MATH 52 FINAL EXAM DECEMBER 7, 2009

Size: px
Start display at page:

Download "MATH 52 FINAL EXAM DECEMBER 7, 2009"

Transcription

1 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- LEM PAGE. SO EXTRA WORK FOR PROBLEM 1 WOULD GO ON THE BACK OF THIS COVER SHEET, ETC. MOST PROBLEMS ARE WORTH 10 POINTS. PROBLEM TEN IS 12 POINTS AND PROBLEM ELEVEN IS 8 POINTS. THE TOTAL IS 120 POINTS. THE TERMS C 1 OR C 2 FOR FUNCTIONS AND VECTOR FIELDS MEANS THAT THEY HAVE CONTINUOUS FIRST PARTIAL DERIVATIVES OR CONTINUOUS FIRST AND SECOND PARTIAL DERIVATIVES, RESPECTIVELY, ON THEIR DO- MAINS. Please sign the following, and make sure your name is legible: On my honor, I have neither given nor received any aid on this examination. I have furthermore abided by all other aspects of the honor code with respect to this examination. SIGNATURE: NAME: Prob/[pts] Score Prob/[pts] Score Prob/[pts] Score 1 [10] 5 [10] 9 [10] 2 [10] 6 [10] 10 [12] 3 [10] 7 [10] 11 [8] 4 [10] 8 [10] 12 [10] 1

2 1(a)[4]. Sketch the region of integration in the plane for 1 1+ x 0 1+x 3 f(x, y) dydx. (b)[6]. Reverse the order of integration for the integral in part (a). 2

3 2(a)[6]. Let W be the solid object in space bounded by the xy-plane, the cylinder x 2 +y 2 = 4, and the paraboloid z = x 2 + y 2. Set up AND EVALUATE a triple integral in cylindrical coordinates in the order dz dr dθ that gives the volume of the object. (b)[4]. Set up but DO NOT EVALUATE an integral in cylindrical coordinates in the order dr dz dθ that gives the volume of the same object from part (a). 3

4 3(a)[4]. A wire is in the shape of a coil described by (cos(2πt), sin(2πt), t), 0 t 4. Suppose the electric charge density along the wire is given by e(x, y, z) = z. Set up an explicit integral that gives the total electric charge on the wire. DO NOT EVALUATE. (b)[6]. Suppose surface S is the portion of the paraboloid x 2 + y 2 = 4z with 1 z 4. Parametrize S by T (s, t) = (2t cos(s), 2t sin(s), t 2 ) with 0 s 2π, 1 t 2. If the mass density at points of S is given by δ = z, Use the parametrization to express the moment of S with respect to the plane z = 0 as an explicit integral over a domain in the st-plane. DO NOT EVALUATE. 4

5 4. Let D be the triangle in the first quadrant of the xy-plane bounded by the x-axis, the y-axis, and the line x + y = 1. The transformation T (u, v) = (x, y) = (1 u, uv) maps the square 0 u 1, 0 v 1 in the uv-plane onto the triangle D in the xy-plane. (a)[3]. Find the Jacobian determinant (x, y) (u, v). (b)[3]. Fill in the limits of integration asterisks for (sin((1 x) 2 ) + y 1 x ) da = D (sin((1 x) 2 ) + y ) dx dy 1 x (c)[4]. Use the change of variables theorem to express D (sin((1 x)2 ) + y 1 x ) dxdy as an explicit integral D g(u, v) dudv over a region D in the uv-plane. DO NOT EVALUATE. Which looks easier to evaluate, the integral in (b) or (c)? 5

6 5(a)[3]. Find a potential function for the vector field F (x, y, z) = 1 + x 2 + y 2 + z 2 (x, y, z) on R 3 or give a reason no such potential function exists. (b)[3]. Is G = 2xy i + (x 2 y) j a conservative vector field on R 2? Why or why not? (c)[4]. Find the work done by the vector field G of part (b) on a particle that moves from ( 1, 1) to (1, 1) clockwise around the ellipse 2x 2 + 3y 2 = 5. 6

