x + ye z2 + ze y2, y + xe z2 + ze x2, z and where T is the


 Naomi Wilkinson
 10 months ago
 Views:
Transcription
1 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 in the xy plane. Let D be the unit disk in the xy plane. Then T D is the boundary of a solid E whose volume is π. The orientation on T is outward with respect to E. You are looking at the picture from underneath. This problem will require some thought since all you know about T and E are what you see in the picture plus the volume of E. (a) 6π (b) π (c) 6π (d) π(e e) (e) π
2 .(8pts) Let T be the surface r(u, θ) = u, u cos(θ), u sin(θ) for u 1, θ π. This is the surface you get by rotating y = x, x 1, around the y axis. If the density at a point on this surface is given by 4x 5 + 4x 3, find the mass. Remark: The density has been chosen so that a simple substitution can be used to evaluate the integral you should get. (a) 4π (b) π(4) 3 (c) π (d) π 8 (e) π ( 4 )
3 3.(8pts) Let C be the curve parametrized by r(t) = cos t, sin t, 1. Stokes theorem tells us that for a differentiable vector field F F dr = (curl F) ds C for any oriented smooth surface S. The surfaces below are possible choices for S where the vectors indicate the direction of the normal field. Which of the following is an appropriate choice for S? The curve C is drawn in bold in each picture and the positive directions on the axes are labeled. Hint: Be mindful of the orientation of C! S (a) (b) (c) (d) (e)
4 4.(7pts) We are asked to find absolute maximum and minimum values of f(x, y, z) with respect to the constraints g(x, y, z) = 1 and h(x, y, z) = e. We are given that the five points (1, 1, 1), (, 1, 1), (, 1, 1), (1, 1, 1) and ( 1, 1, 1) satisfy both constraint equations. Each of the five points satisfies an additional condition: f(1, 1, 1) = g(1, 1, 1) + 3 h(1, 1, 1) f(, 1, 1) = 6 g(, 1, 1) f(, 1, 1) ( ( g(, 1, 1) h(, 1, 1) ) = f( 1, 1, 1) ( ( g( 1, 1, 1) h( 1, 1, 1) ) f(1, 1, 1) = Which point can not be an absolute extremum? (a) (1, 1, 1). (b) (1, 1, 1). (c) ( 1, 1, 1). (d) (, 1, 1). (e) (, 1, 1). 5.(7pts) Compute the angle between the vectors 1, 1, and, 1, 3. (a) π/4 (b) π (c) π/ (d) arccos ( ) 1 (e) arccos 8 ( ) 1 8
5 6.(7pts) Compute the flux F ds, where F = 4y, xy, xz and S is the surface z = ye x, x 1, y 1, with the upward orientation. S (a) e (b) 1 e (c) e 1 (d) e (e) 1 + e 7.(7pts) Find F dr where F = x, z, y and C is parametrized by C r(t) = cos t, t, sin t, t π. (a) π + (b) π 1 (c) π (d) 1 (e) π
6 8.(7pts) Compute x(t) dt where x(t) = sin(t)i + e t k. (a) cos t + e t + C (c) ( cos t + C 1 )i + (e t + C )k (b) ( cos t)i + e t k (d) ( cos t + C)i + Cj + (e t + C)k (e) ( cos t + C 1 )i + C j + (e t + C 3 )k 9.(7pts) If f(x, y) = sin(xy) find the directional derivative D u f at the point (x, y), where 1 1 u =,. (a) (d) x + y cos(xy) x + y sin(xy) (b) x + y sin(xy) (c) x y cos(xy) (e) x + y cos(xy)
7 1.(7pts) Determine the surface area of the part of z = xy that lies inside the cylinder x +y = 1. (a) π 3 ( 3 + 1) (b) π 3 ( 3 1) (c) π (d) π 3 ( 3 + 1) (e) π 3 ( 3 1) 11.(7pts) Find F ds where F = x, yz, z and D is the surface of the cube E with D vertices: (,, ), (,, ), (, 3, ), (,, 4), (, 3, ), (, 3, 4), (,, 4), and (, 3, 4), so x, y 3 and z 4. (a) (b) 4 (c) 48 (d) 36 (e) 4
8 1.(7pts) Which of the following is perpendicular to both a and b if a b = 1 and a b,,. (a) (a b) b (b) (a b) a (c) (a + b) (a b) (d) (a b)(a b) (e) a + b 13.(7pts) Compute the curl of the vector field F(x, y, z) = e x sin(y), e y cos(z), e x. (a) e y sin(z), e x, e x cos(y) (c) e y sin(z), e x, e y cos(y) (b) e x sin(z), e x, e x cos(y) (d) e y sin(z), e x, e x cos(y) (e) e y sin(z), e x, e x cos(y)
9 14.(7pts) Find the center of mass of the thin plate D which has the shape of a halfdisk consisting of the region below the semicircle bounded by x + y = 9 and above the xaxis. Assume that D has density ρ(x, y) = x + y and mass 9π. ( ) ( (a), (b), π ) ( ) ( ) ( ) 9 3 (c), (d), (e), 9π 3 π 3π π 15.(7pts) Let f(x, y) = x + 6xy 3y. Find and classify all critical points. (a) (, ), saddle point. (c) (, ) and (1, 1), both saddle points. (b) ( 1, 1), local minima, (, ), local maximum. (d) (, ), local maximum. (e) (, ), local maximum.
