::::l<r/ L 11>(=ft\iir(~1J~:::: Fo. l. AG =(0,.2,L}> M  &c == < ) I) ~...::.1 ( \ I 0. /:rf!:,t f1c = <I _,, 2...


 Willa Beasley
 3 years ago
 Views:
Transcription
1 Math 3298 Exam 1 NAME: SCORE: l. Given three points A(I, l, 1), B(l,;2, 3), C(2,  l, 2). (a) Find vectors AD, AC, nc. (b) Find AB+ DC, AB  AC, and 2AD. >,,. /:rf!:,t f1c = <I _,, 2...J I) ~..::. M  &c == < ) I)..) l. AG =(0,.2,L}> (c) Find length of AB and area of the parallelogram spanned by AB, AC. ~...::.1 ( \ I 0 (}:rev.. = I A ~ >' kc. = ( _ v 1 J , ::::l<r/ L 11>(=ft\iir(~1J~:::: Fo 2. Given two unit vectors u, v. The angle between them is 7r/4. What is u v? 3. Find the equation of a line passing point P(l, 1, 5) and parallel to another line whose equation is x = l + t, y = 3  St, z = 9 + 2t. Jv;( v = <. l / a/ r. v >
2 4. find equation of the plane which pa:;s through point P(2, 1, L) n.nd parallel to another plane x + 2y  z = o. ~ ~\ JJ< n ~(1,, 2) \> LI ( x  v) + z..{ 'd I ) I. ( i' ~1) ~ 5. Fin<l the point in which the line x = l  t, y = l + 3t, z = t intersects the plane 2x y + z = 2. (n tk. 1 \J.e~\qn_; 2(1t).._ (I 15 f:) t f ~ 1 ~ t== i. L+ l ~ 0 >< " ~  i poi..\. C:.L 1:;= Make a rough sketch of tlie quadratic surfaces, and give their names (i) z = 4x 2 + 9y 2. (ii) x2 + y2  z2 = 0. :/::, l / 7. ind a vector function r(t) = (x(t), y(t), z(t)) for the curve which is the intersection of surfaces 4 = x 2 + y 2 and z 2 = x 2 + y 2.
3 8. Suppose a =f. 0. If ax (b  c) = 0, can we claim that b = c? If Yes, prove it. If 110, give a.11 example to support your conclusion. U. A particle moves with position function r(t) = (cosl,sint,t). Find its velocity, speed, an<l acceleration. vb < = Sv\,,\,. ('.,oyt,, 1/ / / \ n x ~ 11 L /;) lta T ~ / ~ I? c.."' ~ k' v,_ "' t ~ I s~itj ~ l (sf (I ~ (: ~ :_~ ; ~ r; k ::: ( ~ f ;!;v l.;f ' 0 ) 11. particle starts at the origin with initial \I locity v(o) = (0, 0, 0). Its acceleration is a(t) = (2, 1, 0). Find the distance it travels from t = 0 to t = 1. Vu,= ) ' ~(h~(; ~ ) \>,I, o).iv~ ~ <!. = (lk, t ' o) + c::...:... ~ ~ S"vL...> \} Ci. ; :::. Ct> ; o ) ~ "=3> G :: ( o; li 1 o> ~ \J (:h ~z)/t ' {; / D> \/ lh ~ J~t)"'+t 1t l)~ ~ (? ~ I S:::: ( fr\;~\ :; Jitvl Jo v o I
4 L2. Fi11d the local max am! min values a11d :;addle poi11t::; of the f1111ctio11 J(.c, y) =.c 1 +xy+ y 1 1.i:. 2 { ' ~ *' u = ( 0, L3. (i) Find V' J for J(x, y, z) = :;in x ~ os ~ ez. (ii) Find the directional derivative of J along 72). (3) At point (rr, Jl', 0), find the direction which will yield the largest directional derivative of J c c :t; \7 f = (l<s\x Of\'(} e /  s. h.;< s,k ~ ~ ; Svtv:>< (,../) 'O.e, ) j) ~ J_?.L ~. 7(; IA f ::.. \7 f. la ::. r1))~ K (ra d  ~ f rv S ..ivx (fa. D e = it 6<5 j ('_ t(c.ri,x + <;,), x) "Tf, <f '" )J < ~ lig( ( t1) (I) ~, 0 ' 0 > ~ (I / D, 0 > j)j, r rul M"'XI ~" la = i1}~(if,of, o) = < \ 1 D, 0,) 14. Find absolute max and min off (x, y) = x + y  xy on the closed region D bounded by the triangle with vertices (0, 0),(0, 2),(4,0) '. ('. f \\, \f( D I ), =/ I, l ) ::c I + I  I ~
5 Math 3298 Exam 2 NAME: SCORE: 1. Evaluate the iterated integral r212y Jo Y x 2 ydxdy. 2. Sketch the region D of the double integral associated with the iterated integral in last problem. Then change of order of the iterated integral to dydx. Evaluate it. ~ ) ~ f: \. JA )74 (clyd# j) 0 f"i ~  "'f 1\ I\  4'. l r 1. ;:;  1
6 4. A plate lies on the region D bounded between the circles x 2 + y 2 = 11 and x 2 + y 2 = 4. The density is p(x, y) = h Find the mass of the plate. Hint: switch to polar coordinates. x +y L)\ 2)T ( z._ 2, Ry,/wo\,(9. 5. Find the volume of the solid under the surface z = 9  x 2  y 2, and over the rectangle D = [O, 2] x [O, 2]. ~ ~ \'J1<'1") JA ~ \, \. (1~~~')h ""7< j) l ~ \ 0 ( 9 '6  )<"'" f J' ) \ J.,.  )}c~:)  t)e  ~ 'l Ji } 6. Find the surface area of the paraboloid z = x 2 + y 2 between z = 1 and z = 4. _j) ll J_ ( I + ~ ~ 1) 2. \ i.
