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Reduce
School: Purdue
function AEQ = reduce(A) %last updated 5/7/94%REDUCE Perform row reduction on matrix A by explicitly choosing% row operations to use. A row operation can be "undone", but% this feature cannot be used in succession.% Use in the form => reduce(A) <=% By: Da

Ans261FES2002
School: Purdue
Course: Calculus III
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

Ans261FEF2008
School: Purdue
Course: Calculus III
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

261FES2008
School: Purdue
Course: Calculus III
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

261FES2001
School: Purdue
Course: Calculus III
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 marksense sheet (answer sheet). 2. On the marksense sheet, fill in the instructor's name and the

261FES2000
School: Purdue
Course: Calculus III
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 marksense sheet (answer sheet). 2. If you have test 01, mark 01 and blacken the corresponding circl

261FES1999
School: Purdue
Course: Calculus III
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

Sol261E2S2000
School: Purdue
Course: Calculus III
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 25x2 0 25r2 (c) 0 cos r 3 dr d 25x2 y 2 8. (a) 1 5 0 z dz dy dx (b) /2 0 3 62x 0 0 zrdz dr d. 62xy 9. 0 0 dz dy dx 1

Sol261E2S1999
School: Purdue
Course: Calculus III
SPRING 1999 ANSWERS FOR EXAM II: 1. E 2. B 3. B 4. B 5. B 6. E 7. D 2 1/2 1r2 8. 0 0  1r2 r dz dr d 9. 3/2 6 (13  1) 10. (1) (3, 3/2) minimum (2) (2, 1) saddle 1

153E1F00
School: Purdue
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

153E2F00
School: Purdue
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

153E3S02O
School: Purdue
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

Project
School: Purdue
function project(u,w) %last updated 5/9/94 %PROJECT Projecting vector U onto vector W orthogonally. Vectors % U and W can be either a pair of 2D or 3D vectors. A sketch % showing U being projected onto W is displayed sequentially. % % Use in the form => p

Practicefinal7
School: Purdue
MA 261 PRACTICE PROBLEMS 1. If the line x1 2 has symmetric equations y 3 = = z+2 7 , find a vector equation for the line A. r = (1 + 2t)i  3tj + (2 + 7t)k that contains the point (2, 1, 3) and is parallel to . B. r = (2 + t)i  3j + (7  2t)k D. r =

Practicefinal
School: Purdue

Pp
School: Purdue
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

Num
School: Purdue
Numerical Methods & .m Files In order to use Matlab routines for the Euler, Improved Euler and RungeKutta 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

Lsqgame
School: Purdue
%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

Groundbase
School: Purdue

FEx
School: Purdue

Df
School: Purdue
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

Ci2
School: Purdue

Basic
School: Purdue

Assignment3
School: Purdue

Ans153E3F01
School: Purdue
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 

Ans153E1S03O
School: Purdue
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 yx 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

Ans153E1F01
School: Purdue
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

Sol261E2F1998
School: Purdue
Course: Calculus III
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 32x2 5x2 10.  3/2  32x2 dz dy dx 2+x2 +y 2 1

261E1S2001
School: Purdue
Course: Calculus III
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

261E1S2000
School: Purdue
Course: Calculus III
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

261E1S1998
School: Purdue
Course: Calculus III
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

261E1F2000
School: Purdue
Course: Calculus III
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

Sol261FES2000
School: Purdue
Course: Calculus III
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

Ans261E2S2008
School: Purdue
Course: Calculus III
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

261E2S2001
School: Purdue
Course: Calculus III
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

261E2S2000
School: Purdue
Course: Calculus III
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

261E2S1999
School: Purdue
Course: Calculus III
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

Ans261FES2008
School: Purdue
Course: Calculus III
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

Quiz4Solutions
School: Purdue
Course: Calculus
Quiz 4 Solutions David Imberti November 28, 2012 (1) (this problem changed slightly for each section) a2 2 xy lim(x,y)(0,0) x +2+y+y x 2 = lim(x,y)(0,0) (x+y) x+y = lim(x,y)(0,0) x + y =0 b2 2xy lim(x,y)(0,0) x x+y+2y x = y, x 0: 2 +2 = limx0 x2x x = 0 x

Quiz5
School: Purdue
Course: Calculus
Quiz 5 David Imberti September 24, 2012 1Find the tangent plane to x2 + y 2 + z 2 = 1 at (0, 0, 1). 2The volume of a piece of track is given by V = xyz , where x = 2 in. y = 4 in., and z = 3600 in., you estimate in the heat that the track could expand a

Quiz5Solutions
School: Purdue
Course: Calculus
Quiz 5 Solutions David Imberti November 28, 2012 (1) This is the top of the sphere. The easiest way to do this is to draw the picture and conclude that z = 1 is the tangent plane. Alternatively: z = + 1 x2 y 2 (positive since the point being considered is

Quiz6
School: Purdue
Course: Calculus
Quiz 6 David Imberti November 28, 2012 (1) Use Lagrangian Multipliers to nd MAXimum of 4x2 + 2y 2 along x2 + y 2 = 1. Calculate (2) (a) 11 exp(2x + 3y )dxdy 00 (b) 11 exp( x )dydx y 0x (Hint: reverse the integration order) 1

