Calculus III 2004 Summer Practice Final 8/3/2004
|
|
- Josephine Elliott
- 5 years ago
- Views:
Transcription
1 .. Calculus III 4 ummer Practice Final 8/3/4. Compute the following limits if they exist: (a) lim (x,y) (,) e xy x+. cos x (b) lim x. (x,y) (,) x 4 +y 4 (a) ince lim (x,y) (,) exy and lim x + 6 in a (x,y) (,) neighbourhood of (, ), we use quotient property of limits to get lim (x,y) (,) e xy x +. (b) If we approaches (, ) along y, we have, by l hospital rule, cos x x sin x x cos x lim lim lim. (x,) (,) x 4 x 4x 3 x x This last limit tends to as x. Thus, the lim (x,y) (,) does not exist.. Find the absolute maximum and minimum for the function f (x, y, z) x + y + z on the ball B (x, y, z) x + y + z ª. Write B U B where U {(x, y, z) x + y + z < } and B {(x, y, z) x + y + z }. ince f (,, ) 6 (,, ), wehavenocriticalpointson U. cos x x x 4 +y 4
2 To find the critical points on B, letf (x, y, z) x + y + z and g (x, y, z) x + y + z. By the method of Lagrange multiplier, we have f λ g, that is, (,, ) λ (x, y, z). o, we have the following system of equations: λx λy λz x + y + z. By λx,weknowλ 6. We have x y z. By λ the last equation, we have 3.Thisimpliesthatλ ± 3 4λ ³. Moreover, we have two critical points (x, y, z) 3, 3, 3 or (x, y, z) ³ 3, 3, 3. ince we have only two candidates of absolute maximum and minimum, one must be the absolute maximum and another ³ must be the absolute minimum. It is easy to see that f 3, 3, is the absolute maximum and f ³ 3, 3, is the absolute minimum. 3. Evaluate x + y + z dxdydz over the sphere {(x, y, z) x + y + z }. Using spherical coordinates, let x ρ sin φ cos θ, y ρ sin φ sin θ and z ρ cos φ where ρ, φ π and θ π.
3 3 o, x + y + z ρ and the Jacobian is ρ sin φ. Therefore, x + y + z dxdydz π π π π µ µ π ρ 4 dρ Ã! ρ 5 5 ρ ρ ρ sin φdθdφdρ ρ4 sin φ dθdφdρ µ π sin φdφ dθ ³ cos φ π φ θ π θ ( ( ) ( )) π 5 4π etermine whether the integral xy dxdy exists where [, ] [, ]. If it exists, compute its value.
4 4 This integral is improper whenever xy.let δ, [δ, ] [ε, ]. Wehave δ,ε as δ and ε and xy is welldefined and integrable in δ,ε. Therefore, lim δ, δ lim xy dxdy lim δ δ lim δ δ lim dx δ δ x h 4 lim x δ h lim 4 4 i δ δ 4. xy dydx " µ # xy dx x y x x lim x dx x xδ i ln δ ince lim δ ln δ lim RR δ δ δ x+y dxdy ln. x +xy+y lim δ δ δ lim δ δ,wehave 5. Evaluate y e x dxdy.
5 By drawing the graph of the region, we change the order of integration into 5 y e x dxdy x ex xe x dx x e e. e x dydx 6. Evaluate z d where is the portion of the cylinder x + y 4between the planes z and z x +3. Use cylindrical coordinates for the cylinder. Let x cosθ, y sinθ and z z where z x +3cosθ +3 and θ π. o, the parametrization Φ of is Φ (θ, z) (cosθ, sinθ, z) where θ π and z cosθ +3. We calculate o, we have T θ T z ( sinθi +cosθj) (zk) cosθi +sinθj. kt θ T z k.