7 7. Given the following double integrals f 1 Jx xydydx + r.,/2 rx xydydx + J 2 11;../2 ~ 11 lo.,/2 lo Sketch all three regions of integration on one XY plane. { J 4 x 2 xydydx Use an easy way to calculate the integral in last problem. 9. Rewrite the integral x2 lo rly J(x, y, z)dzdydx as an iterated integral in the order dxdydz. Sketch the region first. /
8 10. Calculate the triple integral j j f 2 1 dv by converting it to spherical coordinates. }EX +y 2 +z 2 Here E is the region between sphere x 2 + y 2 + z 2 = 1 and x 2 + y 2 + z 2 = 4. \L: ~~ \ L t1. ( S 'k c\( Jc~ ctr C ~~ s, ~4  f \ ~ l~ ct~  f ~ ' If (;v) ~ \ cj1t ~ ~ ~ Yi\ ( l + I)  ~If
9 Math 32D8 Exam [[I NAME: SCORE: l. Compute the line in tegral 1 yz cos xcl.s, c C : :i; = t, y = 3 cost, z = :3 sin t, 0 ~ t ~ 1r. 2. Use Green's Theorem to evaluate k F cir = k J l + x:lclx + 2xycly where C is the triangle with vertices (0, 0), (1, 0), an<l (1, 3) with positive orientation.
10 :.l. Find a parametric equation for the curve!j = sin x starting from (0, 0) and ending at (7r, 0). Specify the parameter region. '' Find a parametric equation for the surface which is the pnrt of the sphere x 2 + y 2 + z 2 = 4 and above the cone z = j x 2 + y 2. Specify the parameter region. x ~ 2 (' 0 \} $ " f d ~ 0~4) S Vf c 9 / q ) Ej) ~ [ G) vrr) X ( d, l j t: ~ ~~1 5. Given a parametric surface ) z_ 6L= tt ~ :C :::fil r ~ ~ 1 ~\ == ~ r (u,v) = (ucosv,usinv,sinu), 'U E [Jr, Jr], v E [0, 27rj. Find its Cartesian equation 'l '}, , \ v '){ +J = t.t ~ \J.+\j\ "'\!\ \) 6. Given a vector field F = (p(x, y, z), Q(x, y, z), R(x, y, z)). Define curlf and divf. > 9rz w JL.d.R ~ _ ll> c(a,'< \ f :::: ~  oc ) 0 c  ())I). / 'O )\ 60 t
11 7. Suppose that yon know that J(:c, y) = e.i: +.1: 2 y'2 a nd 'V f = F = (c'" + '2.c.1/, 2.c 2 y). Eval11ate the li11c integral ( F dr where C: is r(l) = (l +sin rct, '2t +cos rct), O :::; t :::; 1. le L~Jy " f(yl1l]f(ylo\) =:,f ( f) I)  f ( o 1 1) 8. Evaluate the surface integral j h F ds, where F(x, y, z) = (xz,  2y, 3x), S: r(u,v) = (2s in ucosv,2s in usi ~ v,2cosu) au<l 0 :::; u ~ 71', and 0 :::; v :::; 27!'. ) J ~~r.js s F. (j If i '(.i:) J.A  ~) <4»lu. C,.,,U."" I/ I  (j. S.\.I<_". v' G ~ 'kvl "''">. v
12 F = (l11y 1'2x!i,3. i:~.1/ 1:!.:) y is conservati\'p. If so, fi11cl the potential fu11ctio 11 f (:i:, y) such that F = 'V f. P 1 ~ df'.l f L :::: {V~ t ))('0 ~= V\ t b '/.. J ..) u & == 3 x~ir ~ ~ ~ ::: l t t J ir\ > r ~~ivt_ fu<, ~l "" ) (.e"' ~+v\<'.'j') J..1< = lz.q"' ~ + i<. ~ ~" + ~ { ~ ),... ol v i. i. I 1!,.,_, ;L ~ qr ft = 1 + j x ~ + d { ~ ) = Cl ;;= 3 i 0 t 'a ( u l 'J ) ~ 0 ~ cri~)~lj. 10. Find the surface area of the upper half sphere :i: 2 + y 2 + z 2 = 4 and z ~ 0 by surface integral. s ; 0( (! 1f \ c=q [0~,\Cf ' 2. s.\,, ~),\cf " AC'>)= ))1Js = c):~y6 n~\clt\ 2.(,U\{\J >., tj. E:[ 0, ), TI] ' p E[o,~ l "' \": \ ~~ 23,'Ml s, ve. ~ It ' '1, 0 > ){ < µ..;~ un1 u... ~ ~ :''..~A"" ;r =: [ \~ } 51~ q jfj cl Cf ' $TI (  Gr> <I')\: :=: g Tl
13 t l l. Use th0 Lagnu1µ;p M11lt.ipli<'r nwt.hod to find t.h1 ~ whose s111n is 100. maxirn11n1 procl11rt. of thrc'c' posit.ivc' mtmlwrs ~~' 1 h :: )( ~ i ~t< X~c ~,t ~ ~ t' ~ ={,(JO,X. 1  =t?v 12 3 L~~ ~I 4t_,_,, """ ~ ~?; =\ l (L_ Xe ~ ~., Q2 Xb ~ ;\I Q Xt~ t l =(ITT> G x. t: ~ 1 l :) re )( J) = o? x ~ ~ 't:=o w,~l\ ~\J.l(_ X~t=V vw~ ~146 (  'fil = )( ~ ;> )( ( l  ~ ) ~ ~ ~ ~ l ')( =1> W "vi,(!r V "l 'l( ~ ~ ~ 0 I VU. f 'i;vu'')<. ')( =: ~ = r e (QI) X=~=:i~3 ( (J1j 0 tl lrb ~7
M273Q Multivariable Calculus Spring 2017 Review Problems for Exam 3
M7Q Multivariable alculus Spring 7 Review Problems for Exam Exam covers material from Sections 5.5.4 and 6.6. and 7.. As you prepare, note well that the Fall 6 Exam posted online did not cover exactly
More informationEXAM 3, MATH 233 FALL, 2003
