No calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk of everything excepts pens, pencils and erasers.


 Brett Pope
 3 years ago
 Views:
Transcription
1 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 your desk of everything excepts pens, pencils and erasers. If you need scratch paper, use the back of the previous page. Without fully opening the exam, check that you have pages 1 through 12. Fill in your name, etc. on this first page. Show all your work. Write your answers clearly! Include enough steps for the grader to be able to follow your work. Don t skip limits or equal signs, etc. Include words to clarify your reasoning. Do first all of the problems you know how to do immediately. Do not spend too much time on any particular problem. Return to difficult problems later. If you have any questions please raise your hand and a proctor will come to you. There is no talking allowed during the exam. You will be given exactly 90 minutes for this exam. Remove and utilize the formula sheet provided to you at the end of this exam. This is a practice exam. The actual exam may differ significantly from this practice exam because there are many varieties of problems that can test each concept. I have read and understand the above instructions:. SIGNATURE Page 1 of 12
2 Standard Response Questions. Show all work to receive credit. Please BOX your final answer. 1. (14 points) Find the volume of the sphere x 2 + y 2 + z 2 64 that lies between the cones z = x 2 + y 2 x 2 + y 2 and z =. 3 Page 2 of 12
3 2. Evaluate the following integrals (a) (7 points) x e y4 dy dx. (b) (7 points) 2 8 y 2 (x 2 + y 2 ) dx dy 0 y Page 3 of 12
4 3. (14 points) Find the absolute maximum and minimum values of f(x, y) = 8 3 x3 + 2y 26 on the set D where D is the closed region bounded by y = 0 and y = 1 x 2 Page 4 of 12
5 4. Let F(x, y) = 2xy + 5 3y, x 2 10x be a vector field. y 2 y 3 (a) (4 points) Show that F is conservative. (b) (5 points) Find a potential function for F, that is, a function f such that f = F. (c) (5 points) Evaluate the line integral F dr, where C is the curve parametrized by r(t) = t sin(t), πe t cos(t) for 0 t π. C Page 5 of 12
6 Multiple Choice. Circle the best answer. No work needed. No partial credit available. No credit will be given for choices not clearly marked. 5. (4 points) Find the curl of the vector field F(x, y, z) = x + yz, y + xz, z + xy. A. 0, 1, 0 B. 0, 0, 0 C. 1, 1, 1 D. 1, 1, 1 E. 0, 0, 1 6. (4 points) Find the divergence of F(x, y, z) = (3x + yz) i + ( y xz) j + (4z + 6xy) k A. 3 B. 6 C. 12 D. 0 E. None of the above 7. (4 points) In cylindrical coordinates, what does z = r represent? A. Cone B. Elliptical Paraboloid C. Sphere D. Hyperbolic Paraboloid E. Ellipsoid Page 6 of 12
7 8. (4 points) In spherical coordinates, what does r = 3 represent? A. Sphere B. Cone C. Line D. Plane E. Hyperbolic paraboloid 9. (4 points) Convert A. B. C. D. 4 4 x x 2 2π x 2 dy dx x 2 2 x 2 dy dx x 4 x 2 dy dx 2 4 x dy dx E. None of the above r dr dθ from polar to Cartesian coordinates: 10. (4 points) Find the gradient of f(x, y, z) = 6xy 2 z 3 3xyz A. 6y 2 z 3 6yz, 12xyz 3 6xz, 18xy 2 z 2 6xy B. 3y 2 z 3 3yz, 6xyz 3 3xz, 9xy 2 z 2 3xy C. 6y 2 z 3 3yz, 12xyz 3 3xz, 18xy 2 z 2 3xy D. 12xyz 3 3yz, 6y 2 z 3 3xz, 18xy 2 z 2 3xy E. None of the above Page 7 of 12
8 11. (4 points) Which of the following functions has a constant gradient vector field? A. f(x, y) = 2x 2 + y B. f(x, y) = y 2 C. f(x, y) = x y D. f(x, y) = sin(xy) E. None of the above 12. (4 points) Evaluate dy dz dx. A. π(1) 2 B. 1 C. xyz D. 0 E. None of the above 13. (4 points) Evaluate the line integral ( 1, 1), ( 1, 1), (1, 1) and (1, 1). A. B. C. D. 0 cos(2)+1 2 (3) cos(2)+1 2 (6) cos(2)+1 2 (12) E. None of the above C 3 cos(2y) dx+3x 2 sin(2y) dy where C is the square with vertices Page 8 of 12
9 Challenge Question(s). Show all work to receive credit. 14. (14 points) Find the area of the part of the paraboloid x = y 2 + z 2 that lies inside the cylinder y 2 + z 2 = 49. Page 9 of 12