6 6 Therefore, z d π cosθ+3 π π 3 6π. z () dzdθ 3 ( cos θ +3)3 dθ 8cos 3 θ +36cos θ +54cosθ +7 dθ 7. Auniformfluid that flows vertically downward (like, heavy rain) is described by the vector field F (x, y, z) (,, ). Find the total flux through the cone z p x + y,x +y. Parametrize the cone by letting x r cos θ, y r sin θ and z p q x + y (r cos θ) +(rsin θ) r where r and θ π. o, the parametrization Φ of is Φ (r, θ) (r cos θ, r sin θ, r) where r and θ π. We calculate (since our fliud flows downward, we calculate T θ T r instead of T r T θ.) T θ T r ( r sin θi + r cos θj) (cos θi +sinθj + k) r cos θi + r sin θj rk. The flux is the surface integral of F over, thatis, RR F d. Therefore, we have F d F (T θ T r ) dθdr π. Φ π π (,, ) (r cos θ, r sin θ, r) drdθ rdrdθ
7 8. Evaluate C x ydx + ydy where C is the bournary of the region between the curve y x and y x 3 where x. By Green s theorem, (y) x ydx + ydy C x (x y) dxdy y where C is the boundry of with counterclockwise orientation. We can describe as x and x 3 y x. Hence, (y) x (x y) dxdy y x. x dydx x h 3 x y x i dx yx 3 x 3 + x 5 dx 9. Let F yzi +( x +3y +)j + x + z k. Evaluate ( F) d where is the cylinder x + y a, z (without the top and bottom). We have i j k F x y z yz x +3y + x + z (x y) j +( z) k. We can parametrize as x a cos θ, y a sin θ and z z where θ π and z. Thus, the parametrization Φ 7
8 8 of is Φ (θ, z) (a cos θ, a sin θ, z). where θ π and z. We calculate T θ T z ( a sin θi + a cos θj) (zk) a cos θi + a sin θj. Therefore, ( F) d ( F) (T θ T z ) dθdz π π (, (a cos θ)+a sin θ, z) (a cos θ, a sin θ, ) dzdθ a sin θ cos θ +a sin θ dθ π a sin θ sin θ cos θ dθ a π. By tokes theorem, ( F) d F ds where is the boundary of the surface. Therefore, we have two parts of. One is x + y a and z. Another is x + y a and z. We parametrize x + y a by letting x a cos θ and y a sin θ where θ π. Hence, dx a sin θdθ and dy a cos θdθ. Moreover, in both cases, dz.
9 Thus, for the first one, we have F ds (yz) dx +( x +3y +)dy + x + z dz π π ( a cos θ +3a sin θ +)(a cos θdθ) a cos θ +3a sin θ cos θ +a cos θ dθ a π. For the second one, we have F ds (yz) dx +( x +3y +)dy + x + z dz π π π ( a sin θ ) ( a sin θdθ)+( a cos θ +3a sin θ +)(a cos θdθ) a sin θ a cos θ +3a sin θ cos θ +a cos θ dθ a sin θ a +3a sin θ cos θ +a cos θ dθ 3a π. Therefore, ( F) d F ds a π 3a π 4a π. Why do we get a different answer? oes this mean that the tokes theorem is not valid in this case? If so, why? Or, do we need to use different orientations on two parts of boundary of?. (a) how that F 6xy (cos z) i +3x (cos z) j 3x y sin z k is conservative. 9
10 (b) Find f such that F f. (c) Evaluate the integral of F along the curve x cos 3 θ, y sin 3 θ, z where θ π. (a) ince F i j k x y z 6xy (cos z) 3x (cos z) (3x y sin z), F is conservative. (b) Assume F is the gradient of f (x, y, z). Then, f must satisfy f 6xy cos z x f 3x cos z y f 3x y sin z. z Integrate the firstonewithrespecttox,wehavef (x, y, z) 3x y cos z + h (y, z). Hence, we have 3x cos z f y 3x cos z+ h h. Therefore, we have,thatis,h (y, z) y y g (z). Thus,f (x, y, z) 3x y cos z+g (z). o, 3x y sin z f z 3x y cos z + g (z). We conclude that g (z),that is, g (z) C where C is a constant. This implies that f (x, y, z) 3x y cos z + C. (c) ince F is a gradient, we have F ds c f ds c ³ ³ π f 3.. Let c ³ cos 3 π f (c ()) ³ sin 3 π cos 3 cos 3 sin 3 cos F x yi + z 8 j xyzk. Evaluate the integral of F over the surface of the unit cube.