EXAM 3, MATH 233 FALL, 2003 This examination has 20 multiple choice questions, and two essay questions. Please check it over and if you find it to be incomplete, notify the proctor. Do all your supporting
More informationPractice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr.
1. Let F(x, y) xyi+(y 3x)j, and let be the curve r(t) ti+(3t t 2 )j for t 2. ompute F dr. Solution. F dr b a 2 2 F(r(t)) r (t) dt t(3t t 2 ), 3t t 2 3t 1, 3 2t dt t 3 dt 1 2 4 t4 4. 2. Evaluate the line
More information1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is
1. The value of the double integral (a) 15 26 (b) 15 8 (c) 75 (d) 105 26 5 4 0 1 1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 2. What is the value of the double integral interchange the order
More informationSolutions to old Exam 3 problems
Solutions to old Exam 3 problems Hi students! I am putting this version of my review for the Final exam review here on the web site, place and time to be announced. Enjoy!! Best, Bill Meeks PS. There are
More informationf. D that is, F dr = c c = [2"' (a sin t)( a sin t) + (a cos t)(a cost) dt = f2"' dt = 2
SECTION 16.4 GREEN'S THEOREM 1089 X with center the origin and radius a, where a is chosen to be small enough that C' lies inside C. (See Figure 11.) Let be the region bounded by C and C'. Then its positively
More informationA L A BA M A L A W R E V IE W
A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N
More informationMath 233. Practice Problems Chapter 15. i j k
Math 233. Practice Problems hapter 15 1. ompute the curl and divergence of the vector field F given by F (4 cos(x 2 ) 2y)i + (4 sin(y 2 ) + 6x)j + (6x 2 y 6x + 4e 3z )k olution: The curl of F is computed
More informationDEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS. MATH 233 SOME SOLUTIONS TO EXAM 1 Fall 2018
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS MATH SOME SOLUTIONS TO EXAM 1 Fall 018 Version A refers to the regular exam and Version B to the makeup 1. Version A. Find the center
More informationPage Problem Score Max Score a 8 12b a b 10 14c 6 6
Fall 14 MTH 34 FINAL EXAM December 8, 14 Name: PID: Section: Instructor: DO NOT WRITE BELOW THIS LINE. Go to the next page. Page Problem Score Max Score 1 5 5 1 3 5 4 5 5 5 6 5 7 5 8 5 9 5 1 5 11 1 3 1a
More informationName: SOLUTIONS Date: 11/9/2017. M20550 Calculus III Tutorial Worksheet 8
Name: SOLUTIONS Date: /9/7 M55 alculus III Tutorial Worksheet 8. ompute R da where R is the region bounded by x + xy + y 8 using the change of variables given by x u + v and y v. Solution: We know R is
More informationPage Problem Score Max Score a 8 12b a b 10 14c 6 6
Fall 2014 MTH 234 FINAL EXAM December 8, 2014 Name: PID: Section: Instructor: DO NOT WRITE BELOW THIS LINE. Go to the next page. Page Problem Score Max Score 1 5 2 5 1 3 5 4 5 5 5 6 5 7 5 2 8 5 9 5 10
More informationMATH 52 FINAL EXAM SOLUTIONS
MAH 5 FINAL EXAM OLUION. (a) ketch the region R of integration in the following double integral. x xe y5 dy dx R = {(x, y) x, x y }. (b) Express the region R as an xsimple region. R = {(x, y) y, x y }
More informationMath 23b Practice Final Summer 2011
Math 2b Practice Final Summer 211 1. (1 points) Sketch or describe the region of integration for 1 x y and interchange the order to dy dx dz. f(x, y, z) dz dy dx Solution. 1 1 x z z f(x, y, z) dy dx dz
More information12.1. Cartesian Space
12.1. Cartesian Space In most of your previous math classes, we worked with functions on the xyplane only meaning we were working only in 2D. Now we will be working in space, or rather 3D. Now we will
More informationI1. rei. o & A ;l{ o v(l) o t. e 6rf, \o. afl. 6rt {'il l'i. S o S S. l"l. \o a S lrh S \ S s l'l {a ra \o r' tn $ ra S \ S SG{ $ao. \ S l"l. \ (?
