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Unformatted text preview: Yea P Math 200 Calculus Midterm Exam I January 31, 2007 Name: _ ' , 's (3 .13) ""L'. _ Ct' Hui—‘1 \
Lecture Hour: .._l'?'. 033 . Section Hour: at” Ll ‘ Guidelines for the test: 0 N0 books, notes, or calculators are allowed. a You may leave answers in symbolic form, like m, unless they simplify
further, like \/§ = 3,60 = 1, or cos(3ir/4) = —\/§/2. 0 Use the space provided. If necessary, write “see other side" and continue
working on the back of the same sheet. I Circle your ﬁnal answers when relevant. 0 Show all steps in your solutions and make your reasoning clear. Answers
with no explanation will receive no credit, even if they are correct. 0 No credit will be given for work/answers that are illegible 0 You have 50 minutes. Question Perfect Score Your Score /0 (l) (a) (i) Show that the following equation represents a Sphere.
2:1:24— 23:2 +222 —4x+4z = 5 2Cx1+vi+313Lx Has 5 zfxtZx +\ +y1' .;2"'+‘—:?.2“t"I '5 4 l "A" 2([ x— iii.4. x/z +(2+\\)‘)= Z
‘Z. '2 (x 431+ {14414 Q": % "Evhs '15. om Gdtuocﬂrimﬁ 3" 951,4qu rel/v“)
tar(mat 'Wi Wuw “Hoax—i
(v—a31+(\J'b3L+L1'C317V1 (T) an—c,
Anus :3ng fawn—{1m ‘15 ‘5‘ 'i'Dlw V‘k—
(ii) What is the center and radius of the sphere ? U. gemkw (\,o,I‘j r: \?X (b) The point (1, 2, 3) rests on the surface of a Sphere of radius r centered
at the origin. Find r. ‘14, 27431 : \l Page I N chum. 4
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A. >95 14.) :3 .. 533 +0515uﬁ>m5 mjabu n .m: Wm ﬁﬁ me x m H m x m ”was scam my and by $0ng can we wwumﬁﬁ
.88 may 3.3m 93 6:3 .323 macaw mm 8 .H ,3 N m. «E: amonESm 6 6
T4.) m mam: a 3 98%?" ﬁg.» c2323 3. wagoam bug? m +16
Baum» 23 $595 .333 650% mm E309» v.3 m was: a Raﬁ wmowmnm n3 AWIV u T hwy u x .MIMN BSQEOO 8
h. .. I w nm E3 x +% um 35 389% E E (3) (a) Match the foliowing equations (i)(iii) with the graphs (I)—(III). Y0u
need not explain your decisions. $21$=cost y=sint, z=sin5t @ (ii) 3 = cos 2Ut, y = 81112015, 2 = logt (11) (III)
(b) Sketch the curve with the vector equation
ﬁt) = t§+ e“ :3. Indicate with an arrow the direction in which 15 increases. Page 3 (4) A particle moves along a. curve C with velocity at time t given by
"3(t) = (4cos t) 3+ 3}" — (4 sin 15) E.
Suppose that at time t = 0, the particle is at location given by the vector 13:41:. (3.) Find the vector functiori F(t) which gives the position of the particle
at any time t 2 0. 3le .— F H)
Swarml: +3, arcv if: 3 (enm+E)
if;
= “gm\IPE'l'j L‘lCOE:"l’l¢ ...—'I {ﬁr‘1‘" em; crow/7: + big «(”03“ +26%
“7.. .V Reparametrize C with respect to arc length 5 measured from the
‘ point where t = 0 in the direction of increasing t. 3H): 3; lF‘Cm ldu (c) How far is this particle from its initial position when time t = 7: ? a a “were! , a (3%)(3):(3%) rLhliils'inﬂxi'Biif/«tiblcos 7r+qlic 0 <4 44 M“ 01" ”Tr/35+ fawn: W
TCﬂZOA4EW§4o'£ NW .: Sf _
) Find the tangential (gorﬁponent of the acceleration vector when t = ar. 21‘  v‘ “if + k: :3
Wi an r, C'L'I'ti‘xr'K V':1V'H)\;_&%fuwb+/1 + 3/:  (Me‘mﬂlc v l~43m+f+ a; e 4co5+ 12/ Page 4 v': l'LiS'lﬂ‘HC # qustgl
V‘ ; ”(ilel'l'xt +CO§+TZI> (5) Suppose the planes P, Q are given by
P : :1: + y — z = 2;
Q : 23—y+32=1.
(a) Let L denote the line of intersection of P and Q. Find a. point on L. (Mi) (2“) < mam > <2 ) '15 ;'?7> _ _2_ ‘
2 3 ".5 (a) (b) Find a vector 11‘ which is parallel to L. .i H\.
U=(*!0
‘Lo (0) Find a. plane which is perpendicular to L and contains the point
(1,0,1). —‘éb><+8 +\/ +72 7=CJ r9x+xi +7L:*I
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