(—(:q+5
2,:L‘a+b \UQ Subf/oc‘l' 4. Find the point(s) on the graph of f = m2 + 1 Where the tangent line passes through
(2,4). (7 points). ¥‘(X): 1% / 50 “>de7 )3»! (i'ﬂc’j wukrm bevwepn ( X/ x1+\) Omci (21W), S/a/DE 2K / ‘1‘
9 ZXiLJLLL 1‘)‘ 3 or 5. Find the equation of the normal line (line perpendicular to the tangent) to
$312 + 2y  a: : 2 at (1,1). (6 points). d ,.
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{3&7 M ((,I.)3 d d
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K (73’ would 69 Molizonmklj . y 6. Find the following derivatives. (2 points each). Dx[a:sin(2x2+1)]= X cos(2x2+i) + Si/l(Z)<Li~l) Dm [cos(sin(x))] = ~5 n (5; a X) C0 '3 X D [ 300w ]_ (44 SM ><~*) (‘35MX) ‘ 71a)st
I (WEAK—{)1 4sinw — 1 7. A particle is at position s(t) = t3 — 22—1t2 + 3015+ 12 at time t. It momentarily comes
to rest twice, Find its acceleration at each such time. (6 points). v (6): 5m: 36 ~24t + 30 =3(€~— 7t+/0)=3(t“2)(£~s‘) Comes +0 r951” whet“ i=2 or {25‘ C((ﬁ): S”(€> 26f. ‘2‘ 8. Use differentials / tangent line approximations to estimate \/ 121.1. (6 points). Let y : J;— “Gaeid Pro" ((21, 1:) ~
clv _ .L ’ Eli ____ __l‘
E; ” 2‘}; dx “m N 21
ﬂqh line :‘l(x(2() 9. 10. Water is being poured into a cone of height 12 inches and diameter 6 inches at a rate
of 10 in3 per minute, but the cone also leaks water at 2 in3 per minute. How quickly is the water depth increasing when the water depth is 5 inches? (7 points).
3 3 What is the largest area of a rectangle with two corners on the :caxis inscribed inside
the parabola y = 16 — x2? (6 points). D 3 Co, q j x2 11. For = m2+2, down, has inﬂection points, critical points, local min, local max, asymptotes. State
any symmetry of H. Sketch its graph. (11 points). ﬁnd where H is increasing, decreasing, concave up, concave We: 2mm ' 1*} 2*
m: A
($4 +1)2~ {Walk
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H ()0 >0 whom ><>0 ma: (0,“)
H’m <0 mm Mo decr (“°(°) ,—>o~ (’x‘)co 6’ 50 Y3( '(s q Lodz} zomr —~ W (5312)
HUM 1%
{9 ; XY+HX1+LQ~§XL1~MK1
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., \ Y __, 1 r.
‘” v?» (3 1+9 1v
{X l “ 1)q (leerl >( x
H//(><l30 => Xi; Hybmx ~“(tS’ A 3
6 R g 3 “"3 12. Suppose that if a business sells :3 units of squash, it has revenue $4 + 4x2 and total
costs of x4 + $53. How many units should the business produce to maximize proﬁt
and What is the maximum proﬁt? (10 points). 3
X P: Proer :— Reumue“ C0“ 2 lez” ‘3'
D: [0100) Filx): gx ~><2 :0 =7 XlS’K)=O =>X10 or x=
FII(K) : 8 ’2X/ Fl/(O): ? >0 I 5'0 O
’P”(8>=sz<o Wé‘o W
,x‘qulos may % (SW? M CWWM “Wild/"r5 NEW—25o 53%? 1:11, PM: ~09) le The busing» manual sell 8 qnl'rs (w 67“
Q MK meﬁnk d; 1:55 7 13. Evaluate the following antiderivatives. (5 points each). 3
/(x2+cosx)dx= 2—; +54% Xf‘C, facsin3(5a:2+1)cos(5m2+1)d$= (7'6 Sngxafll “LC,
Le} Q: glow—KIM)
do : ﬁx Cd${§><l*~‘)d>< )9 ‘2
X<OS(9X “ldx: % du § I , l I? g
0 {X 9”“ng F‘>C05(5x Hid“ : éiuiclq :LTlOCAL/f—C
l L
: 95 §inl(§><z+()7¢(f 14. Use the Mean Value Theorem (MVT) to Show that f = 51:3 + x + 7 has exactly one
real root. (Hint: Try showing that it has at least one and then showing that it has
no more than one). (10 points). 70,; (pm, am, 9 NW“ ma fur (rm as my: 63/1 0,47 1h berm, )K/w‘e ﬂw) <0, $00) > 0 So“ £97 TI 3N ‘(“(0,(O) 5.t~ :O. rIf we and” warm" ropich Mumbws, coach Say (I'm x—am 79M 2 ~60 , 50 70 MWMW <0 1} Kﬂmoo : Q>0 5'0 @aowf’aaﬂr >0 M ‘0 r wk
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Q XQCHy 'p .3" 130$ 1L / {1070 we ‘4
7’0! (45 10 Am; a 7L /eas+ a n e [cad ‘ MOM 10m mmwv mics xt Min hm‘h) Le 9593” REFO‘ I
m Mm, 31¢eron 5+ ﬂcr—M 50 ymza (BL/(Jr $‘(C52 3C1 +( is never 26/0 (Wu'n Uq/ug ,‘5 (I)
go 1mg "(3 ;mfoss§~‘>“°~ Th“) 30 C(oc‘s “07‘ have MO/e fact” one rec1+. .‘ ﬂ [4&5 €><0£Cf(\/ cane mgof‘ ...
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 Summer '08
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 Calculus, lim, Inch, Lori ganfd/, ropich Mumbws, 26f. ‘2‘

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