>. 1! = * l >'r : ^, :  fr). ;1,!/!i ;(?= f: r*. fl J :!= J; J >. Vf i  ) CJ ) ṯ, ( r k : ( l i ( l 9 ) ( ;l fr i) rf,? l i =r, [l CB i.l.!.) i l.l l.!. * (.1 (..i .1.! r ).!,l l.r l ( i b i i '9,
More informationMath 234. What you should know on day one. August 28, You should be able to use general principles like. x = cos t, y = sin t, 0 t π.
Math 234 What you should know on day one August 28, 2001 1 You should be able to use general principles like Length = ds, Area = da, Volume = dv For example the length of the semi circle x = cos t, y =
More information1 4 (1 cos(4θ))dθ = θ 4 sin(4θ)
M48M Final Exam Solutions, December 9, 5 ) A polar curve Let C be the portion of the cloverleaf curve r = sin(θ) that lies in the first quadrant a) Draw a rough sketch of C This looks like one quarter
More informationDepartment of Mathematical and Statistical Sciences University of Alberta
MATH 214 (R1) Winter 2008 Intermediate Calculus I Solutions to Problem Set #8 Completion Date: Friday March 14, 2008 Department of Mathematical and Statistical Sciences University of Alberta Question 1.
More informationMPM 2D Final Exam Prep 2, June b) Y = 2(x + 1)218. ~..: 2. (xl 1:'}")( t J') ' ( B. vi::: 2 ~ 1'+ 4 1<. t:2 ( 6! '.
MPM 2D Final Exam Prep 2 June 2017 1. Express each equation in standard form and factored form: ~ ~ +et's 'leu t W (.. ".>tak( a) y = (x + 5)2 + 1 on ::t~'t.{1'" ~heeh v 1' K 1 C'. T.) '. (J. lr lov J
More informationr(j) ::::. X U.;,..;...h_D_Vl_5_ :;;2.. Name: ~s'~o=i Class; Date: ID: A
Name: ~s'~o=i Class; Date: U.;,..;...h_D_Vl_5 _ MAC 2233 Chapter 4 Review for the test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the derivative
More informatione x2 dxdy, e x2 da, e x2 x 3 dx = e
STS264 Calculus II: The fourth exam Dec 15, 214 Please show all your work! Answers without supporting work will be not given credit. Write answers in spaces provided. You have 1 hour and 2minutes to complete
More informationMTH 234 Solutions to Exam 2 April 13, Multiple Choice. Circle the best answer. No work needed. No partial credit available.
MTH 234 Solutions to Exam 2 April 3, 25 Multiple Choice. Circle the best answer. No work needed. No partial credit available.. (5 points) Parametrize of the part of the plane 3x+2y +z = that lies above
More informationAPPH 4200 Physics of Fluids
APPH 42 Physics of Fluids Problem Solving and Vorticity (Ch. 5) 1.!! Quick Review 2.! Vorticity 3.! Kelvin s Theorem 4.! Examples 1 How to solve fluid problems? (Like those in textbook) Ç"Tt=l I $T1P#(
More informationMath 210, Final Exam, Practice Fall 2009 Problem 1 Solution AB AC AB. cosθ = AB BC AB (0)(1)+( 4)( 2)+(3)(2)
Math 2, Final Exam, Practice Fall 29 Problem Solution. A triangle has vertices at the points A (,,), B (, 3,4), and C (2,,3) (a) Find the cosine of the angle between the vectors AB and AC. (b) Find an
More informationT h e C S E T I P r o j e c t
T h e P r o j e c t T H E P R O J E C T T A B L E O F C O N T E N T S A r t i c l e P a g e C o m p r e h e n s i v e A s s es s m e n t o f t h e U F O / E T I P h e n o m e n o n M a y 1 9 9 1 1 E T
More information********************************************************** 1. Evaluate the double or iterated integrals:
Practice problems 1. (a). Let f = 3x 2 + 4y 2 + z 2 and g = 2x + 3y + z = 1. Use Lagrange multiplier to find the extrema of f on g = 1. Is this a max or a min? No max, but there is min. Hence, among the
More informationMa 227 Final Exam Solutions 5/9/02
Ma 7 Final Exam Solutions 5/9/ Name: Lecture Section: I pledge m honor that I have abided b the Stevens Honor Sstem. ID: Directions: Answer all questions. The point value of each problem is indicated.