7 6. Let S be the portion of the sphere x 2 +y 2 +z 2 = 25 that lies on or below the plane z = 4. (So, S is most of the sphere.) Let F = (x + 3y 2 + z 2 ) i + (x 2 + y z 2 ) j + (xy 2z) k. (a)[2]. Calculate div( F ). (b)[8]. Compute the flux S F n ds, where n is the unit normal given by n = 1 5 (x, y, z) at points of S. Explain your work clearly. For your own protection, NO CREDIT will be given for attempts that try to parametrize S. [Hint: S is not a closed surface. But S together with some other surface does form a closed surface. Use part (a). You can also use symmetry and knowledge of certain areas to evaluate certain integrals without actually integrating.] 7

8 7. In this problem you will use the graph parametrization of a hemisphere given by T (x, y) = (x, y, a 2 x 2 y 2 ), for (x, y) in the disk D described by x 2 + y 2 a 2 in the xy-plane. (a)[3]. Give the standard normal vector N = T x T y for this parametrization and find the length N. [You can just give the answers if you remember, otherwise work it out.] (b)[3]. The surface area of S is given by the integral D N da. Is this integral an improper integral or a proper integral? Give a reason for your answer. (c)[4]. Showing your work, EVALUATE the integral D [You know the answer, but you must show work here.] N da. 8

9 8. In this problem, we consider the plane vector field y F = (P (x, y), Q(x, y)) = ( x 2 + y 2, x x 2 + y 2 + 2x) which satisfies Q/ x P/ y = 2 at all points where F is defined. (a)[5]. If C 1 is the circle x 2 + y 2 = 1 oriented counterclockwise, calculate C 1 F ds. (b)[5]. If C 2 is the ellipse x y2 9 = 1 oriented counterclockwise, calculate C 2 F ds. [You may use the formula for the area inside an ellipse, but NO CREDIT for attempts that parametrize the ellipse. Remember Q/ x P/ y = 2 at all points where F is defined.] 9

10 9[10]. Write T (True) or F(False) in the margin to the left of each statement. There are ten statements, continuing on the next page. One point for each correct answer. (i). If f(x) and g(y) are two continuous functions of one variable then the double integral of the product f(x)g(y) over an x-simple region a x b, p(x) y q(x) can always be computed as the product ( b a f(x)dx)( q(x) p(x) g(y)dy). (ii). The region in the plane bounded by the two circles x 2 + y 2 = 4 and x 2 + (y 1) 2 = 1 is neither an x-simple nor a y-simple region. (iii). If S is a surface with boundary in space and if S has constant density, then the y coordinate of the center of mass of S is the average value of the function y on S. (iv). If T (u, v) = (x(u, v), y(u, v), z(u, v)) is a linear transformation mapping R 2 onto a plane in R 3 and if N = T u T v is the standard normal vector to T (R 2 ), then for any region D in the uv-plane that has an area, the area of T (D) is N area(d). (v). ( f) = 0 for all C 2 functions f(x, y, z) on R 3. 10

11 (vi). If f(x, y, z) is a C 2 function on R 3 then Div Grad(f) = 2 f x f y f z 2. (vii). If F = (P, Q) is a C 1 vector field on the domain W = R 2 (0, 0), [that is, W is R 2 with the origin removed], and if Q x = P y on domain W, then C P dx + Qdy = 0 for all simple closed curves C in W. (viii). If S is the hemisphere x 2 + y 2 + z 2 = 4, z 0 in R 3 with upward pointing unit normal n and if F = ( yz, y 2, z) then S F n ds < 0. (ix). On every smooth surface with or without boundary in R 3 there are two continuous unit normal vector fields. (x). If f(u) is any continuous function for u > 0, and if r = x 2 + y 2 + z 2, then the vector field f(r)(x, y, z) is conservative in the region R 3 (0, 0, 0). 11

