This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Exam #1
an
IIIIII. — 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 twodimensional 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 xzplane, 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’], ...
View
Full
Document
This note was uploaded on 03/16/2012 for the course MATH 1B taught by Professor Reshetiken during the Spring '08 term at University of California, Berkeley.
 Spring '08
 Reshetiken
 Math

Click to edit the document details