More informationG G. G. x = u cos v, y = f(u), z = u sin v. H. x = u + v, y = v, z = u v. 1 + g 2 x + g 2 y du dv
1. Matching. Fill in the appropriate letter. 1. ds for a surface z = g(x, y) A. r u r v du dv 2. ds for a surface r(u, v) B. r u r v du dv 3. ds for any surface C. G x G z, G y G z, 1 4. Unit normal N
More informationFigure 25:Differentials of surface.
2.5. Change of variables and Jacobians In the previous example we saw that, once we have identified the type of coordinates which is best to use for solving a particular problem, the next step is to do
More informationMultiple Choice. 1.(6 pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.
Multiple Choice.(6 pts) Find smmetric equations of the line L passing through the point (, 5, ) and perpendicular to the plane x + 3 z = 9. (a) x = + 5 3 = z (c) (x ) + 3( 3) (z ) = 9 (d) (e) x = 3 5 =
More informationPRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives.
PRACTICE PROBLEMS Please let me know if you find any mistakes in the text so that i can fix them. 1.1. Let Show that f is C 1 and yet How is that possible? 1. Mixed partial derivatives f(x, y) = {xy x
More informationP a g e 3 6 of R e p o r t P B 4 / 0 9
P a g e 3 6 of R e p o r t P B 4 / 0 9 p r o t e c t h um a n h e a l t h a n d p r o p e r t y fr om t h e d a n g e rs i n h e r e n t i n m i n i n g o p e r a t i o n s s u c h a s a q u a r r y. J
More informationPHYS 232 QUIZ minutes.
/ PHYS 232 QUIZ 1 02182005 50 minutes. This quiz has 3 questions. Please show all your work. If your final answer is not correct, you will get partial credit based on your work shown. You are allowed
More informationIn general, the formula is S f ds = D f(φ(u, v)) Φ u Φ v da. To compute surface area, we choose f = 1. We compute
alculus III Test 3 ample Problem Answers/olutions 1. Express the area of the surface Φ(u, v) u cosv, u sinv, 2v, with domain u 1, v 2π, as a double integral in u and v. o not evaluate the integral. In
More informationFinal Exam. Monday March 19, 3:305:30pm MAT 21D, Temple, Winter 2018
Name: Student ID#: Section: Final Exam Monday March 19, 3:305:30pm MAT 21D, Temple, Winter 2018 Show your work on every problem. orrect answers with no supporting work will not receive full credit. Be
More informationMath 350 Solutions for Final Exam Page 1. Problem 1. (10 points) (a) Compute the line integral. F ds C. z dx + y dy + x dz C
Math 35 Solutions for Final Exam Page Problem. ( points) (a) ompute the line integral F ds for the path c(t) = (t 2, t 3, t) with t and the vector field F (x, y, z) = xi + zj + xk. (b) ompute the line
More informationJim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt
Jim Lambers MAT 28 ummer emester 2121 Practice Final Exam olution 1. Evaluate the line integral xy dx + e y dy + xz dz, where is given by r(t) t 4, t 2, t, t 1. olution From r (t) 4t, 2t, t 2, we obtain
More information176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s
A g la di ou s F. L. 462 E l ec tr on ic D ev el op me nt A i ng er A.W.S. 371 C. A. M. A l ex an de r 236 A d mi ni st ra ti on R. H. (M rs ) A n dr ew s P. V. 326 O p ti ca l Tr an sm is si on A p ps
More information1. If the line l has symmetric equations. = y 3 = z+2 find a vector equation for the line l that contains the point (2, 1, 3) and is parallel to l.
. If the line l has symmetric equations MA 6 PRACTICE PROBLEMS x = y = z+ 7, find a vector equation for the line l that contains the point (,, ) and is parallel to l. r = ( + t) i t j + ( + 7t) k B. r
More informationMATH 332: Vector Analysis Summer 2005 Homework
MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,
More informationMath 6A Practice Problems II
Math 6A Practice Problems II Written by Victoria Kala vtkala@math.ucsb.edu SH 64u Office Hours: R : :pm Last updated 5//6 Answers This page contains answers only. Detailed solutions are on the following
More informationCalculus with Analytic Geometry 3 Fall 2018
alculus with Analytic Geometry 3 Fall 218 Practice Exercises for the Final Exam I present here a number of exercises that could serve as practice for the final exam. Some are easy and straightforward;
More informationSOME PROBLEMS YOU SHOULD BE ABLE TO DO
OME PROBLEM YOU HOULD BE ABLE TO DO I ve attempted to make a list of the main calculations you should be ready for on the exam, and included a handful of the more important formulas. There are no examples
More informationFinal exam (practice 1) UCLA: Math 32B, Spring 2018
Instructor: Noah White Date: Final exam (practice 1) UCLA: Math 32B, Spring 218 This exam has 7 questions, for a total of 8 points. Please print your working and answers neatly. Write your solutions in
More informationWithout fully opening the exam, check that you have pages 1 through 12.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 12. Show all your work on the standard
More informationNo calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk of everything excepts pens, pencils and erasers.
