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Purdue | MA 261
MULTIVARIATE CALCULUS
Professors
- Stefanov,
- Becker,
- Bauman,
- Santos,
- Tarfulea,
- Dadarlat,
- Drasin,
- Phillips,
- Brown,
- Gabreliev,
- Jacobs,
- Jonathan Casillas Chavez Ta,
- Turkay,
- Daniel Phillips,
- Marius Darladat,
- Jason Patel,
- Yongheng Zhang,
- Prof. Chen,
- Arthur,
- Andrei Gabrielov
100 sample documents related to MA 261
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MA 16200 FINAL EXAM PRACTICE PROBLEMS E. E. 1 3 1. If a = i + j + k and b = 2i - k, find the vector projection of b onto a, proja b. 1 1 1 B. 3 a C. 5 a D. 3 b A. 1 a 3 2. Find the angle between the vectors a = -i + 2j and b = i + 3j B. C. 2 D. A. 3 4 4 3
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MA 261 - Fall 2012 Study Guide # 1 1. Vectors in R2 and R3 (a) = a, b, c = a + b + c ; vector addition and subtraction geometrically using paralv i j k lelograms spanned by and ; length or magnitude of = a, b, c, | | = a2 + b2 + c2 ; u v v v directed vect
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Quiz 1, Section 171, T 2:30 pm P1. Find two unit vectors that make an angle of 120 with v =< 1, 1 >. Solution: We rst name the vectors we are looking for: a = (a1 , a2 ) and b = (b1 , b2 ). What we know is the following: a = a2 + a2 = 1. 1 2 b = b2 + b2
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Quiz 2, Section 171, T 2:30 pm P1. Reduced the equation to one of the standard forms and classify the surface. 4 x2 + y2 + 4z2 4y 24z + 36 = 0 Solution: Grouping terms with the same variable, we have: 4 x2 + (y2 4y) + (4z2 24z) = 36 Completing squares: 4
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Quiz 3, Section 171, T 2:30 pm P1. (a) Find the arclenght of the curve: r(t) = (2 sin(t), 5t, 2 cos(t), 0 t 2. (b) If you started at the point (0, 0, 2), and moved 2 29 units along the previous curve, where are you now? Solution: (a) First, lets calculate
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Basic Vector Algebra Denition of a Vector: A vector is an object which is characterized by two properties: length (magnitude) and direction. Vector Algebra: 1. Vector Addition u + v : the result is obtained by means of the triangular- or parallelogram- la
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MA 261 - Spring 2011 Study Guide # 2 f f 0. Gradient vector for f (x, y ): f (x, y ) = , , properties of gradients; gradient points in x y direction of maximum rate of increase of f ; The maximum value of the directional derivative is equal to f ; f (x0 ,
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Lagrange Multipliers 1. Use Lagrange multipliers to find the point on the given plane that is closest to the following point. (Enter your answer as a fraction.) x - y + z = 3; (2, 1, 1) ( 7/3 , 2/3 , 4/3 ) 2. Use Lagrange multipliers to find the volume of
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MA 261 - Spring 2011 Study Guide # 3 You also need Study Guides # 1 and # 2 for the Final Exam 1. Line integral of a function f (x, y ) along C , parameterized by x = x(t), y = y (t) and a t b, is f (x, y ) ds = ( b f (x(t), y (t) C a dx dt )2 ( + dy dt )
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MA261 0021&0022 Quiz 1 Spring 2011 Problem 1. Find the angle between the vectors 1, 1, 1 and 1, 1, 2 . (Hint: Look at the dot product of these two vectors.) Since 1, 1, 1 1, 1, 2 = 1 + 1 2 = 0, we know these two vectors are perpendicular. Thus, the angle
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MA261 0021&0022 Quiz 2 Spring 2011 Problem 1. Given two planes x + y + z = 2 and x + y = 1, (a) Find the two normal vectors associated to each plane, respectively. (Notice that x + y = 1 is the same thing as x + y + 0 z = 1) Solution. We read the normal v
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MA261 0021&0022 Quiz 3 Spring 2011 Problem. Consider the ellipse x2 y 2 + = 1, z = 0 4 1 in the three dimensional space. One system of parametric equations of it is r(t) = x(t), y (t), z (t) where x(t) = 2 cos t y (t) = sin t , t [0, 2 ]. z (t) = 0 (a) Fi