12 10(a)[6]. For each of the regions W in R 3 described below, suppose F is a C 1 vector field with Curl F = 0 throughout W. In each case, decide if you can you be certain that F is a conservative vector field throughout W? Answer YES or NO. There are six regions. One point for each correct answer. (i). R 3 with the x axis removed. (ii). R 3 with the circle x 2 + y 2 = 1 in the xy-plane removed. (iii). R 3 with the origin removed. (iv). R 3 with the plane z = 0 removed. (v). The octant x > 0, y > 0, z > 0 in R 3. (vi). The region in R 3 described as all (x, y, z) with 4 < x 2 + y 2 + z 2 < 25. (b)[6]. Write T(true) or F(false) in the margin to the left of each of the six statement below. One point for each correct answer. (i). If F is a C 1 vector field on all of R 2 with Div F = 0 then the flux of F across any simple closed curve C in R 2 is always zero. (ii). There exists a vector field G on R 3 with Div( G) = z 2 y 2 x 2. (iii). If F is a C 1 vector field on all of R 3 and if F = 1 at all points of a solid region W in R 3 and if S is the boundary of W then the outward flux S F n ds always equals the volume of W. (iv). If F is a C 1 vector field on all of R 3 and if C C in R 3, then F = 0. F ds = 0 for all simple closed curves (v). If F is a C 2 vector field on R 3 (0, 0, 0), that is, everywhere except the origin, and if S is the sphere of radius 1 and center (0, 0, 0), then S Curl( F ) ds = 0. (vi). If F is a conservative vector field, C 1 on all of R 3, and if S is a closed oriented surface, that is, with no boundary, then S F ds = 0. 12

13 11. Consider the vector field on R 3 (0, 0, 0) given by F = 1 (x, y, z) where r = r3 x 2 + y 2 + z 2. Let S a be the sphere of radius a given by x 2 + y 2 + z 2 = a 2 with outward unit normal n. (a)[4]. Calculate the flux integral S a F n ds, WITHOUT using any parametrization. SHOW WORK. [You may assume the surface area of S a is 4πa 2. ] (b)[2]. Routine derivative calculations yield Div( F ) = 0 on the domain of F. Why doesn t Gauss s theorem give the result S a F n ds = 0? (c)[2]. Is it possible that F = Curl( G) for some vector field G on R 3 (0, 0, 0)? Give a reason why or why not. 13

14 12[10]. The graph on the next two pages plots a vector field F (x, y) = (P (x, y), Q(x, y)), oriented curves A, B, C, and points M, N, R, T. One assumes F is a C 1 vector field. Answer the following ten questions. There are two copies of the graph so you can tear off the last page while contemplating the questions. One point for each correct answer. (i). Is the circulation of F around curve A positive, negative, or 0? (ii). If n is the continuous unit normal to curve C pointing downward (except near N), is the flux across C in the n direction positive, negative, or 0? (iii). Is the quantity C F ds positive, negative, or 0? (iv). Is the value of p x + Q y at T most likely positive, negative, or 0? (v). Is the quantity Q x P y at M most likely positive, negative, or 0? (vi). Is the divergence of F at N most likely positive, negative, or 0? (vii). If D is the plane region bounded by curve A, is the quantity D positive, negative, or 0? Div( F ) dxdy (viii). If F is a fluid flow velocity field, would a small paddle at N most likely spin clockwise, counterclockwise, or not at all? (ix). If F is a force field, is the work done by F on a particle moving around directed curve B positive, negative, or 0? (x). Is it possible that F = f for some function f(x, y)? 14

15 y R T A B M N C x 15

16 y R T A B M N C x 16

One side of each sheet is blank and may be used as scratch paper.

One 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

MATH 52 FINAL EXAM SOLUTIONS

MATH 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 information

1. 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.

1. 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 information

PRACTICE 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. 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 information

Calculus III. Math 233 Spring Final exam May 3rd. Suggested solutions

Calculus 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 information

Math Review for Exam 3

Math 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 information

McGill University April 16, Advanced Calculus for Engineers

McGill 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 information

SOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)

SOLUTIONS 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 information

Practice problems **********************************************************

Practice 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 information

Page Problem Score Max Score a 8 12b a b 10 14c 6 6

Page 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 information

Math 234 Exam 3 Review Sheet

Math 234 Exam 3 Review Sheet Math 234 Exam 3 Review Sheet Jim Brunner LIST OF TOPIS TO KNOW Vector Fields lairaut s Theorem & onservative Vector Fields url Divergence Area & Volume Integrals Using oordinate Transforms hanging the

More information

1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is

1 + 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 information

In 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

In 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 information

Name: Date: 12/06/2018. M20550 Calculus III Tutorial Worksheet 11

Name: 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 information

Page Problem Score Max Score a 8 12b a b 10 14c 6 6

Page 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 information

MATH 332: Vector Analysis Summer 2005 Homework

MATH 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 information

7a3 2. (c) πa 3 (d) πa 3 (e) πa3

7a3 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 information

MATH 2400: Calculus III, Fall 2013 FINAL EXAM

MATH 2400: Calculus III, Fall 2013 FINAL EXAM MATH 2400: Calculus III, Fall 2013 FINAL EXAM December 16, 2013 YOUR NAME: Circle Your Section 001 E. Angel...................... (9am) 002 E. Angel..................... (10am) 003 A. Nita.......................

