Calc 3 Exam 1 Mitchell - NAME Calculus III MAC2813(3196...

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Unformatted text preview: NAME Calculus III: MAC2813 (3196) First Midterm Exam W. Mitchell Friday September 16, 2011 Start your answers on the worksheet; if you need more space, 1. (12 ) continue them on the back. Be sure to show your work to receive 2. (12 ) — full credit. 3 13 — No calculators or notes are allowed on this test. ' ( ) 4. (13 ) 5. (12 ) 6. (13 ) 7. (12 ) 8. (13 ) Tot. (100) 1. (12 points) (a) Sketch the vector pro ja b in the figure below. / - 2') )5; \ (b) Find |projab| = K 5 6 \ 2. (12 points) For the vector function r(t) = <71— cos(t), sin(t)>, f0 0 S t 3 7r. (a) Sketch a graph of the curve parameterized by r(t ) indicatin farting and end— ingpoints.’L'x1-+7 ”I (flb’fyt 5/ 1 ......”(P0,f b) I? (b) Give a vector tangent to the curve at r(7r / 4), and also sketch this vector on your graph of the curve. 7’: <..# 5/0“) (:21ij > 5/. 74.9“] ”If?" 41111111161“ 6 A 5”,“ Zxé’)1— C 41(41111") (0“) a) ‘7 ””7 ’1) 5/741“; Joe-{MUM (“0) .— 2—- : (a! (fiYC)) £7 3. (13 points) For the points A(0,1,1), B(1,0, 1) and C ' _...w .u" l ’L 1 1- : LV’I‘ (a) Find the area of the triangle ABC: 5: ('3) <f' 3 'f (4-) 2. “($- .2»- 2' (b) Find a vector perpendicular to the plane containing the points A, B and C: <i,ii> /3 A ,r 7 ; ’3'? f Jrgi'"; "a“; I do"? "14‘. £_}Jr ..... 13.] “lam" 4. (13 points) Find the distance from the point (1, 1, 1) to the plane with equation I} + 2y + 32 = 4: (a) Find a vector perpendicular to the plane: < / , Z— , .5 > bu+ Mai 4085/1" WWW <4) 5‘7 “Miefim‘” (b) Find a vector from some point in the plane to (1, 1, 1): <___ , ~/ , ’ > (c) Find the (perpendicular) distance from (1, 1, 1) to the plane. fl 2 3>< 3 “I -/>/ [3’2 3} m//:,..—w M /<113>/ :35“; b” r"! 6095/ 5. (12 points) Sketch and describe the surface with equation —x2+y2—2y—z2 = —5 6. (13 points) Find the angle between the vector (2, 2, 1) and the plane with equa— tion 2:17—3y+z=5. (2/19 6, ”(2—39 / C050”): /” ”260/ /fi//<2HZ/> \fifi/ 4” 6+I/ ~x’. _ W3 “JV/2: 7m; §:g_;e:b ozfiwgmwt (w—Cof’j—u’r/jjl J” M m -'(3—j M14: (ZIMWW/ < 2, fié gar J 7‘0 'I’Le PW 2,1017% P/M: H A 7. (12 points) Mark each of the following if it is ALWAYS true, and otherwise. (a) For any two vectors u and v in V3, 11 - v = v - u. or False (b) If u(t ) and v are differentiable vector functions then 1%:(2mvf: )2 (;,1Zx V0152 Caz/(t x u(t). True or (c) For any two vectors u and v in V3, (11 + v) X v = u x v. -or False V x V : 3 . (d) Suppose that the curve parameterized by the vector function r(t) is a strai_ht line then r(t ) is constant True or ‘ Cold/1L1 far oxovvfé FU‘)- ~f<f E Z) w (e) The cross product of two unit vectors is a unit vector. True or 05/ /7£f?m-€ /./ 8. (13 points) Short answer questions. A (a) What is the direction of a X b? PM " “”172; %£ a (b) What is the magnitude of the cross product |a X bl? Describe it geometrically; a don’t give a formula. A Pa % 47 V, 4 : . , 7 ”a in» my "m w W recur/x; ()c If a X b: 0 then What can you say about the vectors a and b? f3 W WW [W W/flf’é 4/7“};sz No Malta moor/[0n 791$ W e f I/ (”7/ 779150772519 4 77F [ma/14;: ’w, m W53; 27;; “79.42641 a!) ((1) Find the point of intersection (if any) of the curves parameterized by the equa— tions r1(t) =ti+(2t+1)j+tk and r2(t )2 (t — 1)i+tj. ’7“):<5 25+/fl75 W7 5/ 0 > m“%’/:MM¢M/flflw 799 0(5)’ <5—j 001wa WWfl M)” 5”)“5‘MW W [:0 fl) 7" coo/“<1 ”Q L'm' )7» ;;/ 7&1" X ”1rd fglo+/=//,Cr/ WWWVW ...
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