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MA261 0021&0022 Quiz 3 Spring 2011 Problem. Consider the ellipse x2 y 2 + = 1, z = 0 4 1 in the three dimensional space. One system of parametric equations of it is r(t) = x(t), y (t), z (t) where x(t) = 2 cos t y (t) = sin t , t [0, 2 ]. z (t) = 0 (a) Fi
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MA261 0021&0022 Quiz 4 Spring 2011 Problem 1. The position function of a particle is given by r(t) = 3 sin t, 3 cos t, 4t . (1) Find the velocity v(t). Solution. v(t) = r(t) = 3 cos t, 3 sin t, 4 . (2) Find the speed |(t)|. v Solution. |v(t)| = 32 + 42 =
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MA261 0021&0022 Quiz 5 Spring 2011 2 Problem 1 (Spring 2006). If u(x, y ) = yexy , (a) nd ux . 2 2 Solution. ux = yy 2 exy = y 3 exy . (b) nd uxy . 2 2 2 Sollution. uxy = (ux )y = 3y 2 exy + y 3 2xyexy = y 2 exy (3 + 2xy 2 ). Problem 2 (Spring 2001). Give
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MA261 0021&0022 Quiz 6 Spring 2011 Problem 1. Suppose y is a function of x and they satisfy F (x, y ) = 0. Take the partial derivative with respect to x of both sides of the above equation (Use Chain Rule ) to show that F dy x = F . dx y Proof. Given F (x
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MA261 0021&0022 Quiz 7 Spring 2011 Problem 1. Given f (x, y ) = x2 y 2 , (a) Find the critical point of f (x, y ). Solution. Setting f (x, y ) to be zero, we have f (x, y ) = fx (x, y ), fy (x, y ) = 2x, 2y = 0, 0 . So, (x, y ) = (0, 0) is the only critic
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MA261 0021&0022 Quiz 8 Spring 2011 Problem 1 (modied from Problem 2 in the second midterm of Spring 2 2009). We will nd the maximum of f (x, y ) = xy on the ellipse x + y 2 = 1 using the 4 Lagrange multiplier method. 2 (a) Let g (x, y ) = x + y 2 . Write
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MA261 0021&0022 Quiz 9 Spring 2011 Problem 1 (Test II.8, Spring 2008). [Warning: No use of cell phone browsing the internet during the quiz!] Let D be the part of disk centered at 0 with radius 2 that lies to the right of the line x = 1. Then which of the
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MA261 0021&0022 Quiz 10 Spring 2011 Problem 1. Find the length of a wire C in the shape of a helix described by the parametric equation C : x = cos t, y = sin t, z = t, 0 t 4 Solution. To get the length of the wire, let us integrate the constant function
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MA261 0021&0022 Quiz 11 Spring 2011 Problem 1. (a) Given F(x, y ) = y exy , xexy , nd a function f (x, y ) such that f (x, y ) = F(x, y ). Solution. Since we are told f (x, y ) = F(x, y ), we have fx = yexy and fy = xexy . After integrating the rst equali
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MA261 0021&0022 Quiz 12 Spring 2011 Problem 1. (a) Given a function f (x, y, z ) = xyz , nd its gradient f (x, y, z ) and then the curl of the gradient, namely curl( f (x, y, z ). Solution. Given f (x, y, z ) = xyz , we know f (x, y, z ) = fx , fy , fz =
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MA 261 NAME STUDENT ID # INSTRUCTOR INSTRUCTIONS FINAL EXAM Spring 1999 Page 1/14 1. There are 14 different test pages (including this cover page). Make sure you have a complete test. 2. Fill in the above items in print. I.D.# is your 9 digit ID (probably
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MATH 261 - SPRING 2000 Name Signature Div. Sect. No. Instructor (Test 01) Recitation Instructor FINAL EXAM INSTRUCTIONS 1. You must use a #2 pencil on the mark-sense sheet (answer sheet). 2. If you have test 01, mark 01 and blacken the corresponding circl