Name: Section: Recitation Instructor: READ THE FOLLOWING INSTRUCTIONS. Do not open your exam until told to do so. No calculators, cell phones or any other electronic devices can be used on this exam. Clear
More informationSOLUTIONS TO SECOND PRACTICE EXAM Math 21a, Spring 2003
SOLUTIONS TO SECOND PRACTICE EXAM Math a, Spring 3 Problem ) ( points) Circle for each of the questions the correct letter. No justifications are needed. Your score will be C W where C is the number of
More informationMATH 2203 Final Exam Solutions December 14, 2005 S. F. Ellermeyer Name
MATH 223 Final Exam Solutions ecember 14, 25 S. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of presentation. In
More informationLB 220 Homework 4 Solutions
LB 220 Homework 4 Solutions Section 11.4, # 40: This problem was solved in class on Feb. 03. Section 11.4, # 42: This problem was also solved in class on Feb. 03. Section 11.4, # 43: Also solved in class
More informationNote: Each problem is worth 14 points except numbers 5 and 6 which are 15 points. = 3 2
Math Prelim II Solutions Spring Note: Each problem is worth points except numbers 5 and 6 which are 5 points. x. Compute x da where is the region in the second quadrant between the + y circles x + y and
More informationMath 53: Worksheet 9 Solutions
Math 5: Worksheet 9 Solutions November 1 1 Find the work done by the force F xy, x + y along (a) the curve y x from ( 1, 1) to (, 9) We first parametrize the given curve by r(t) t, t with 1 t Also note
More informationOne side of each sheet is blank and may be used as scratch paper.
Math 244 Spring 2017 (Practice) Final 5/11/2017 Time Limit: 2 hours Name: No calculators or notes are allowed. One side of each sheet is blank and may be used as scratch paper. heck your answers whenever
More information(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere.
MATH 4 FINAL EXAM REVIEW QUESTIONS Problem. a) The points,, ) and,, 4) are the endpoints of a diameter of a sphere. i) Determine the center and radius of the sphere. ii) Find an equation for the sphere.
More informationLet s estimate the volume under this surface over the rectangle R = [0, 4] [0, 2] in the xyplane.
Math 54  Vector Calculus Notes 3.  3. Double Integrals Consider f(x, y) = 8 x y. Let s estimate the volume under this surface over the rectangle R = [, 4] [, ] in the xyplane. Here is a particular estimate:
More informationMath 32B Discussion Session Week 10 Notes March 14 and March 16, 2017
Math 3B iscussion ession Week 1 Notes March 14 and March 16, 17 We ll use this week to review for the final exam. For the most part this will be driven by your questions, and I ve included a practice final
More informationCalculus III. Math 233 Spring Final exam May 3rd. Suggested solutions
alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.
More informationP a g e 5 1 of R e p o r t P B 4 / 0 9
P a g e 5 1 of R e p o r t P B 4 / 0 9 J A R T a l s o c o n c l u d e d t h a t a l t h o u g h t h e i n t e n t o f N e l s o n s r e h a b i l i t a t i o n p l a n i s t o e n h a n c e c o n n e
More informationAPPH 4200 Physics of Fluids
APPH 4200 Physics of Fluids Rotating Fluid Flow October 6, 2011 1.!! Hydrostatics of a Rotating Water Bucket (again) 2.! Bath Tub Vortex 3.! Ch. 5: Problem Solving 1 Key Definitions & Concepts Ω U Cylindrical
More informationMathematics Extension 1
BAULKHAM HILLS HIGH SCHOOL TRIAL 04 YEAR TASK 4 Mathematics Etension General Instructions Reading time 5 minutes Working time 0 minutes Write using black or blue pen Boardapproved calculators may be used
More informationPractice Problems for the Final Exam
Math 114 Spring 2017 Practice Problems for the Final Exam 1. The planes 3x + 2y + z = 6 and x + y = 2 intersect in a line l. Find the distance from the origin to l. (Answer: 24 3 ) 2. Find the area of
More informationMATHEMATICS EXTENSION2
Student Number: Class: TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION 015 MATHEMATICS EXTENSION General Instructions: Total Marks 100 Reading Time: 5 minutes. In Question 1116, show all relevant mathematical
More informationf dr. (6.1) f(x i, y i, z i ) r i. (6.2) N i=1
hapter 6 Integrals In this chapter we will look at integrals in more detail. We will look at integrals along a curve, and multidimensional integrals in 2 or more dimensions. In physics we use these integrals
More informationMTH 234 Exam 2 November 21st, Without fully opening the exam, check that you have pages 1 through 12.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 12. Show all your work on the standard
More information,\ I. . < c}. " C:)' ) I p od ;::: 'J. d, cl cr  I. ( I) Cl c,\. c. 1\'0\ ~ '~O'_. e ~.\~\S
Math 3306  Test 1 Name: An d {"0v\ ( _ roj ~ ed Date: l'( ~0 { 1\ Fall 2011 1. (to Pts) Let S == {I, 2, 3,4, 5,6,7,8,9, 10}. of each of the following types of mappings, provide justification for why the
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 2/13/13, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.2. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241sp13/241.html)
More informationSection 14.1 Vector Functions and Space Curves
Section 14.1 Vector Functions and Space Curves Functions whose range does not consists of numbers A bulk of elementary mathematics involves the study of functions  rules that assign to a given input a
More informationMAY THE FORCE BE WITH YOU, YOUNG JEDIS!!!