More information

( ) ( ) ( ) ( ) Calculus III - Problem Drill 24: Stokes and Divergence Theorem

( ) ( ) ( ) ( ) 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 information

MATHS 267 Answers to Stokes Practice Dr. Jones

MATHS 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 information

Math 233. Practice Problems Chapter 15. i j k

Math 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 information

Vector Calculus handout

Vector Calculus handout Vector Calculus handout The Fundamental Theorem of Line Integrals Theorem 1 (The Fundamental Theorem of Line Integrals). Let C be a smooth curve given by a vector function r(t), where a t b, and let f

More information

Jim 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 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 information

DO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START

DO 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 information

Math 265H: Calculus III Practice Midterm II: Fall 2014

Math 265H: Calculus III Practice Midterm II: Fall 2014 Name: Section #: Math 65H: alculus III Practice Midterm II: Fall 14 Instructions: This exam has 7 problems. The number of points awarded for each question is indicated in the problem. Answer each question

More information

Practice problems. m zδdv. In our case, we can cancel δ and have z =

Practice 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 information

MAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative.

MAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. MAC2313 Final A (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. ii. The vector field F = 5(x 2 + y 2 ) 3/2 x, y is radial. iii. All constant

More information

Name: Instructor: Lecture time: TA: Section time:

Name: 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 information

Ma 1c Practical - Solutions to Homework Set 7

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

Print Your Name: Your Section:

Print 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 information

Practice 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.

Practice 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 information

Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.

Instructions: 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 information

2. Below are four algebraic vector fields and four sketches of vector fields. Match them.

2. Below are four algebraic vector fields and four sketches of vector fields. Match them. Math 511: alc III - Practice Eam 3 1. State the meaning or definitions of the following terms: a) vector field, conservative vector field, potential function of a vector field, volume, length of a curve,

More information

Note: Each problem is worth 14 points except numbers 5 and 6 which are 15 points. = 3 2

Note: 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 information

(a) 0 (b) 1/4 (c) 1/3 (d) 1/2 (e) 2/3 (f) 3/4 (g) 1 (h) 4/3

(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 information

Math 41 Final Exam December 9, 2013

Math 41 Final Exam December 9, 2013 Math 41 Final Exam December 9, 2013 Name: SUID#: Circle your section: Valentin Buciumas Jafar Jafarov Jesse Madnick Alexandra Musat Amy Pang 02 (1:15-2:05pm) 08 (10-10:50am) 03 (11-11:50am) 06 (9-9:50am)

More information

Sections minutes. 5 to 10 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed.

Sections 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 information

Math 11 Fall 2007 Practice Problem Solutions

Math 11 Fall 2007 Practice Problem Solutions Math 11 Fall 27 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 information

MATH 0350 PRACTICE FINAL FALL 2017 SAMUEL S. WATSON. a c. b c.

MATH 0350 PRACTICE FINAL FALL 2017 SAMUEL S. WATSON. a c. b c. MATH 35 PRACTICE FINAL FALL 17 SAMUEL S. WATSON Problem 1 Verify that if a and b are nonzero vectors, the vector c = a b + b a bisects the angle between a and b. The cosine of the angle between a and c

More information

MATH 2203 Final Exam Solutions December 14, 2005 S. F. Ellermeyer Name

MATH 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 information

The Divergence Theorem Stokes Theorem Applications of Vector Calculus. Calculus. Vector Calculus (III)

The Divergence Theorem Stokes Theorem Applications of Vector Calculus. Calculus. Vector Calculus (III) Calculus Vector Calculus (III) Outline 1 The Divergence Theorem 2 Stokes Theorem 3 Applications of Vector Calculus The Divergence Theorem (I) Recall that at the end of section 12.5, we had rewritten Green

More information

Green s, Divergence, Stokes: Statements and First Applications

Green 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 information

Page Points Score Total: 210. No more than 200 points may be earned on the exam.

