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MATH 2010 - Exam#1 an III-III — 1 t2#1(12 pt_=2 Suppose...

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Unformatted text preview: Exam #1 an III-III. — 1+t2 #1 (12 pt, _=2). Suppose g (t) = ( 1 — t2 ) describes the motion of a particle. 3t + 2 (a) At what time does the particle intersect the plane 21: + 3y + 42 = 13? 71 2pm?) + 37/4?) + [wt/+2) if /s> '2 +2? + 3~3{’ H21! + 9.7 z /3 ~£/£— /2) 1 0 W”; a), ) 2431‘8’3/3 V 6' ~/¢3+/‘52 v /31/ (b) At what time is the velocity parallel to the plane 2x + 3y + 42 = 13? ./ a 3g — n 3 t2 + 1 #2 (10 pt, __:1.5). Let (.1 (t) = < 4t3 ) . Find the line tangent to the curve at the point t 2 < —4 ) . Where does the tangent line hit the soy—plane? _1 73/") t iii/“263;? $74) Bryan! [745 is Li“) 3 5 t 141/715 xyydémc e74 7” ’ w/m/q 2'20: "Hf—$20 =5 262/, w #5). #3 (5 pt, _=1). Consider the two-dimensional vectors a = 01 , b = b1 shown _’ a2 _" 52 below. g 40/7 (a) Geometrically, what do each of the following represent. det( al b1 ) represents A rem [a (:21?) 2 ‘4’"60‘ [34.139 :3 I ar'o’“ of Pam/gal (12 b2 0 _fl0 ’2'”), det( :1 :1 > represents Av I‘mzé [‘12 t 4/2“ I 2!). w/ 2 2 //,, a1 b1 / A g) <a2 > X < b2> represents ’47 0! g . 0 0 b1 a1 <b2 >>< < a2> represents 4PM ( A {ail , 0 0 (b) One of these four quantities is different from the other three. Which one? Explain Why. Jeff/é [g 0, S/fgnep/ 01/35? an/ %/)¢ V60 {firs are offgn—flc/ C/OClgM/I‘SC, 2 #4 (10 pt, _=1.5). (3) Find an implicit equation for the sphere with center < radius 4. 0 —3 2 )and (b) Find where this intersects the xz-plane, and parametrize the resulting curve. < ‘44: — >- = «’27 43> ’ 3 L_,.__J v 13 2—4t 3t+1 9(t)=( 1 3 7 > in the direction of the line '0 3/5 #6 (10 pt, _:1). Find an equation for line through the point ( 9W} <§>u the plane (.1 (u, v) = < 4m : 7 8 9 ) and perpendicular to #7 (10 pt, _=1.5). Find an equation (implicit or parametric) of the plane that passes 1 t -—- 2 through the point ( 2 ) and contains the line 9 (t) = < 2 —— t ) . 3 3t (gm : + «9 ~ A?» «35> a m So 7 #8 (10 pt, _:1.5). For all parts, you must show your work/logic and make the sketches absolutely clear (this may require additional perspectives or description). (3) Sketch, in detail, the surface given by z = 4x2 + 3/2. (b) Sketch, in detail, the surface given by z = (c) Sketch the contour plot for the function f (2:, y) = 42:2 + y2. ((1) Sketch the contour plot for the function g (11:, y) = W. (a) Wfflow‘ San/Bail? /00/é5 M/A‘WWL Sea/434% Ava/{s Ufa Zsrz P r ‘2 e {)0 79c '5' f3 mra [z 0554/ 0 rain t2 #9 (9 pt, _=1). Suppose f (x, y, z) = my2 and C is acurve parametrized by g (t) = (t3 >, 2 for —6 S t g 6. Set up the integral f0 f (x, y, 2) ds. Do not evaluate the integral. 3/4] : 326252 0 [cf/5 “ I: New/337M ,7/11 a. 6 [( ({Z)({3)7‘7/(2{)Z +51!sz 6/24 #10 (8 pt, _=1). A curve is described in polar coordinates by r = 3t and 0 = Int, for 1 g t S 6. Find the arc length of this curve. [as :f/(m/z :[6313/6 =1 $5 #11 (4 pt, _=1). Suppose f(z, y, z) is a scalar field and C is a parametrized curve with 0 S t S 4. Let g ($,y, z) = 3f (m, y, 2). Let 02 be the same curve as C, but parametrized so that we mOVe through exactly twice as fast (0 S t S 2) and in the opposite direction. {:0 f=2 Suppose f0f(x,y,z)ds=12. Then [029(x,y,z)ds=_3_{_. Explain. Trip/inf ILA: SCa/nf' {xxe/p/ 790445 71/; Vat/hew (yam Can mafia/0' pit/‘57! fagfop [,1 0“/ 011‘ 7%e [inlay/VJ.) 6/1009}? a a/fzffifirem/ farame/ffgg/én [as 1‘70 6707re671~ of/fiCPu/gjc flue WAD/‘5 Wow/o/ A z Mia/4717 4253‘ . #12 (4 pt, _=1). Suppose g (t) is a. function that outputs the velocity at each time t. (a) Is f: g) (t) dt 3. point, a vector, or a scalar? (b) Precisely what does the quantity f: _g (t) dt represent? a» a7) 7716, 51/”5/o/acaman7z ocCurcc/ 29/41- fle I‘Vitcr W/ [I u ‘2’], ...
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