Quiz6Solutions
School: Purdue
Course: Calculus
Quiz 6 Solutions David Imberti November 28, 2012 (1) f = 4x2 + 2y 2 , g = x2 + y 2 1 f =< 8x, 4y >, g =< 2x, 2y > f = g 4x = x 2y = y x2 + y 2 = 1 If x = 0 then since x2 + y 2 = 1, y = 0 and thus = 2 but y 2 = 1, y = 1. Likewise if y = 0 then x = 1. If bo

Quiz7
School: Purdue
Course: Calculus
Quiz 7 David Imberti November 28, 2012 (1) Convert to polar coordinates, and then calculate the integral. 2 1 2y (x + y )dxdy 0y (2) Calculate the surface area of the part of the plane 3x + 2y + z = 6 that lies in the rst octant. 1

Quiz7Solutions
School: Purdue
Course: Calculus
Quiz 7 Solutions David Imberti November 28, 2012 (1) 1 2y 2 (x + y )dxdy 0y 22 = 04 0 r (cos + sin)drd 2 = ( 04 cos + sin)( 0 r2 ) 3 4 = (sin cos)0 ( r3 )sqrt2 0 3 22 =3 (2) z = 6 3x 2y z z x = 3, y = 2 2 2 1 + (3) + (2) dA = 14 dA The area on the xy

Quiz8Solutions
School: Purdue
Course: Calculus
Quiz 8 Solutions David Imberti November 30, 2012 (1) This problem could be done many ways. One possible way: (a) 2 1 1 2 1 1z rdrdzd 0 0 0 rdrdzd 0 00 (b) You can just use general expressions for the volume of a cone and a cylinder if you dont want to eva

Quiz9Solutions
School: Purdue
Course: Calculus
Quiz 9 Solutions David Imberti November 28, 2012 We use Greens Theorem directly for both questions. (1) C F dr = A(C ) P + Q dA y x = A(C ) 0dA =0 (2) C F dr = A(C ) P + Q dA y x = A(C ) x2 + y 2 dA Polar Coordinates: 2 100 = 0 0 r3 drd = 2 1 (100)4 4 1

Quiz10
School: Purdue
Course: Calculus
Quiz 10 David Imberti November 27, 2012 (1) Give a parametrization of the shell of a sphere in the rst octant with radius 1. (2) Find F dS with S the surface as in 1 and F = xi + y j + z k (i.e., nd the linear heat ow across S the octalsphere) 1

Quiz10Solutions
School: Purdue
Course: Calculus
Quiz 10 Solution David Imberti November 28, 2012 (1) This is just the vector equation of the spherical coordinates. r(, ) =< cossin, sinsin, cos > To force this into the rst octant: 0 ,0 . 2 2 (2) This problem is the last example from section 16.7 in the

Vectoralgebra
School: Purdue
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

SolutionsQ5
School: Purdue

SolutionsQ4
School: Purdue

SolutionsQ3
School: Purdue
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

Quiz4
School: Purdue
Course: Calculus
Quiz 4 David Imberti November 28, 2012 1Find if it exists, or show that it does not. 2 2 xy a lim(x,y)(0,0) x +2+y+y x 2 2xy b lim(x,y)(0,0) x x+y+2y 2Sketch the level curves of f (x, y ) = c, for f (x, y ) = x2 + y 2 , c = 1, 4. 1

Quiz3Solutions
School: Purdue
Course: Calculus
Quiz 3 Solutions David Imberti November 28, 2012 (1) One way to do this is to recognize that the rst part of the track is a straight line from (0, 0) to (1, 1) which is thus of length 2, and the other piece is an eighth part of a circle of radius 1 which

Quiz3
School: Purdue
Course: Calculus
DavidImberti September 10, 2012 Quiz 3show work You are helping design a railroad for good ol Union Pacic, you approximate a model parametric equation for the track as: r(t) = tikm + tjkm for 0 t 1 r(t) = (cos( 34 (t 1) + 1)ikm + (sin( 34 ) (t 1) + 1)jkm

MA261quiz_1
School: Purdue
Course: Multivariate Calculus
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

MA261quiz_2
School: Purdue
Course: Multivariate Calculus
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

MA261quiz_3_solution
School: Purdue
Course: Multivariate Calculus
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

MA261quiz_4
School: Purdue
Course: Multivariate Calculus
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 =

MA261quiz_5
School: Purdue
Course: Multivariate Calculus
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

MA261quiz_6
School: Purdue
Course: Multivariate Calculus
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

MA261quiz_7
School: Purdue
Course: Multivariate Calculus
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

MA261quiz_8
School: Purdue
Course: Multivariate Calculus
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

MA261quiz_9
School: Purdue
Course: Multivariate Calculus
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

MA261quiz_10
School: Purdue
Course: Multivariate Calculus
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

MA261quiz_11
School: Purdue
Course: Multivariate Calculus
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