10 Congratulations you are now done with the exam! Go back and check your solutions for accuracy and clarity. Make sure your final answers are BOXED. When you are completely happy with your work please bring your exam to the front to be handed in. Please have your MSU student ID ready so that is can be checked. DO NOT WRITE BELOW THIS LINE. Page Points Score Total: 106 No more than 100 points may be earned on the exam. Page 10 of 12
11 FORMULA SHEET PAGE 1 Vectors in Space Curves and Planes in Space Suppose u = u 1, u 2, u 3 and v = v 1, v 2, v 3 : Line parallel to v: r(t) = r 0 + tv Unit Vectors: Length of vector u Dot Product: Cross Product: i = 1, 0, 0 j = 0, 1, 0 k = 0, 0, 1 u = u 12 + u 22 + u 3 2 u v = u 1 v 1 + u 2 v 2 + u 3 v 3 = u v cos θ u v = Vector Projection: i j k u 1 u 2 u 3 v 1 v 2 v 3 Partial Derivatives proj u v = u v u 2 Chain Rule: Suppose z = f(x, y) and x = g(t) and y = h(t) are all differentiable then dz dt = f dx x dt + f dy y dt u Plane normal to n = a, b, c : a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0 Arc Length of curve r(t) for t [a, b]. L = b a r (t) dt Unit Tangent Vector of curve r(t) T(t) = r (t) r (t) More on Surfaces Directional Derivative: D u f(x, y) = f u Second Derivative Test Suppose f x (a, b) = 0 and f y (a, b) = 0. Let D = f xx (a, b)f yy (a, b) [f xy (a, b)] 2 (a) If D > 0 and f xx (a, b) > 0, then f(a, b) is a local minimum. (b) If D > 0 and f xx (a, b) < 0, then f(a, b) is a local maximum. (c) If D < 0 then f(a, b) is a saddle point. Geometry / Trigonometry Area of an ellipse x2 a + y2 = 1 is A = πab 2 b2 sin 2 x = 1 (1 cos 2x) 2 cos 2 x = 1 (1 + cos 2x) 2 sin(2x) = 2 sin x cos x Page 11 of 12
12 FORMULA SHEET PAGE 2 Multiple Integrals Area: A(D) = 1 da D Volume: V (E) = Transformations D E 1 dv Polar/Cylindrical r 2 = x 2 + y 2 x = r cos θ y = r sin θ y/x = tan θ f(x, y) da = f(r cos θ, r sin θ) r dr dθ f(x, y, z) dv = E D f(r cos θ, r sin θ, z) r dz dr dθ E Spherical Additional Definitions curl(f) = F div(f) = F F is conservative if curl(f) = 0 Line Integrals Fundamental Theorem of Line Integrals f dr = f(r(b)) f(r(a)) C Green s Theorem P dx + Q dy = (Q x P y ) da C D Transformations x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ ρ 2 = x 2 + y 2 + z 2 E E f(x, y, z) dv = f(ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ)(ρ 2 sin φ) dρ dφ dθ Page 12 of 12
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 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 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 informationPage 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 informationWithout fully opening the exam, check that you have pages 1 through 12.
MTH 34 Solutions to Exam April 9th, 8 Name: Section: Recitation Instructor: INSTRUTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show
More informationWithout fully opening the exam, check that you have pages 1 through 11.
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 11. Show all your work on the standard
More informationMTH 234 Exam 1 February 20th, Without fully opening the exam, check that you have pages 1 through 11.
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 11. Show all your work on the standard
More informationMTH 234 Solutions to Exam 1 Feb. 22nd 2016
MTH 34 Solutions to Exam 1 Feb. nd 016 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
More informationMTH 234 Exam 2 April 10th, Without fully opening the exam, check that you have pages 1 through 12.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in our name, etc. on this first page. Without full opening the eam, check that ou have pages 1 through 12. Show all our work on the standard response
More informationMTH 234 Solutions to Exam 2 April 10th, Without fully opening the exam, check that you have pages 1 through 12.
MTH 34 Solutions to am April 1th, 17 Name: Section: Recitation Instructor: INSTRUTIONS Fill in your name, etc. on this first page. Without fully opening the eam, check that you have pages 1 through 1.
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 informationWithout fully opening the exam, check that you have pages 1 through 12.