11 Let be the surface of the unit cube W. By Gauss divergence theorem, we have F d ( F) dv. We calculate W F xy ++( xy). Thus, we have F d () dv. W
Ma 1c Practical - Solutions to Homework Set 7
Ma 1c Practical - olutions to omework et 7 All exercises are from the Vector Calculus text, Marsden and Tromba (Fifth Edition) Exercise 7.4.. Find the area of the portion of the unit sphere that is cut
More information( ) ( ) ( ) ( ) Calculus III - Problem Drill 24: Stokes and Divergence Theorem
alculus III - Problem Drill 4: tokes and Divergence Theorem Question No. 1 of 1 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as needed () Pick the 1. Use
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 informationSOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)
SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please
More informationName: Instructor: Lecture time: TA: Section time:
Math 222 Final May 11, 29 Name: Instructor: Lecture time: TA: Section time: INSTRUCTIONS READ THIS NOW This test has 1 problems on 16 pages worth a total of 2 points. Look over your test package right
More informationAnswer sheet: Final exam for Math 2339, Dec 10, 2010
Answer sheet: Final exam for Math 9, ec, Problem. Let the surface be z f(x,y) ln(y + cos(πxy) + e ). (a) Find the gradient vector of f f(x,y) y + cos(πxy) + e πy sin(πxy), y πx sin(πxy) (b) Evaluate f(,
More informationHOMEWORK 8 SOLUTIONS
HOMEWOK 8 OLUTION. Let and φ = xdy dz + ydz dx + zdx dy. let be the disk at height given by: : x + y, z =, let X be the region in 3 bounded by the cone and the disk. We orient X via dx dy dz, then by definition
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 x-simple region. R = {(x, y) y, x y }
More informationPractice problems **********************************************************
Practice problems I will not test spherical and cylindrical coordinates explicitly but these two coordinates can be used in the problems when you evaluate triple integrals. 1. Set up the integral without
More informationName: Date: 12/06/2018. M20550 Calculus III Tutorial Worksheet 11
1. ompute the surface integral M255 alculus III Tutorial Worksheet 11 x + y + z) d, where is a surface given by ru, v) u + v, u v, 1 + 2u + v and u 2, v 1. olution: First, we know x + y + z) d [ ] u +
More informationf(p i )Area(T i ) F ( r(u, w) ) (r u r w ) da
MAH 55 Flux integrals Fall 16 1. Review 1.1. Surface integrals. Let be a surface in R. Let f : R be a function defined on. efine f ds = f(p i Area( i lim mesh(p as a limit of Riemann sums over sampled-partitions.
More informationSolutions to Sample Questions for Final Exam
olutions to ample Questions for Final Exam Find the points on the surface xy z 3 that are closest to the origin. We use the method of Lagrange Multipliers, with f(x, y, z) x + y + z for the square of the
More informationMATHS 267 Answers to Stokes Practice Dr. Jones
MATH 267 Answers to tokes Practice Dr. Jones 1. Calculate the flux F d where is the hemisphere x2 + y 2 + z 2 1, z > and F (xz + e y2, yz, z 2 + 1). Note: the surface is open (doesn t include any of the
More informationMATH2000 Flux integrals and Gauss divergence theorem (solutions)
DEPARTMENT O MATHEMATIC MATH lux integrals and Gauss divergence theorem (solutions ( The hemisphere can be represented as We have by direct calculation in terms of spherical coordinates. = {(r, θ, φ r,
More informationMathematics 205 Solutions for HWK 23. e x2 +y 2 dxdy
Mathematics 5 Solutions for HWK Problem 1. 6. p7. Let D be the unit disk: x + y 1. Evaluate the integral e x +y dxdy by making a change of variables to polar coordinates. D Solution. Step 1. The integrand,
More information2( 2 r 2 2r) rdrdθ. 4. Your result fits the correct answer: get 2 pts, if you make a slight mistake, get 1 pt. 0 r 1
Page 1 of 1 112 微甲 7-11 班期末考解答和評分標準 1. (1%) Find the volume of the solid bounded below by the cone z 2 4(x 2 + y 2 ) and above by the ellipsoid 4(x 2 + y 2 ) + z 2 8. Method 1 Use cylindrical coordinates:
More information(b) Find the range of h(x, y) (5) Use the definition of continuity to explain whether or not the function f(x, y) is continuous at (0, 0)
eview Exam Math 43 Name Id ead each question carefully. Avoid simple mistakes. Put a box around the final answer to a question (use the back of the page if necessary). For full credit you must show your
More informationMcGill University April Calculus 3. Tuesday April 29, 2014 Solutions
McGill University April 4 Faculty of Science Final Examination Calculus 3 Math Tuesday April 9, 4 Solutions Problem (6 points) Let r(t) = (t, cos t, sin t). i. Find the velocity r (t) and the acceleration
More informationVector Calculus. Dr. D. Sukumar. January 31, 2014
Vector Calculus Dr. D. Sukumar January 31, 2014 Green s Theorem Tangent form or Ciculation-Curl form c Mdx +Ndy = R ( N x M ) da y Green s Theorem Tangent form or Ciculation-Curl form c Mdx +Ndy = C F
More informationSolution. This is a routine application of the chain rule.