Page Points Score Total: 210. No more than 200 points may be earned on the exam. Name: PID: Section: Recitation Instructor: DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE. Page Points Score 3 18 4 18 5 18 6 18 7 18 8 18 9 18 10 21 11 21 12 21 13 21 Total: 210 No more than 200

More information

Final exam (practice 1) UCLA: Math 32B, Spring 2018

Final 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 information

Mathematics (Course B) Lent Term 2005 Examples Sheet 2

Mathematics (Course B) Lent Term 2005 Examples Sheet 2 N12d Natural Sciences, Part IA Dr M. G. Worster Mathematics (Course B) Lent Term 2005 Examples Sheet 2 Please communicate any errors in this sheet to Dr Worster at M.G.Worster@damtp.cam.ac.uk. Note that

More information

Practice problems ********************************************************** 1. Divergence, curl

Practice 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 information

Math 11 Fall 2016 Final Practice Problem Solutions

Math 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 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)

(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 information

MATH H53 : Final exam

MATH 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 information

Solutions for the Practice Final - Math 23B, 2016

Solutions for the Practice Final - Math 23B, 2016 olutions for the Practice Final - Math B, 6 a. True. The area of a surface is given by the expression d, and since we have a parametrization φ x, y x, y, f x, y with φ, this expands as d T x T y da xy

More information

14.7 The Divergence Theorem

14.7 The Divergence Theorem 14.7 The Divergence Theorem The divergence of a vector field is a derivative of a sort that measures the rate of flow per unit of volume at a point. A field where such flow doesn't occur is called 'divergence

More information

Math Exam IV - Fall 2011

Math 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 information

HOMEWORK 8 SOLUTIONS

HOMEWORK 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 information

The Divergence Theorem

The 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 information

MA FINAL EXAM Form 01 May 1, 2017

MA FINAL EXAM Form 01 May 1, 2017 MA 26100 FINAL EXAM Form 01 May 1, 2017 NAME STUDENT ID # YOUR TA S NAME RECITATION TIME 1. You must use a #2 pencil on the scantron 2. a. Write 01 in the TEST/QUIZ NUMBER boxes and darken the appropriate

More information

MA FINAL EXAM Form B December 13, 2016

MA FINAL EXAM Form B December 13, 2016 MA 6100 FINAL EXAM Form B December 1, 016 NAME STUDENT ID # YOUR TA S NAME RECITATION TIME 1. You must use a # pencil on the scantron. a. If the cover of your exam is GREEN, write 01 in the TEST/QUIZ NUMBER

More information

Major Ideas in Calc 3 / Exam Review Topics

Major Ideas in Calc 3 / Exam Review Topics Major Ideas in Calc 3 / Exam Review Topics Here are some highlights of the things you should know to succeed in this class. I can not guarantee that this list is exhaustive!!!! Please be sure you are able

More information

Math 23b Practice Final Summer 2011

Math 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 information

Final Exam Review Sheet : Comments and Selected Solutions

Final 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 information

MAT 211 Final Exam. Spring Jennings. Show your work!

MAT 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 information

Math 212-Lecture 20. P dx + Qdy = (Q x P y )da. C

Math 212-Lecture 20. P dx + Qdy = (Q x P y )da. C 15. Green s theorem Math 212-Lecture 2 A simple closed curve in plane is one curve, r(t) : t [a, b] such that r(a) = r(b), and there are no other intersections. The positive orientation is counterclockwise.

More information

Summary of various integrals

Summary of various integrals ummary of various integrals Here s an arbitrary compilation of information about integrals Moisés made on a cold ecember night. 1 General things o not mix scalars and vectors! In particular ome integrals

More information

F ds, where F and S are as given.