MTH 34 Solutions to Exam November 9, 8 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through.
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 informationMTH 133 Final Exam Dec 8, 2014
Name: PID: Section: Recitation Instructor: DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE. Page Problem Score Max Score 1 5 3 2 5 3a 5 3b 5 4 4 5 5a 5 5b 5 6 5 5 7a 5 7b 5 6 8 18 7 8 9 10 11 12 9a
More informationMTH 132 Solutions to Exam 2 Nov. 23rd 2015
Name: Section: 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 your desk
More informationMLC Practice Final Exam. Recitation Instructor: Page Points Score Total: 200.
Name: PID: Section: Recitation Instructor: DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE. Page Points Score 3 20 4 30 5 20 6 20 7 20 8 20 9 25 10 25 11 20 Total: 200 Page 1 of 11 Name: Section:
More informationWithout fully opening the exam, check that you have pages 1 through 10.
MTH 234 Solutions to Exam 2 April 11th 216 Name: Section: Recitation Instructor: INSTRUTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through
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 informationMTH 132 Solutions to Exam 2 Apr. 13th 2015
MTH 13 Solutions to Exam Apr. 13th 015 Name: Section: Instructor: READ THE FOLLOWING INSTRUCTIONS. Do not open your exam until told to do so. No calculators, cell phones or any other electronic devices
More informationMTH 133 Solutions to Exam 1 Feb. 25th 2015
MTH 133 Solutions to Exam 1 Feb. 5th 15 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
More informationDisclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.
Disclaimer: This is meant to help you start studying. It is not necessarily a complete list of everything you need to know. The MTH 234 final exam mainly consists of standard response questions where students
More informationWithout fully opening the exam, check that you have pages 1 through 11.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show all your work on the standard response
More informationMTH 133 Exam 2 November 16th, 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 through 2. Show all your work on the standard response
More informationMTH 133 Solutions to Exam 2 April 19, Without fully opening the exam, check that you have pages 1 through 12.
MTH 33 Solutions to Exam 2 April 9, 207 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through
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 informationWithout fully opening the exam, check that you have pages 1 through 10.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the eam, check that you have pages 1 through 10. Show all your work on the standard
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 informationMA 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 informationWithout fully opening the exam, check that you have pages 1 through 11.
MTH 33 Solutions to Final Exam May, 8 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show
More informationWithout fully opening the exam, check that you have pages 1 through 11.
Name: Section: (Recitation) Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show all your work on the standard response
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 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 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 informationWithout fully opening the exam, check that you have pages 1 through 11.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show all your work on the standard response
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 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 informationMTH 132 Exam 2 November 21st, Without fully opening the exam, check that you have pages 1 through 11.
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 11. Show all your work on the standard
More informationMTH 132 Solutions to Exam 2 November 21st, Without fully opening the exam, check that you have pages 1 through 11.
Name: Section: Recitation/Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show all your work on the standard response
More informationThe University of British Columbia Final Examination  December 17, 2015 Mathematics 200 All Sections
The University of British Columbia Final Examination  December 17, 2015 Mathematics 200 All Sections Closed book examination Time: 2.5 hours Last Name First Signature Student Number Special Instructions:
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 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 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 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 through 2. Show all your work on the standard response
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 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 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 informationMA 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 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 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 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 informationWithout fully opening the exam, check that you have pages 1 through 12.
MTH 33 Exam 2 April th, 208 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through 2. Show all
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 informationMTH 133 Solutions to Exam 2 November 15, Without fully opening the exam, check that you have pages 1 through 13.
MTH 33 Solutions to Exam 2 November 5, 207 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through
More informationWithout fully opening the exam, check that you have pages 1 through 11.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show all your work on the standard response
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 informationAPPM 2350 Final Exam points Monday December 17, 7:30am 10am, 2018
APPM 2 Final Exam 28 points Monday December 7, 7:am am, 28 ON THE FONT OF YOU BLUEBOOK write: () your name, (2) your student ID number, () lecture section/time (4) your instructor s name, and () a grading
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 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 informationMATH UN1201, Section 3 (11:40am 12:55pm)  Midterm 1 February 14, 2018 (75 minutes)
Name: Instructor: Shrenik Shah MATH UN1201, Section 3 (11:40am 12:55pm)  Midterm 1 February 14, 2018 (75 minutes) This examination booklet contains 6 problems plus an additional extra credit problem.
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 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 informationWithout fully opening the exam, check that you have pages 1 through 13.
MTH 33 Solutions to Exam November th, 08 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through
More informationWithout fully opening the exam, check that you have pages 1 through 12.