Final Exam Math 222 Spring 2011 May 11, 2011 Name: Recitation Instructor s Initials: You may not use any type of calculator whatsoever. (Cell phones off and away!) You are not allowed to have any other
More information~,. :'lr. H ~ j. l' ", ...,~l. 0 '" ~ bl '!; 1'1. :<! f'~.., I,," r: t,... r':l G. t r,. 1'1 [<, ."" f'" 1n. t.1 ~ n I'>' 1:1 , I. <1 ~'..
,, 'l t (.) :;,/.I I n ri' ' r l ' rt ( n :' (I : d! n t, :?rj I),.. fl.),. f!..,,., til, ID fi... j I. 't' r' t II!:t () (l r El,, (fl lj J4 ([) f., () :. ,,.,.I :i l:'!, :I J.A.. t,.. p,  ' I I I
More informationLecture Notes for MATH2230. Neil Ramsamooj
Lecture Notes for MATH3 Neil amsamooj Table of contents Vector Calculus................................................ 5. Parametric curves and arc length...................................... 5. eview
More informationPractice problems. 1. Evaluate the double or iterated integrals: First: change the order of integration; Second: polar.
Practice problems 1. Evaluate the double or iterated integrals: R x 3 + 1dA where R = {(x, y) : 0 y 1, y x 1}. 1/ 1 y 0 3y sin(x + y )dxdy First: change the order of integration; Second: polar.. Consider
More informationJim Lambers MAT 280 Fall Semester Practice Final Exam Solution
Jim Lambers MAT 8 Fall emester 67 Practice Final Exam olution. Use Lagrange multipliers to find the point on the circle x + 4 closest to the point (, 5). olution We have f(x, ) (x ) + ( 5), the square
More informationSept , 17, 23, 29, 37, 41, 45, 47, , 5, 13, 17, 19, 29, 33. Exam Sept 26. Covers Sept 30Oct 4.
MATH 23, FALL 2013 Text: Calculus, Early Transcendentals or Multivariable Calculus, 7th edition, Stewart, Brooks/Cole. We will cover chapters 12 through 16, so the multivariable volume will be fine. WebAssign
More informationMath 340 Final Exam December 16, 2006
Math 34 Final Exam December 6, 6. () Suppose A 3 4. a) Find the rowreduced echelon form of A. 3 4 so the row reduced echelon form is b) What is rank(a)? 3 4 4 The rank is two since there are two pivots.
More informationContents. MATH 32B2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9
MATH 32B2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B2 (8W)
More information18.02 Multivariable Calculus Fall 2007
MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. V9. Surface Integrals Surface
More informationPractice problems. 1. Given a = 3i 2j and b = 2i + j. Write c = i + j in terms of a and b.
Practice problems 1. Given a = 3i 2j and b = 2i + j. Write c = i + j in terms of a and b. 1, 1 = c 1 3, 2 + c 2 2, 1. Solve c 1, c 2. 2. Suppose a is a vector in the plane. If the component of the a in
More informationSOLUTIONS TO HOMEWORK ASSIGNMENT #2, Math 253
SOLUTIONS TO HOMEWORK ASSIGNMENT #, Math 5. Find the equation of a sphere if one of its diameters has end points (, 0, 5) and (5, 4, 7). The length of the diameter is (5 ) + ( 4 0) + (7 5) = =, so the
More informationName: ID: Math 233 Exam 1. Page 1
Page 1 Name: ID: This exam has 20 multiple choice questions, worth 5 points each. You are allowed to use a scientific calculator and a 3 5 inch note card. 1. Which of the following pairs of vectors are
More information0 t b r 6, 20 t l nf r nt f th l t th t v t f th th lv, ntr t n t th l l l nd d p rt nt th t f ttr t n th p nt t th r f l nd d tr b t n. R v n n th r
n r t d n 20 22 0: T P bl D n, l d t z d http:.h th tr t. r pd l 0 t b r 6, 20 t l nf r nt f th l t th t v t f th th lv, ntr t n t th l l l nd d p rt nt th t f ttr t n th p nt t th r f l nd d tr b t n.
More informationon an open connected region D, then F is conservative on D. (c) If curl F=curl G on R 3, then C F dr = C G dr for all closed path C.
. (5%) Determine the statement is true ( ) or false ( ). 微甲 4 班期末考解答和評分標準 (a) If f(x, y) is continuous on the rectangle R = {(x, y) a x b, c y d} except for finitely many points, then f(x, y) is integrable
More informationScore: Fall 2009 Name Row 80. C(t) = 30te O. 04t
Math 1410  Test #3A Score: Fall 2009 Name Row 80 Q1: This is a calculator problem. If t, in minutes, is the time since a drug was administered, the concentration, C(t) in ng/ml, of a drug in a patient's
More informationReview for the Final Exam
Calculus 3 Lia Vas Review for the Final Exam. Sequences. Determine whether the following sequences are convergent or divergent. If they are convergent, find their limits. (a) a n = ( 2 ) n (b) a n = n+
More informationMath 114: Makeup Final Exam. Instructions:
Math 114: Makeup Final Exam Instructions: 1. Please sign your name and indicate the name of your instructor and your teaching assistant: A. Your Name: B. Your Instructor: C. Your Teaching Assistant: 2.