EXAM 2 SOLUTIONS 1. If z = e r cos θ, r = st, θ = s 2 + t 2, find the partial derivatives dz ds chain rule. Write your answers entirely in terms of s and t. dz and dt using the Solution. This is a routine
More informationMULTIVARIABLE INTEGRATION
MULTIVARIABLE INTEGRATION (PLANE & CYLINDRICAL POLAR COORDINATES) PLANE POLAR COORDINATES Question 1 The finite region on the x-y plane satisfies 1 x + y 4, y 0. Find, in terms of π, the value of I. I
More informatione x3 dx dy. 0 y x 2, 0 x 1.
Problem 1. Evaluate by changing the order of integration y e x3 dx dy. Solution:We change the order of integration over the region y x 1. We find and x e x3 dy dx = y x, x 1. x e x3 dx = 1 x=1 3 ex3 x=
More informationMAT 211 Final Exam. Spring Jennings. Show your work!
MAT 211 Final Exam. pring 215. Jennings. how your work! Hessian D = f xx f yy (f xy ) 2 (for optimization). Polar coordinates x = r cos(θ), y = r sin(θ), da = r dr dθ. ylindrical coordinates x = r cos(θ),
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 informationMath 212-Lecture Integration in cylindrical and spherical coordinates
Math 22-Lecture 6 4.7 Integration in cylindrical and spherical coordinates Cylindrical he Jacobian is J = (x, y, z) (r, θ, z) = cos θ r sin θ sin θ r cos θ = r. Hence, d rdrdθdz. If we draw a picture,
More informationReview for the Final Test
Math 7 Review for the Final Test () Decide if the limit exists and if it exists, evaluate it. lim (x,y,z) (0,0,0) xz. x +y +z () Use implicit differentiation to find z if x + y z = 9 () Find the unit tangent
More informationPractice problems. m zδdv. In our case, we can cancel δ and have z =
Practice problems 1. Consider a right circular cone of uniform density. The height is H. Let s say the distance of the centroid to the base is d. What is the value d/h? We can create a coordinate system
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 informationx + ye z2 + ze y2, y + xe z2 + ze x2, z and where T is the
1.(8pts) Find F ds where F = x + ye z + ze y, y + xe z + ze x, z and where T is the T surface in the pictures. (The two pictures are two views of the same surface.) The boundary of T is the unit circle
More informationMATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS
MATH 228: Calculus III (FALL 216) Sample Problems for FINAL EXAM SOLUTIONS MATH 228 Page 2 Problem 1. (2pts) Evaluate the line integral C xy dx + (x + y) dy along the parabola y x2 from ( 1, 1) to (2,
More informationSolutions to the Final Exam, Math 53, Summer 2012
olutions to the Final Exam, Math 5, ummer. (a) ( points) Let be the boundary of the region enclosedby the parabola y = x and the line y = with counterclockwise orientation. alculate (y + e x )dx + xdy.
More informationProblem 1. Use a line integral to find the plane area enclosed by the curve C: r = a cos 3 t i + b sin 3 t j (0 t 2π). Solution: We assume a > b > 0.