10 16.(7pts) Find the curvature when t = of r(t) = cos t, sin t, e t. (a) 3 (b) 1 (c) 3 (d) 1 (e) 3 17.(7pts) Which of the following is a minimum of f(x, y) = x + y subject to the constraint x + y = 1? ( ) ( ) 1 3 (a) ( 1, ) (b), (c), (d) (, 1) (e) None of these points is a minimum.
11 18.(7pts) Find the direction of fastest increase of the function f(x, y) = x y 3 at the point (, 1). (a) 4, 1 (b) 3 1,, (c) 3,, (d) 4 5, 3 5 (e) 4 5, (7pts) Find the equation of the tangent plane to the surface z = x + ln(x + y) at the point ( 1, 3, 1). (a) 3x + y z = 1 (b) x + y 3z = 1 (c) 1 + 3t, 3 + t, 1 t (d) x + 3y z = 11 (e) 3x y + z = 7
12 .(7pts) Define a transformation x(u, w) = u w, y(u, w) = w u for < u <, < w <. (x, y) Compute the Jacobian of this transformation. (u, w) Recall: d ax dx = ax ln(a). (a) u w ln(u) + w u ln(w) (b) u w w u( 1 ln(u) ln(w) ) (c) (u + w )u w 1 w u 1 (d) (u + w)u w 1 w u 1 (e) 1.(7pts) Find the volume that lies inside the sphere x + y + z = and outside the cone z = x + y. (a) 3 8 π (b) π (c) π (d) 8 π (e) π 3
13 1. Solution. div F = 3 so div F dv = 6π. Hence E F ds = F ds = F ds + F ds = 6π E T D T T is oriented the way we want and D is oriented so that the normal vector points down: the unit normal is,, 1 and the field restricted to D is x + y, y + x, since this is what we get when z =. Then x + y, y + x,,, 1 ds = ds = D Hence T D F ds = 6π. x +y 1
14 . Solution. Hence To do 1 r u = 1, u cos(θ), u sin(θ) r θ =, u sin(θ), u cos(θ) i j k r u r θ = det 1 u cos(θ) u sin(θ) u sin(θ) u cos(θ) = u 3, u cos(θ), u sin(θ) r u r θ = 4u 6 + u 4 Mass = (4x 5 + 4x 3 ) ds = (4u 5 + 4u 3 ) 4u 6 + u 4 da = T π dθ 1 u 1 θ π (4u 5 + 4u 3 ) 4u 6 + u 4 du (4u 5 +4u 3 ) 4u 6 + u 4 du use the substitution v = 4u 6 +u 4, dv = (4u 5 +4u 3 ) du. Now isn t that handy. 1 (4u 5 + 4u 3 ) 4u 6 + u 4 du = Hence the mass is π = 4 3 π v dv = v 3 5 = Solution. This question boils down to figuring out whether the given surface has C as its boundary and has orientation consistent with the orientation on C. Since the annulus has both C and the inner circle as its boundary, it is immediately out. The sphere has no boundary, so that is out. Now, for a surface to have orientation consistent with the orientation of its boundary, if you walk around the boundary in the direction of its orientation with your head pointing in the direction of the normal vector field, then the surface will be on your left. This leaves the disk with upward normals as the correct answer.