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MATH 261 - SPRING 2001 FINAL EXAM Name Signature Div. Sect. No. Instructor Recitation Instructor FINAL EXAM INSTRUCTIONS 1. You must use a #2 pencil on the mark-sense sheet (answer sheet). 2. On the mark-sense sheet, fill in the instructor\'s name and the
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MA 261 FINAL EXAM Form A Spring 2008 1. Find an equation of the plane that contains the point (2, 1, 1) and the line x = 1 + 3t, y = 2 + t, z = 4 + t. A. 3x + y + z = 8 B. 2x + y + z = 6 C. x + 2y + 4z = 8 D. x - 5y + 2z = -1 E. x - 2y + z = 1 2. Compute
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MA26100 FinalExamKey 1. E 2. C 3. A 4. D 5. E 6. B 7. E 8. C 9. A 10. E 11. C 12. B 13. C 14. D 15. D 16. B 17. D 18. D 19. E 20. A 21. B 22. A Fall08
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MA261 Final Exam Spring 2002 1. D 2. A 3. B 4. C 5. E 6. D 7. B 8. A 9. D 10. E 11. C 12. A 13. A 14. C 15. A 16. B 17. A 18. B 19. E 20. D
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MA261 Final Exam Spring 2003 1. D 2. A 3. D 4. B 5. E 6. A 7. C 8. D 9. D 10. E 11. E 12. A 13. C 14. C 15. D 16. C 17. D 18. E 19. B 20. C 21. D 22. C 23. E 24. A
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MA 261 FINAL EXAM Form A Spring 2008 Answer Key: 1D 2A 3D 4B 5C 6D 7B 8D 9B 10A 11A 12E 13A 14A 15E 16B 17B 18B 19A 20E 21B 22B 1. Find an equation of the plane that contains the point (2, 1, 1) and the line x = 1 + 3t, y = 2 + t, z = 4 + t. A. 3x + y + z
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Answers to MA261 Final (Spring 2000) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. D E C C B A D A D D B C C C C E D E A E
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MATH 261 FALL 2000 FIRST EXAM September 26, 2000 STUDENT NAME -STUDENT ID -RECITATION HOUR -RECITATION INSTRUCTOR - INSTRUCTIONS: 1. This test booklet has 5 pages including this one. 2. Fill in your name, your student ID number, your recitation hour and y
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MA 261 NAME STUDENT ID # INSTRUCTOR INSTRUCTIONS EXAM I Spring 1998 Page 1/6 1. There are 6 different test pages (including this cover page). Make sure you have a complete test. 2. Fill in the above items in print. I.D.# is your 9 digit ID (probably your
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MA 261 NAME STUDENT ID # Exam 1 Spring 2000 Page 1/9 RECITATION INSTRUCTOR RECITATION TIME DIRECTIONS 1) Fill in the above information. Also write your name at the top of each page of the exam. 2) The test has 9 pages, including this one. 3) Problems 1 th
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MA 261 NAME STUDENT ID EXAM 1 Spring 2001 Page 1/6 Page 1 Page 2 Page 3 Page 4 /12 /7 /18 /18 /27 /18 /100 RECITATION INSTRUCTOR RECITATION TIME Page 5 Page 6 TOTAL DIRECTIONS 1. Write your name, student ID number, recitation instructor\'s name and recitat
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MA 261 NAME STUDENT ID # INSTRUCTOR INSTRUCTIONS EXAM II Spring 1999 Page 1/7 1. There are 7 different test pages (including this cover page). Make sure you have a complete test. 2. Fill in the above items in print. I.D.# is your 9 digit ID (probably your
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MA 261 NAME STUDENT ID # Exam 2 Spring 2000 Page 1/8 RECITATION INSTRUCTOR RECITATION TIME DIRECTIONS 1) Fill in the above information. Also write your name at the top of each page of the exam. 2) The test has 8 pages, including this one. 3) Problems 1 th
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MA 261 NAME STUDENT ID # Exam 2 Spring 2001 Page 1/6 RECITATION INSTRUCTOR RECITATION TIME DIRECTIONS 1) Fill in the above information. Also write your name at the top of each page of the exam. 2) The exam has 6 pages, including this one. 3) Problems 1 th
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Exam 2 Answer Key MA 262 Spring 2008 1. C 2. E 3. D 4. B 5. B 6. A 7. D 8. B 9. C 10. E
