HW1_solutions - MATH 2401 Due in class on September 14,...

Info iconThis preview shows pages 1–14. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 14
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 2401 Due in class on September 14, 2009 Sections F4 & F5 Homework Assignment 1 Name: gtID#: Note: There are 12 problems in this assignment, and eight of them Will be graded (you won’t know which ones beforehand). Write out your solutions neatly and explain ' your work. Please add extra sheets if you need more space, but do not use the back sides of the pages. Problem 1. Determine the following for t = 1: ea TTDM M fifiMAWMawaarub. {3? LC ensign—h) $2.2M <'0 “i=1, i><\:=£, Laws—=0 L mimech , Problem 2. (a) Find the position vector of the trajectory of circular motion in the plane around the origin starting at (—1,0) going clockwise at unit speed. (b) Find the position vector of the trajectory of circular motion in the plane around the origin starting at (10,0) going counterclockwise at speed 60. (c) Do part (b) again but this time at speed 60 rounds per unit time. @ :05: cos(/\T—t)£+ sin (“Tr—1:38 :2—costC + slnts / Mu (imam 4920. 11(le = 10 {refit}: + ${n@t>S/X / “C20 mm H (a 2 :Lo L—e; m (cat + egos-mg f I 50 (» s‘m (egtj cps (éfihgpx Sc: ’W Snead. LS, as Obi—9 ma” tam : e0 W = $9 \.-Q. © \NE M 41; \AON‘C QC VOL-“WAS , \ l QAT": \QO/Fr racUOxx/LS wuvx “5:1... WW5 MM HQ = in Lass (\aw‘tMfi 5‘“ (“O/MA Q / t 2C3 . Problem 3. Consider the curve $2/3 + y2/3 = 4 (a) Sketch the curve and find a vector valued function f that traces it out. (b) Determine all the points on the curve Where the tangent vector is zero. 1/3 3 @ X:O$\$=(+ z» VA=1=$ '3: 6:0 => xzzua x=1=8 wvm o»\eow4' \oo+\r\ GOALS filVLQQ' Problem 4. Find the unit tangent vector, the principal normal vector, and an equation in x, y, z for the osculating plane for the curve r(t) = 2ti + th + 1n tk, t > 0, at the point P(2, 1, 0). We have L’La-z 1L+lt3+J{E , Se \(r't’c3\\=*<i LH— LLJelJci} =-\i (fit +‘fiY : 936+:(E 093+ 5%? m S.\:I\C.Qa t>O so (kt-Ezle Paid-va, TMaL—Qow I . ~ .x. Tcazwc : 91*1tr‘tb Q2133 e __ we)“ 1%? _ at; +1JJ3+¥1 Oil—H IFS Ska “GE-fr \Arer Comgvite.» ‘ ‘ lb ‘\’ At [3361- B _. ("z-t ‘35 1t £135» 0H5 E 2 "_ TM” “WWW” - I q i r ‘ 1++Qvutvus gem e) Jaw-tr m filtwfiflggflfmfl” P \s T03: 33(169333 “ML U _ Tm {ml‘gjfli :43, (.L—t1y25} E \— \WLAU \SWMW (,5 M omictffi WW\ 07% ?‘ m M 5 Problem 4 continued. . . ’1’; ’Rkfi-J a «QWV‘X‘R vafl‘ wt W 40 at 06:05an (“42% \s vusw\ 3m M ska/1.0.. We COM (MOOSan (m gem Q) +0 06% via ofi \oa—w A: Qacaw E = <7 Tum mufl . :: (1525+ >< C.L+l§:l\3) L3k\ L :2 (—— $153+ (4 +103’Jr UHQN: SB mmmmmmmmm Problem 5. A metal ball rolls down an “ice cream cone” along a path that has the parametrization 1.3 7: r(t) = (10w — t) [cos ti + sin tj + k], t E [0, 1071'], Sketch the path and find the distance that the ball travels. “sokwt’m \Lwajfl/x Mgkvve, {n a‘n MMMLO Frogm at W CMVV‘C owwk {S “\AcLLQQnaLLWt (rpm eamvmjsk‘éFbcch‘om ‘ So huts ‘VEW‘P M emmmfikest‘tawt 9°- M+ ATM \OQ‘M (kw-ANA? we. : fifitntl I tELO‘ ' ” ‘ L —» t‘ 4 (Qt): [Cast L— 33th Adil Art ~S\V\’C __ Cos _— :: (Cast-13 SWTESE— ('S'NVEJC—tC—OS‘EB ‘Q—Jc \S— L '(Jc) “7-: (cost #:5an + (sint ACT/cost?) + L — Cosuc 'ltcggtsm‘b {it smlie L L i + slm1t+lts{y%c>s‘b PLROS 1c + L " j,+-E‘+1, = 93ft \N‘k‘!