15 4. Solution. The points where a possible absolute maximum and minimum can occur are points on the two constraints such that f is in the plane formed by g and h. So f is a linear combination of g and h f ( g h ) =, scalar triple product of f, g, h is zero, or the hessian of f, g, h is zero. All those occur expect for the point ( 1, 1, 1). 5. Solution. The desired angle is given by θ = arccos where stands for the dot product. 1, 1,, 1, 3 14 ( ) 1 = arccos 8 6. Solution. The parametrization is r(x, y) = x, y, ye x, x 1, y 1. The normal vector has positive zcoordinate so the orientation is up. Hence F ( r(x, y) ) = 4y, xy, xye x i j k and ds = r x r y da = det 1 ye x 1 e x da = yex, e x, 1 da. Hence Flux = 4y, xy, xye x ye x, e x, 1 da = ( 4y 3 e x xye x + xye x ) da = x 1 y y 3 e x dy dx = 4 y x 1 y 1 1 e x dx = e x dx = e + 1
16 7. Solution. F(r(t)) = cos t, t, sin t r (t) = sin t, cos t, 1 F(r(t)) r (t) = sin t cos t + t cos t + sin t C π π F dr = π F(r(t)) r (t) dt = 1 π sin t cos t dt = u du = 1 (where u = sin t) π t cos t + sin t dt = t sin t = π Thus F dr = 1 + π. ( sin t cos t + t cos t + sin t) dt 8. Solution. x(t) dt = (sin t)i + e t k ) dt sin = t,, e t dt = sin t dt, dt, e t dt = cos t + C 1, C, e t + C 3 = ( cos t + C1 )i + C j + (e t + C 3 )k 9. Solution. The gradient is given by f = y cos(xy), x cos(xy) so D u f = f u = x + y cos(xy).
17 1. Solution. The partial derivatives are given by f x = y and f y = x. The surface area is given by S = x + y + 1 da. After converting to polar coordinates we obtain π 1 D r r + 1 dr dθ = π 1 3 ( 3 π 1) dθ = 3 ( 3 1). 11. Solution. Use the Divergence Theorem to converge this into a triple integral, x dv. This becomes 3 4 x dx dz dy = 48. E
18 1. Solution. We re looking for a vector which is parallel to a b, and it s obvious that (a b)(a b) is parallel to (a b). (because a b ) However, a+b, (a b) b and (a b) a are perpendicular to a b, and (a+b) (a b) is not a vector. 13. Solution. i j k curl F = det = e y sin(z), e x, e x cos(y) x y z e x sin(y) e y cos(z) e x
19 14. Solution. The region and the density are symmetric about the yaxis, hence the x coordinate of the center of mass is. To find the ycoordinate, we compute the moment about the xaxis. In polar coordinates, π ( π ) ( 3 ) M x = x + y y da = r(r sin(θ))r dr dθ = sin(θ) dθ r 3 dr = x +y 9 The mass is Hence, M = x +y 9 ( π) ( ) cos θ r x + y da = ȳ = π rr dr dθ = M x mass(d) = = 81 4 = 81 π π = 9 π r dr dθ = π r3 3 3 = 9π 15. Solution. We first compute f x = x + 6y, f y = 6x 6y, f xx =, f xy = 6, f yy = 6. The critical points are the solutions to the equations x + 6y = 6x 6y = The only solutions is x = y =. Thus (, ) is the only critical points. The value of the Hessian 48 and we conclude that (, ) is a saddle point
20 16. Solution. κ(t) = r (t) r (t). r (t) = sin t, cos t, e t and r (t) =, 1, 1. r (t) = r (t) 3 i j k cos t, sin t, e t and r (t) = 1,, 1., 1, 1 1,, 1 = det 1 1 = 1, 1, , 1, 1 = 3 and, 1, 1 = 3. κ() =. 17. Solution. Letting g(x, y) = x + y, the equations to solve are: { f = λ g g(x, y) = 1 which, written out are: x = λx 4y = λy x + y = 1 The first equation says that either x = or λ = 1. If x =, then the third equation gives that y = ±1; so (, 1) and (, 1) are possible extrema. If λ = 1, then the second equation gives that y = which, by the third equation, gives that x = ±1. This gives (1, ) and ( ) ( ) ( 1, ) as two more possible extrema. This eliminates (, ),, 1 3, and, as possible answers. Plugging in the possible extrema, we find that f(1, ) = f( 1, ) = 1 f(, 1) = f(, 1) = and so (1, ) and ( 1, ) are the minima of f subject to x + y = 1. Thus ( 1, ) is the correct answer.