MATH 64: FINAL EXAM olutions Problem 1. Use a line integral to find the plane area enclosed by the curve C: r = a cos 3 t i + b sin 3 t j ( t π). olution: We assume a > b >. A = 1 π (xy yx )dt = 3ab π
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 informationPractice problems ********************************************************** 1. Divergence, curl
Practice problems 1. Set up the integral without evaluation. The volume inside (x 1) 2 + y 2 + z 2 = 1, below z = 3r but above z = r. This problem is very tricky in cylindrical or Cartesian since we must
More informatione x2 dxdy, e x2 da, e x2 x 3 dx = e
STS26-4 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 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 Exam IV - Fall 2011
Math 233 - Exam IV - Fall 2011 December 15, 2011 - Renato Feres NAME: STUDENT ID NUMBER: General instructions: This exam has 16 questions, each worth the same amount. Check that no pages are missing and
More informationMath 263 Final. (b) The cross product is. i j k c. =< c 1, 1, 1 >
Math 63 Final Problem 1: [ points, 5 points to each part] Given the points P : (1, 1, 1), Q : (1,, ), R : (,, c 1), where c is a parameter, find (a) the vector equation of the line through P and Q. (b)
More information7a3 2. (c) πa 3 (d) πa 3 (e) πa3
1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin
More information1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r (t) = 3 cos t, 0, 3 sin t, r ( 3π
1. a) 5 points) Find the unit tangent and unit normal vectors T and N to the curve at the point P 3, 3π, r t) 3 cos t, 4t, 3 sin t 3 ). b) 5 points) Find curvature of the curve at the point P. olution:
More information1. For each function, find all of its critical points and then classify each point as a local extremum or saddle point.
Solutions Review for Exam # Math 6. For each function, find all of its critical points and then classify each point as a local extremum or saddle point. a fx, y x + 6xy + y Solution.The gradient of f is
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 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 informationReview problems for the final exam Calculus III Fall 2003
Review problems for the final exam alculus III Fall 2003 1. Perform the operations indicated with F (t) = 2t ı 5 j + t 2 k, G(t) = (1 t) ı + 1 t k, H(t) = sin(t) ı + e t j a) F (t) G(t) b) F (t) [ H(t)
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 informationMath Review for Exam 3
1. ompute oln: (8x + 36xy)ds = Math 235 - Review for Exam 3 (8x + 36xy)ds, where c(t) = (t, t 2, t 3 ) on the interval t 1. 1 (8t + 36t 3 ) 1 + 4t 2 + 9t 4 dt = 2 3 (1 + 4t2 + 9t 4 ) 3 2 1 = 2 3 ((14)
More informationFinal Exam Review Sheet : Comments and Selected Solutions
MATH 55 Applied Honors alculus III Winter Final xam Review heet : omments and elected olutions Note: The final exam will cover % among topics in chain rule, linear approximation, maximum and minimum values,
More informationMarking Scheme for the end semester examination of MTH101, (I) for n N. Show that (x n ) converges and find its limit. [5]
Marking Scheme for the end semester examination of MTH, 3-4 (I). (a) Let x =, x = and x n+ = xn+x for n N. Show that (x n ) converges and find its limit. [5] Observe that x n+ x = x x n [] The sequence
More informationMAT 211 Final Exam. Fall Jennings.
MAT 211 Final Exam. Fall 218. Jennings. Useful formulas polar coordinates spherical coordinates: SHOW YOUR WORK! x = rcos(θ) y = rsin(θ) da = r dr dθ x = ρcos(θ)cos(φ) y = ρsin(θ)cos(φ) z = ρsin(φ) dv
More informationM273Q 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 informationSections minutes. 5 to 10 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed.