F ds, where F and S are as given. Math 21a Integral Theorems Review pring, 29 1 For these problems, find F dr, where F and are as given. a) F x, y, z and is parameterized by rt) t, t, t t 1) b) F x, y, z and is parameterized by rt) t,

More information

16.5 Surface Integrals of Vector Fields

16.5 Surface Integrals of Vector Fields 16.5 Surface Integrals of Vector Fields Lukas Geyer Montana State University M73, Fall 011 Lukas Geyer (MSU) 16.5 Surface Integrals of Vector Fields M73, Fall 011 1 / 19 Parametrized Surfaces Definition

More information

51. General Surface Integrals

51. General Surface Integrals 51. General urface Integrals The area of a surface in defined parametrically by r(u, v) = x(u, v), y(u, v), z(u, v) over a region of integration in the input-variable plane is given by d = r u r v da.

More information

Past Exam Problems in Integrals, Solutions

Past Exam Problems in Integrals, Solutions Past Exam Problems in Integrals, olutions Prof. Qiao Zhang ourse 11.22 December 7, 24 Note: These problems do not imply, in any sense, my taste or preference for our own exam. ome of the problems here

More information

Sept , 17, 23, 29, 37, 41, 45, 47, , 5, 13, 17, 19, 29, 33. Exam Sept 26. Covers Sept 30-Oct 4.

Sept , 17, 23, 29, 37, 41, 45, 47, , 5, 13, 17, 19, 29, 33. Exam Sept 26. Covers Sept 30-Oct 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 information

Solutions to old Exam 3 problems

Solutions 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 information

G 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

G 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 information

Math 234 Final Exam (with answers) Spring 2017

Math 234 Final Exam (with answers) Spring 2017 Math 234 Final Exam (with answers) pring 217 1. onsider the points A = (1, 2, 3), B = (1, 2, 2), and = (2, 1, 4). (a) [6 points] Find the area of the triangle formed by A, B, and. olution: One way to solve

More information

The University of British Columbia Final Examination - December 16, Mathematics 317 Instructor: Katherine Stange

The University of British Columbia Final Examination - December 16, Mathematics 317 Instructor: Katherine Stange The University of British Columbia Final Examination - December 16, 2010 Mathematics 317 Instructor: Katherine Stange Closed book examination Time: 3 hours Name Signature Student Number Special Instructions:

More information

MLC Practice Final Exam

MLC 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 information

Review Questions for Test 3 Hints and Answers

Review 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 information

Math 223 Final. July 24, 2014

Math 223 Final. July 24, 2014 Math 223 Final July 24, 2014 Name Directions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Total 1. No books, notes, or evil looks. You may use a calculator to do routine arithmetic computations. You may not use your

More information

McGill University April 20, Advanced Calculus for Engineers

McGill University April 20, Advanced Calculus for Engineers McGill University April 0, 016 Faculty of Science Final examination Advanced Calculus for Engineers Math 64 April 0, 016 Time: PM-5PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer Student

More information

Solutions to Sample Questions for Final Exam

Solutions 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 information

Final exam (practice 1) UCLA: Math 32B, Spring 2018

Final exam (practice 1) UCLA: Math 32B, Spring 2018 Instructor: Noah White Date: Final exam (practice 1) UCLA: Math 32B, Spring 2018 This exam has 7 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions

More information

(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere.

(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 information

ARNOLD PIZER rochester problib from CVS Summer 2003

ARNOLD PIZER rochester problib from CVS Summer 2003 ARNOLD PIZER rochester problib from VS Summer 003 WeBWorK assignment Vectoralculus due 5/3/08 at :00 AM.( pt) setvectoralculus/ur V.pg onsider the transformation T : x 8 53 u 45 45 53v y 53 u 8 53 v A.

More information

53. Flux Integrals. Here, R is the region over which the double integral is evaluated.

53. Flux Integrals. Here, R is the region over which the double integral is evaluated. 53. Flux Integrals Let be an orientable surface within 3. An orientable surface, roughly speaking, is one with two distinct sides. At any point on an orientable surface, there exists two normal vectors,

More information

MULTIVARIABLE INTEGRATION

MULTIVARIABLE 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 information

Math 32B Discussion Session Week 10 Notes March 14 and March 16, 2017

Math 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 information

Problem Points S C O R E

Problem Points S C O R E MATH 34F Final Exam March 19, 13 Name Student I # Your exam should consist of this cover sheet, followed by 7 problems. Check that you have a complete exam. Unless otherwise indicated, show all your work