21 18. Solution. The gradient is f = x, 3y which at (, 1) is 4, 3. The direction is 4 5, x + y, Solution. Let f(x, y, z) = x + ln x + y z. Then f = x + y, 1 so that f( 1, 3, 1) = 3, 1, 1. Since f( 1, 3, 1) defines a normal vector to the surface, an equation of the tangent plane is given by or 3x + y z = 1. 3, 1, 1 x + 1, y 3, z + 1 =. Solution. x x u w u w (x, y) (u, w) = det u w y y = det u w w u w u = det w u w 1 u w ln(u) w u ln(w) = u w u w u w w u u w w u ln(u) ln(w) = u w w u( 1 ln(u) ln(w) )
22 1. Solution. In spherical coordinates the sphere becomes ρ =. To convert the cone into spherical coordinates we add z to both sides of the equation defining the cone so that z = x + y + z which becomes ρ cos φ = ρ ( ) 1 Therefore φ = arccos which gives φ = π 4 or φ = 3π 4. To find the volume we compute V = 4 π dθ = π π 3π 4 π 4 ρ sin φ dρ dφ dθ = 3 π 3π 4 π 4 sin φ dφ dθ =
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 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 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 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 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 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 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 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 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 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 informationPractice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr.
1. Let F(x, y) xyi+(y 3x)j, and let be the curve r(t) ti+(3t t 2 )j for t 2. ompute F dr. Solution. F dr b a 2 2 F(r(t)) r (t) dt t(3t t 2 ), 3t t 2 3t 1, 3 2t dt t 3 dt 1 2 4 t4 4. 2. Evaluate the line
More informationMath 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 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 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 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 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 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 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 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 informationFaculty of Engineering, Mathematics and Science School of Mathematics
Faculty of Engineering, Mathematics and Science School of Mathematics GROUPS Trinity Term 06 MA3: Advanced Calculus SAMPLE EXAM, Solutions DAY PLACE TIME Prof. Larry Rolen Instructions to Candidates: Attempt
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 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 informationf(p i )Area(T i ) F ( r(u, w) ) (r u r w ) da
MAH 55 Flux integrals Fall 16 1. Review 1.1. Surface integrals. Let be a surface in R. Let f : R be a function defined on. efine f ds = f(p i Area( i lim mesh(p as a limit of Riemann sums over sampledpartitions.
More informationReview Sheet for the Final
Review Sheet for the Final Math 64 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 informationMultiple Choice. Compute the Jacobian, (u, v), of the coordinate transformation x = u2 v 4, y = uv. (a) 2u 2 + 4v 4 (b) xu yv (c) 3u 2 + 7v 6
.(5pts) y = uv. ompute the Jacobian, Multiple hoice (x, y) (u, v), of the coordinate transformation x = u v 4, (a) u + 4v 4 (b) xu yv (c) u + 7v 6 (d) u (e) u v uv 4 Solution. u v 4v u = u + 4v 4..(5pts)
More informationMath 263 Final. (b) The cross product is. i j k c. =< c 1, 1, 1 >
Math 63 Final Problem 1: [ points, 5 points to each part] Given the points P : (1, 1, 1), Q : (1,, ), R : (,, c 1), where c is a parameter, find (a) the vector equation of the line through P and Q. (b)
More informationThe Divergence Theorem
Math 1a The Divergence Theorem 1. Parameterize the boundary of each of the following with positive orientation. (a) The solid x + 4y + 9z 36. (b) The solid x + y z 9. (c) The solid consisting of all points
More 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 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 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 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 informationSOLUTIONS TO PRACTICE EXAM FOR FINAL: 1/13/2002
SOLUIONS O PRACICE EXAM OR INAL: 1/13/22 Math 21a Name: MW9 Sasha Braverman MW1 Ken Chung MW1 Jake Rasmussen MW1 WeiYang Qui MW1 Spiro Karigiannis MW11 Vivek Mohta MW11 Jake Rasmussen MW12 Ken Chung H1
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 informationMATH 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 informationMATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS
MATH 228: Calculus III (FALL 216) Sample Problems for FINAL EXAM SOLUTIONS MATH 228 Page 2 Problem 1. (2pts) Evaluate the line integral C xy dx + (x + y) dy along the parabola y x2 from ( 1, 1) to (2,
More 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 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 informationVector 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 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 1 through 12. Show all your work on the standard
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 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 informationMath 210, Final Exam, Spring 2012 Problem 1 Solution. (a) Find an equation of the plane passing through the tips of u, v, and w.