MTH 34 Review for Exam 4 ections 16.1-16.8. 5 minutes. 5 to 1 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed. Review for Exam 4 (16.1) Line
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 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 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 212-1 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 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 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 multi-dimensional integrals in 2 or more dimensions. In physics we use these integrals
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 informationS12.1 SOLUTIONS TO PROBLEMS 12 (ODD NUMBERS)
OLUTION TO PROBLEM 2 (ODD NUMBER) 2. The electric field is E = φ = 2xi + 2y j and at (2, ) E = 4i + 2j. Thus E = 2 5 and its direction is 2i + j. At ( 3, 2), φ = 6i + 4 j. Thus the direction of most rapid
More informationMATH 280 Multivariate Calculus Fall Integration over a surface. da. A =
MATH 28 Multivariate Calculus Fall 212 Integration over a surface Given a surface S in space, we can (conceptually) break it into small pieces each of which has area da. In me cases, we will add up these
More information234 Review Sheet 2 Solutions
4 Review Sheet Solutions. Find all the critical points of the following functions and apply the second derivative test. (a) f(x, y) (x y)(x + y) ( ) x f + y + (x y)x (x + y) + (x y) ( ) x + ( x)y x + x
More informationReview Sheet for the Final
Review Sheet for the Final Math 6-4 4 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And the absence
More informationLine and Surface Integrals. Stokes and Divergence Theorems
Math Methods 1 Lia Vas Line and urface Integrals. tokes and Divergence Theorems Review of urves. Intuitively, we think of a curve as a path traced by a moving particle in space. Thus, a curve is a function
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 informationProblem Set 5 Math 213, Fall 2016
Problem Set 5 Math 213, Fall 216 Directions: Name: Show all your work. You are welcome and encouraged to use Mathematica, or similar software, to check your answers and aid in your understanding of the
More informationMLC Practice Final Exam
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 13. Show all your work on the standard
More informationMcGill University December Intermediate Calculus. Tuesday December 17, 2014 Time: 14:00-17:00
McGill University December 214 Faculty of Science Final Examination Intermediate Calculus Math 262 Tuesday December 17, 214 Time: 14: - 17: Examiner: Dmitry Jakobson Associate Examiner: Neville Sancho
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 informationMcGill University April 16, Advanced Calculus for Engineers
McGill University April 16, 2014 Faculty of cience Final examination Advanced Calculus for Engineers Math 264 April 16, 2014 Time: 6PM-9PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer
More informationLaplace equation in polar coordinates
Laplace equation in polar coordinates The Laplace equation is given by 2 F 2 + 2 F 2 = 0 We have x = r cos θ, y = r sin θ, and also r 2 = x 2 + y 2, tan θ = y/x We have for the partials with respect to
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 informationMath 210, Final Exam, Spring 2012 Problem 1 Solution. (a) Find an equation of the plane passing through the tips of u, v, and w.
Math, Final Exam, Spring Problem Solution. Consider three position vectors (tails are the origin): u,, v 4,, w,, (a) Find an equation of the plane passing through the tips of u, v, and w. (b) Find an equation
More informationGreen s, Divergence, Stokes: Statements and First Applications
Math 425 Notes 12: Green s, Divergence, tokes: tatements and First Applications The Theorems Theorem 1 (Divergence (planar version)). Let F be a vector field in the plane. Let be a nice region of the plane
More informationVector Calculus. Dr. D. Sukumar. February 1, 2016
Vector Calculus Dr. D. Sukumar February 1, 2016 Green s Theorem Tangent form or Ciculation-Curl form c Mdx + Ndy = R ( N x M ) da y Green s Theorem Tangent form or Ciculation-Curl form Stoke s Theorem
More informationInstructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.
Exam 3 Math 850-007 Fall 04 Odenthal Name: Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.. Evaluate the iterated integral
More informationMath 3435 Homework Set 11 Solutions 10 Points. x= 1,, is in the disk of radius 1 centered at origin
Math 45 Homework et olutions Points. ( pts) The integral is, x + z y d = x + + z da 8 6 6 where is = x + z 8 x + z = 4 o, is the disk of radius centered on the origin. onverting to polar coordinates then
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 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 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 informationPeter Alfeld Math , Fall 2005