MTH 33 Exam 2 November 4th, 208 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through 2. Show
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 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 informationMULTIVARIABLE CALCULUS
MULTIVARIABLE CALCULUS JOHN QUIGG Contents 13.1 ThreeDimensional Coordinate Systems 2 13.2 Vectors 3 13.3 The Dot Product 5 13.4 The Cross Product 6 13.5 Equations of Lines and Planes 7 13.6 Cylinders
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 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 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 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 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 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 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 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 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 informationReview for the First Midterm Exam
Review for the First Midterm Exam Thomas Morrell 5 pm, Sunday, 4 April 9 B9 Van Vleck Hall For the purpose of creating questions for this review session, I did not make an effort to make any of the numbers
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 through. Show all your work on the standard response
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 informationMath 11 Fall 2018 Practice Final Exam
Math 11 Fall 218 Practice Final Exam Disclaimer: This practice exam should give you an idea of the sort of questions we may ask on the actual exam. Since the practice exam (like the real exam) is not long
More informationProblem 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 informationGeometry and Motion Selected answers to Sections A and C Dwight Barkley 2016
MA34 Geometry and Motion Selected answers to Sections A and C Dwight Barkley 26 Example Sheet d n+ = d n cot θ n r θ n r = Θθ n i. 2. 3. 4. Possible answers include: and with opposite orientation: 5..
More informationMcGill University December Intermediate Calculus. Tuesday December 17, 2014 Time: 14:0017:00
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 informationDimensions = xyz dv. xyz dv as an iterated integral in rectangular coordinates.
Math Show Your Work! Page of 8. () A rectangular box is to hold 6 cubic meters. The material used for the top and bottom of the box is twice as expensive per square meter than the material used for the
More informationMultiple Choice Answers. MA 114 Calculus II Spring 2013 Final Exam 1 May Question
MA 114 Calculus II Spring 2013 Final Exam 1 May 2013 Name: Section: Last 4 digits of student ID #: This exam has six multiple choice questions (six points each) and five free response questions with points
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 16.116.8. 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 informationDirection of maximum decrease = P
APPM 35 FINAL EXAM PING 15 INTUTION: Electronic devices, books, and crib sheets are not permitted. Write your name and your instructor s name on the front of your bluebook. Work all problems. how your
More informationSOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am12:00 (3 hours)
SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am12:00 (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 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 informationStokes 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 informationPractice Final Solutions
Practice Final Solutions Math 1, Fall 17 Problem 1. Find a parameterization for the given curve, including bounds on the parameter t. Part a) The ellipse in R whose major axis has endpoints, ) and 6, )
More informationMTH132 Exam 1 Covers: Page Total. Max
Name: PID: A Section #: Instructor: Page 3 4 5 6 7 8 Total Score Max 4 4 4 4 1 150 Instructions 1. You will be given exactly 90 minutes for this exam.. No calculators, phones, or any electronic devices.
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 informationMAC2313 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 informationPractice problems for Exam 1. a b = (2) 2 + (4) 2 + ( 3) 2 = 29
Practice problems for Exam.. Given a = and b =. Find the area of the parallelogram with adjacent sides a and b. A = a b a ı j k b = = ı j + k = ı + 4 j 3 k Thus, A = 9. a b = () + (4) + ( 3)
More informationDerivatives and Integrals
Derivatives and Integrals Definition 1: Derivative Formulas d dx (c) = 0 d dx (f ± g) = f ± g d dx (kx) = k d dx (xn ) = nx n 1 (f g) = f g + fg ( ) f = f g fg g g 2 (f(g(x))) = f (g(x)) g (x) d dx (ax
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 informationMATH 2083 FINAL EXAM REVIEW The final exam will be on Wednesday, May 4 from 10:00am12:00pm.
MATH 2083 FINAL EXAM REVIEW The final exam will be on Wednesday, May 4 from 10:00am12:00pm. Bring a calculator and something to write with. Also, you will be allowed to bring in one 8.5 11 sheet of paper
More informationD = 2(2) 3 2 = 4 9 = 5 < 0
1. (7 points) Let f(, ) = +3 + +. Find and classif each critical point of f as a local minimum, a local maimum, or a saddle point. Solution: f = + 3 f = 3 + + 1 f = f = 3 f = Both f = and f = onl at (
More informationSolutions 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 informationMath 116 Second Midterm November 14, 2012
Math 6 Second Midterm November 4, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has pages including this cover. There are 8 problems. Note that
More information