More information

SOME PROBLEMS YOU SHOULD BE ABLE TO DO

SOME 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 information

Math 31CH - Spring Final Exam

Math 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 information

12/19/2009, FINAL PRACTICE VII Math 21a, Fall Name:

12/19/2009, FINAL PRACTICE VII Math 21a, Fall Name: 12/19/29, FINAL PRATIE VII Math 21a, Fall 29 Name: MWF 9 Jameel Al-Aidroos MWF 1 Andrew otton-lay MWF 1 Oliver Knill MWF 1 HT Yau MWF 11 Ana araiani MWF 11 hris Phillips MWF 11 Ethan Street MWF 12 Toby

More information

Name (please print) π cos(θ) + sin(θ)dθ

Name (please print) π cos(θ) + sin(θ)dθ Mathematics 2443-3 Final Eamination Form B December 2, 27 Instructions: Give brief, clear answers. I. Evaluate by changing to polar coordinates: 2 + y 2 3 and above the -ais. + y d 23 3 )/3. π 3 Name please

More information

Final Exam. Monday March 19, 3:30-5:30pm MAT 21D, Temple, Winter 2018

Final Exam. Monday March 19, 3:30-5:30pm MAT 21D, Temple, Winter 2018 Name: Student ID#: Section: Final Exam Monday March 19, 3:30-5:30pm MAT 21D, Temple, Winter 2018 Show your work on every problem. orrect answers with no supporting work will not receive full credit. Be

More information

e x2 dxdy, e x2 da, e x2 x 3 dx = e

e 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 information

1. (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. 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 information

Review Sheet for the Final

Review 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 information

Assignment 11 Solutions

Assignment 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

Look out for typos! Homework 1: Review of Calc 1 and 2. Problem 1. Sketch the graphs of the following functions:

Look out for typos! Homework 1: Review of Calc 1 and 2. Problem 1. Sketch the graphs of the following functions: Math 226 homeworks, Fall 2016 General Info All homeworks are due mostly on Tuesdays, occasionally on Thursdays, at the discussion section. No late submissions will be accepted. If you need to miss the

More information

MATH 263 ASSIGNMENT 9 SOLUTIONS. F dv =

MATH 263 ASSIGNMENT 9 SOLUTIONS. F dv = MAH AIGNMEN 9 OLUION ) Let F = (x yz)î + (y + xz)ĵ + (z + xy)ˆk and let be the portion of the cylinder x + y = that lies inside the sphere x + y + z = 4 be the portion of the sphere x + y + z = 4 that

More information

Double Integrals. Advanced Calculus. Lecture 2 Dr. Lahcen Laayouni. Department of Mathematics and Statistics McGill University.

Double Integrals. Advanced Calculus. Lecture 2 Dr. Lahcen Laayouni. Department of Mathematics and Statistics McGill University. Lecture Department of Mathematics and Statistics McGill University January 9, 7 Polar coordinates Change of variables formula Polar coordinates In polar coordinates, we have x = r cosθ, r = x + y y = r

More information

Midterm 1 practice UCLA: Math 32B, Winter 2017

Midterm 1 practice UCLA: Math 32B, Winter 2017 Midterm 1 practice UCLA: Math 32B, Winter 2017 Instructor: Noah White Date: Version: practice This exam has 4 questions, for a total of 40 points. Please print your working and answers neatly. Write your

More information

Stokes s Theorem 17.2

Stokes s Theorem 17.2 Stokes s Theorem 17.2 6 December 213 Stokes s Theorem is the generalization of Green s Theorem to surfaces not just flat surfaces (regions in R 2 ). Relate a double integral over a surface with a line

More information

MTH 234 Exam 2 November 21st, Without fully opening the exam, check that you have pages 1 through 12.

MTH 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

Multiple Integrals and Vector Calculus: Synopsis

Multiple Integrals and Vector Calculus: Synopsis Multiple Integrals and Vector Calculus: Synopsis Hilary Term 28: 14 lectures. Steve Rawlings. 1. Vectors - recap of basic principles. Things which are (and are not) vectors. Differentiation and integration

More information

Final Review Worksheet

Final 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 information