WeBWorK assignment due 9/2/05 at :59 PM..( pt) Consider the parametric equation x = 2(cosθ + θsinθ) y = 2(sinθ θcosθ) What is the length of the curve for θ = 0 to θ = 7 6 π? 2.( pt) Let a = (-2 4 2) and
More informationMATH 261 FINAL EXAM PRACTICE PROBLEMS
MATH 261 FINAL EXAM PRACTICE PROBLEMS These practice problems are pulled from the final exams in previous semesters. The 2-hour final exam typically has 8-9 problems on it, with 4-5 coming from the post-exam
More informationMath 31CH - Spring Final Exam
Math 3H - Spring 24 - Final Exam Problem. The parabolic cylinder y = x 2 (aligned along the z-axis) is cut by the planes y =, z = and z = y. Find the volume of the solid thus obtained. Solution:We calculate
More informationMath 11 Fall 2016 Final Practice Problem Solutions
Math 11 Fall 216 Final Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,
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 informationMATH H53 : Final exam
MATH H53 : Final exam 11 May, 18 Name: You have 18 minutes to answer the questions. Use of calculators or any electronic items is not permitted. Answer the questions in the space provided. If you run out
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 informationCoordinates 2D and 3D Gauss & Stokes Theorems
Coordinates 2 and 3 Gauss & Stokes Theorems Yi-Zen Chu 1 2 imensions In 2 dimensions, we may use Cartesian coordinates r = (x, y) and the associated infinitesimal area We may also employ polar coordinates
More informationMTH101A (2016), Tentative Marking Scheme - End sem. exam
MTH11A (16), Tentative Marking Scheme - End sem. eam 1. (a) Let f(, y, z) = yz and S be + y + z = 6. Using Lagrange multipliers method, find the maimum and minimum values of f on S. [7] Lag. Eqns.: yz
More informationln e 2s+2t σ(m) = 1 + h 2 x + h 2 yda = dA = 90 da R
olution to et 5, Friday ay 7th ection 5.6: 15, 17. ection 5.7:, 5, 7, 16. (1) (ection 5.5, Problem ) Find a parametrization of the suface + y 9 between z and z. olution: cost, y sint and z s with t π and
More informationMath Review for Exam Compute the second degree Taylor polynomials about (0, 0) of the following functions: (a) f(x, y) = e 2x 3y.
Math 35 - Review for Exam 1. Compute the second degree Taylor polynomial of f e x+3y about (, ). Solution. A computation shows that f x(, ), f y(, ) 3, f xx(, ) 4, f yy(, ) 9, f xy(, ) 6. The second degree
More information1. Find and classify the extrema of h(x, y) = sin(x) sin(y) sin(x + y) on the square[0, π] [0, π]. (Keep in mind there is a boundary to check out).
. Find and classify the extrema of hx, y sinx siny sinx + y on the square[, π] [, π]. Keep in mind there is a boundary to check out. Solution: h x cos x sin y sinx + y + sin x sin y cosx + y h y sin x
More informationMath 234 Review Problems for the Final Exam
Math 234 eview Problems for the Final Eam Marc Conrad ecember 13, 2007 irections: Answer each of the following questions. Pages 1 and 2 contain the problems. The solutions are on pages 3 through 7. Problem
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 informationThe branch of mechanics which deals with the motion of object is called dynamics. It is divided into two branches:
M th KINEMATICS 4 CHAPTER INTRODUCTION The branch of mechanics which deals with the motion of object is called dynamics. It is divided into two branches: Kinematics: (i) Kinematics (ii) Kinetics The branch
More informationMath 10C - Fall Final Exam
Math 1C - Fall 217 - Final Exam Problem 1. Consider the function f(x, y) = 1 x 2 (y 1) 2. (i) Draw the level curve through the point P (1, 2). Find the gradient of f at the point P and draw the gradient
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 informationMathematical Analysis II, 2018/19 First semester
Mathematical Analysis II, 208/9 First semester Yoh Tanimoto Dipartimento di Matematica, Università di Roma Tor Vergata Via della Ricerca Scientifica, I-0033 Roma, Italy email: hoyt@mat.uniroma2.it We basically
More informationTopic 7. Electric flux Gauss s Law Divergence of E Application of Gauss Law Curl of E
Topic 7 Electric flux Gauss s Law Divergence of E Application of Gauss Law Curl of E urface enclosing an electric dipole. urface enclosing charges 2q and q. Electric flux Flux density : The number of field
More informationDO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START
Math 265 Student name: KEY Final Exam Fall 23 Instructor & Section: This test is closed book and closed notes. A (graphing) calculator is allowed for this test but cannot also be a communication device
More informationThe Divergence Theorem
Math 1a The Divergence Theorem 1. Parameterize the boundary of each of the following with positive orientation. (a) The solid x + 4y + 9z 36. (b) The solid x + y z 9. (c) The solid consisting of all points
More informationAssignment 11 Solutions
. Evaluate Math 9 Assignment olutions F n d, where F bxy,bx y,(x + y z and is the closed surface bounding the region consisting of the solid cylinder x + y a and z b. olution This is a problem for which
More information