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Math20c Midterm 1 - Spring 2007

Math20c Midterm 1 - Spring 2007 - Yea P Math 200 Calculus...

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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. 2:1:2-4— 23:2 +222 —4x+4z = 5 2Cx1+vi+313Lx Has-- 5 zfxt-Zx +\ +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'Dl-w V‘k— (ii) What is the center and radius of the sphere ? U. gem-kw (\,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 N 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) Swarm-l: +3, arc-v if: 3 (enm+E) if; = “gm-\I-PE'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%) rL-hlii-ls'inflxi'Biif/«tiblcos 7r+qlic 0 <4 44 M“ 01" ”Tr/35+ fawn: W TCflZOA4EW§4-o'£ NW .: Sf _ ) Find the tangential (gorfiponent of the acceleration vector when t = ar. 21‘ - v‘ “if + k: :3 W-i- an r, C'L'I't-i‘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 # qus-tgl V‘ ; ”(ilel'l'x-t +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 PageE ...
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