More informationExercises for Multivariable Differential Calculus XM521
This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done
More informationMATH 52 FINAL EXAM DECEMBER 7, 2009
MATH 52 FINAL EXAM DECEMBER 7, 2009 THIS IS A CLOSED BOOK, CLOSED NOTES EXAM. NO CALCULATORS OR OTHER ELECTRONIC DEVICES ARE PERMITTED. IF YOU NEED EXTRA SPACE, PLEASE USE THE BACK OF THE PREVIOUS PROB
More information4 Partial Differentiation
4 Partial Differentiation Many equations in engineering, physics and mathematics tie together more than two variables. For example Ohm s Law (V = IR) and the equation for an ideal gas, PV = nrt, which
More informationPrint Your Name: Your Section:
Print Your Name: Your Section: Mathematics 1c. Practice Final Solutions This exam has ten questions. J. Marsden You may take four hours; there is no credit for overtime work No aids (including notes, books,
More informationReview Questions for Test 3 Hints and Answers
eview Questions for Test 3 Hints and Answers A. Some eview Questions on Vector Fields and Operations. A. (a) The sketch is left to the reader, but the vector field appears to swirl in a clockwise direction,
More informationEXAM 2, MATH 132 WEDNESDAY, OCTOBER 23, 2002
EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23, 2002 This examination has 20 multiple choice questions, and two essay questions. Please check it over and if you find it to be incomplete, notify the proctor. Do
More informationn
p l p bl t n t t f Fl r d, D p rt nt f N t r l R r, D v n f nt r r R r, B r f l. n.24 80 T ll h, Fl. : Fl r d D p rt nt f N t r l R r, B r f l, 86. http://hdl.handle.net/2027/mdp.39015007497111 r t v n
More information,.*Hffi;;* SONAI, IUERCANTII,N I,IMITDII REGD 0FFICE: 105/33, VARDHMAN GotD[N PLNLA,R0AD No.44, pitampura, DELHI *ffigfk"
$ S, URCT,,MTD RGD 0C: 10/, VRDM G[ LL,R0D.44, ptmpur, DL114 C: l22ldll98l,c0224gb, eb:.nlmernte.m T, Dte: 17h tber, 201 BS Lmted hre ]eejeebhy Ter Dll Street Mumb 41 The Mnger (Ltng) Delh Stk xhnge /1,
More information,., [~== I ] ~y_/5 = 21 Y / Y. t. \,X ::: 3J ~  3. Test: Linear equations and Linear inequalities. At!$fJJ' ~ dt~  5 = 7C +4 + re t~ +>< 1 )_
CST 11 Math  September 16 th, 2016 Test: Linear equations and Linear inequalities NAME: At!$fJJ' ~ Section: MCU504:   86 1100 1. Solve the equations below: (4 marks) 2 5 a) 3("3 x "3) =  x + 4 /{J1:x
More informationMAC 1147 Final Exam Review
MAC 1147 Final Exam Review nstructions: The final exam will consist of 15 questions plu::; a bonus problem. Some questions will have multiple parts and others will not. Some questions will be multiple
More informationMa 227 Final Exam Solutions 12/13/11
Ma 7 Final Exam Solutions /3/ Name: Lecture Section: (A and B: Prof. Levine, C: Prof. Brady) Problem a) ( points) Find the eigenvalues and eigenvectors of the matrix A. A 3 5 Solution. First we find the
More informationFinal Review Worksheet
Score: Name: Final Review Worksheet Math 2110Q Fall 2014 Professor Hohn Answers (in no particular order): f(x, y) = e y + xe xy + C; 2; 3; e y cos z, e z cos x, e x cos y, e x sin y e y sin z e z sin x;
More informationWest Bengal State University
West Bengal State University MTMG (GEN)Ol B.A./B.Sc./B.Com. ( Honours, Major, General) Examinations, 2015 PART I MATHEMATICS  GENERAL Paper I Duration : 3 Hours l [ Full Marks: 100 The figures in the
More informationFigure 21:The polar and Cartesian coordinate systems.
Figure 21:The polar and Cartesian coordinate systems. Coordinate systems in R There are three standard coordinate systems which are used to describe points in dimensional space. These coordinate systems
More information(a) 0 (b) 1/4 (c) 1/3 (d) 1/2 (e) 2/3 (f) 3/4 (g) 1 (h) 4/3
Math 114 Practice Problems for Test 3 omments: 0. urface integrals, tokes Theorem and Gauss Theorem used to be in the Math40 syllabus until last year, so we will look at some of the questions from those
More informationMath 20C Homework 2 Partial Solutions
Math 2C Homework 2 Partial Solutions Problem 1 (12.4.14). Calculate (j k) (j + k). Solution. The basic properties of the cross product are found in Theorem 2 of Section 12.4. From these properties, we
More information