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FALL 1998 ANSWERS FOR EXAM II: 1. B 2. C 3. E 4. A 5. D 6. B 7. E 8. 2 2 9. 0 1 r 4r 2 + 1 dr d= (173/2 - 53/2 ) 6 3/2 3-2x2 5-x2 10. - 3/2 - 3-2x2 dz dy dx 2+x2 +y 2 1
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SPRING 1999 ANSWERS FOR EXAM II: 1. E 2. B 3. B 4. B 5. B 6. E 7. D 2 1/2 1-r2 8. 0 0 - 1-r2 r dz dr d 9. 3/2 6 (13 - 1) 10. (1) (3, 3/2) minimum (2) (-2, -1) saddle 1
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SPRING 2000 ANSWERS FOR EXAM II: 1. C 2. D 3. B 4. E 5. A 6. (0, 0) saddle, (1, -1) min. /2 1 7. (a) (1/2, 0) (b) r = cos 0 25-x2 0 25-r2 (c) 0 cos r 3 dr d 25-x2 -y 2 8. (a) -1 5 0 z dz dy dx (b) /2 0 3 6-2x 0 0 zrdz dr d. 6-2x-y 9. 0 0 dz dy dx 1
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Some Examples of Arclength parametrization and Curvature Turkay Yolcu tyolcu@math.purdue.edu
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MA 153 Exam 1 Fall 2000 Name: _ Student ID: _ Instructor: _ Class Hour: _ INSTRUCTIONS: (1) There is no credit for guessing. You must show your work to receive credit! (2) Please fill in all the above information and write your name on the top of each of
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MA 153 Exam 2 Fall 2000 Name: _ Student ID: _ Instructor: _ Class Hour: _ INSTRUCTIONS: (1) There is no credit for guessing. You must show your work to receive credit! (2) Please fill in all the above information and write your name on the top of each of
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MA 153 Exam 3 Spring 2002 1. Find the slope, m , and the y -intercept, b , of the line given by the equation 3 x - 4 y = 8 . 3 A. m = - ; b = -2 4 3 B. m = ; b = 8 4 3 C. m = - ; b = 8 4 3 D . m = ; b = -2 4 E . None of the above Use the graph of a functi
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MA 153 Exam 1 Answers Fall 2001 1. 6 x 3 + 19 x2 26 x + 5 2. x + 8 x + 15 x 25 3. y = 4. mW gm x 2 x +2 5 (a) 2(3 x + 4)(2 x 1) (b) (4 x + y )(4 x y)(a + 3c) 6. (a) x= -5, 7 (b) x = 6 7. (a) 20 ab11 (b) 3a3b 4 8. 3:00 p.m. 9. 16.4 cm
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Question # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Orange Form D C D A B C A A B D C Answer y-x x x4y6 3z 4 7 4x 9 5 x 3 - 6 x 2 + 5 x - 14 9 x 3 - 24 x 2 + 16 x ( x + 4) x + 2 ( ) 4x3 + 5 b2 + a2 a ( x + 5)( x 2 + 4) x+2 9t - t 2 (t + 4 )( t - 4) x is betwee
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MA 153 Exam 3 Answers Fall 2001 1. (a) 6 9 (b) 2 (c) 10 x 3 - 3 2. (a) [- 4,2] (b) [- 3,4] 1 (c) (- 4,-2 ) ( ,2] 3. (4,3)and (- 1,-7 ) 4. g -1 (x ) = 2 x+5 5. 14 7 or or 1.75 8 4 y = f (x - 2 )+ 5 6. -13 7. (2,5) (6,5) (4,2) 8. 2a + h - 1 9. y = - 10 (x -
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Direction Fields -deld6 The routine deld6 is already loaded on all PUCC machines as standard software. To access it from any PUCC machine: StartAll ProgramsStandard SoftwareComputational PackagesMATLAB6.1MATLAB6.1 If you are using your own copy of Matlab
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%LSQGAME Least Squares Line Game last updated 2/10/96 % % An interactive \'game\' to select the least squares line % to a set of data. Two guesses for the lsq line can be made % using the mouse to select two points that are then connected. % The \'true\' leas
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Numerical Methods & .m Files In order to use Matlab routines for the Euler, Improved Euler and Runge-Kutta Methods, you will need the les eul.m, rk2.m and rk4.m, respectively. These les are already present on all PUCC machines as standard software. If you
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Phase Portraits - pplane6 The routine pplane6 is already loaded on all PUCC machines as standard software. If you are using your own copy of Matlab you may need to download pplane6. Here is a link : http:/math.rice.edu/dfield/ (Note: pplane5 is an older
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