‘ \W ’— \o’fi‘ == E375 tmaz J: Qn [if tL-VL : 5’\T\i®/1T\L+Q_k+&vx \\o/\T+\5Qcm~§"+1'\ -' ,va-JZ‘ \ [WU + S\Ob’fil'¥’l_l L . : swim/Wm} + INK L / 7 Problem 6. Here, interpret r(t) as the position of a moving object at time t, where 1 . r(t) : t3i + it?) + tk, 256 R. Determine the curvature of the path and the tangential and normal components of acceleration, at time t = ——1. firs-t we tamed“. M \iebcfii QWA Qméhmfifiom 2 git): gee): gem t5 4d: , 9ft) :vWfi: gt mug ack— t': "i—IP -g \ o - ._ CL+éA—VS\<) ; WAHGR WOW mm exit» \\\1xa\\ \WJ 0:53 fit? owe: 74 i " W? / \I’Q -fi. Tana CW“? “PE: " QT: “V \\ :1 m Nom\ Com r£ C\ ___ 1 __.._.———-- ‘3 Q5 \wu W n _/ E E E E E Problem 7. Find the curvature of the curve 1:3 +313 = 3333/ at the point (g, [Hi-n13: It’s not easy to parametize this curve. Instead, think of this as a function y = and use implicit differentiation]. I—F we Luck 03 xi: ag q chfiow 5-9 x t Mm {Q we. ohwnfiofi-e. {Sefi‘zgxfi wH’h msgaefl’ 4:: x/ we (Bait d 1+3 1 '2 3 + X ——A— '5“ “A 3% (‘3 dx = L L A - 4c x g; @ < 7 X +3 ‘ F3 01.x m-hmeam 693): (—33%) w (W: have __ 2 3d = “9 3+3‘fi? Ld z.— :> 1213 T; cifirévMM /\\/UL cuwrfiuwhl Ase. mnedi «we 33d cum «41% {(3273: \Ne Aegeemvxhaskt 0%th (“M-oxaéxx W-r—Ft ~ X 3* d2 4 i 1 0‘1 = 0—H 09 + x4 1x + + ‘3 5632 dx + 3%: fix 1 A l 1&1 '— OL d 4:; 2x»: QBLfiyfl fig 135;“ 3%.; 0L? \pov—‘L \V‘QS » - __ :1 QWA 51" M O a Qflwmg M IX”: 1'; "Adz Ax 3+3H) +£139ée= '10; a. mi 3 d1 ' z. 3 rm .2-“ =7 %:%(%~L%JJ=;%?C 2} 4% q a; W "m =5 I ah»; aw” '72: Mmiw-~*:;Y““WWWMWMEM77{W £— W 1 95°71- : 3N3 : Gwen/twig: 3% : XafiemmfiwwsWm<mfiwwwmwmzwwfiwmmmwxémaacxsa mmmmmw Problem 8. Define the functions g(t) = 1/x/t and h(:z:,y) = (4 — :02)(1 — 3/2). Determine the domain and range of the function f($7y) 2 g 0 Sketch the domain. \N '5 have 1 \ ¥b<l®= 3 (0*meng :m 10 WmmmmnwrwfiKmmmezammnw WWWWWWTW Problem 9. Describe the level surfaces of the function 1 x/4y2+z2’ using equations and sketches (show two explicit level surfaces). m bowel Suv~l§aC$LS at -F are. MWCMOL M: M egfwox‘g‘LOvLI f($ayaz) = -- C70 =3 A; 111:1, < was“ 0 a 1 Wm cure Cflkvxdgyfif L L M' (X 11 Problem 10. Let f and g be differentiable functions on R, and let i W, 1/) = f (502 - y2)g(:vy), (11379) 6 R2- (a) Find the partial derivatives of h. (b) Compute hm(1,0) given that f(1) = 2, f’(1) = —1, 9(0) = 3 and g’(0) = 1. @ \NA «W\AM clerM szL ompk M emdmd nth : W®= “M i ’ (>41— 133%} + Wig) g OS'OWQ NEVA»:— wfi that) aim + wag) m 23‘ (MW ® \NWV XT—ll L320, wime \LL—kflL2’l 064A *tfi;0/ So usifl @I who): gx'toymiqu’ca : Q“ (.6. gt "‘ ‘5': __———-—— 12 Problem 11. (a) Use the contour (level curve) plot above to determine whether fx and fy are >0, = 0, or < 0 at the point (1, (b) Same question at the point (1, 1). On which part(s) of the level curve through (1,1) is fat > 0? (o) Are there any points (approximately) Where fm = fy = O? @ ;X(\,L3<O/ ¥3(\,-§Q>®. ® ¥x(\,I3<®, 1E3<H3=©i We have ‘(lx >0 Wm M om pic-lure 050M @ 1EY=15320 at X’s Meteawfaauflqgove (0.35) 0.234) 2 mg) mg» 13 Problem 12. Define the function 3 2 3 f($7y)={ (may)7é(0,0), 0, (x, y) = (07 0). Is f continuous at (0,0)? :Brmk/L (mfihkma) wawme WW Q’Ql = 343+)41i3tfli \ 2 \X1* 083‘” I) X +\3" X3 (vain “A51: a“ S \ XL+"3L\+ \U wgwah‘M‘WWQ : \><\ \XZ:\§,\ +\Q\ smu- :1 L S\ g \x\+ MM x “5 3mm; W <\x\+\fl\3: CD ( M 15 ...
View Full Document

Page1 / 14

HW1_solutions - MATH 2401 Due in class on September 14,...

This preview shows document pages 1 - 14. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online