Math, Final Exam, Spring Problem Solution. Consider three position vectors (tails are the origin): u,, v 4,, w,, (a) Find an equation of the plane passing through the tips of u, v, and w. (b) Find an equation
More 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 informatione x3 dx dy. 0 y x 2, 0 x 1.
Problem 1. Evaluate by changing the order of integration y e x3 dx dy. Solution:We change the order of integration over the region y x 1. We find and x e x3 dy dx = y x, x 1. x e x3 dx = 1 x=1 3 ex3 x=
More informationMAT 211 Final Exam. 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 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 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 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 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 informationMATH 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 informationWORKSHEET #13 MATH 1260 FALL 2014
WORKSHEET #3 MATH 26 FALL 24 NOT DUE. Short answer: (a) Find the equation of the tangent plane to z = x 2 + y 2 at the point,, 2. z x (, ) = 2x = 2, z y (, ) = 2y = 2. So then the tangent plane equation
More information1. (30 points) In the xy plane, find and classify all local maxima, local minima, and saddle points of the function. f(x, y) = 3y 2 2y 3 3x 2 + 6xy.
APPM 35 FINAL EXAM FALL 13 INSTUTIONS: 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. Show your
More informationName: Date: 12/06/2018. M20550 Calculus III Tutorial Worksheet 11
1. ompute the surface integral M255 alculus III Tutorial Worksheet 11 x + y + z) d, where is a surface given by ru, v) u + v, u v, 1 + 2u + v and u 2, v 1. olution: First, we know x + y + z) d [ ] u +
More 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 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 informationMath 222 Spring 2013 Exam 3 Review Problem Answers
. (a) By the Chain ule, Math Spring 3 Exam 3 eview Problem Answers w s w x x s + w y y s (y xy)() + (xy x )( ) (( s + 4t) (s 3t)( s + 4t)) ((s 3t)( s + 4t) (s 3t) ) 8s 94st + 3t (b) By the Chain ule, w
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 informationMcGill University April Calculus 3. Tuesday April 29, 2014 Solutions
McGill University April 4 Faculty of Science Final Examination Calculus 3 Math Tuesday April 9, 4 Solutions Problem (6 points) Let r(t) = (t, cos t, sin t). i. Find the velocity r (t) and the acceleration
More informationPractice problems **********************************************************
Practice problems I will not test spherical and cylindrical coordinates explicitly but these two coordinates can be used in the problems when you evaluate triple integrals. 1. Set up the integral without
More 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 informationMajor 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 information1. Find and classify the extrema of h(x, y) = sin(x) sin(y) sin(x + y) on the square[0, π] [0, π]. (Keep in mind there is a boundary to check out).
. Find and classify the extrema of hx, y sinx siny sinx + y on the square[, π] [, π]. Keep in mind there is a boundary to check out. Solution: h x cos x sin y sinx + y + sin x sin y cosx + y h y sin x
More informationMath 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 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 3435 Homework Set 11 Solutions 10 Points. x= 1,, is in the disk of radius 1 centered at origin
Math 45 Homework et olutions Points. ( pts) The integral is, x + z y d = x + + z da 8 6 6 where is = x + z 8 x + z = 4 o, is the disk of radius centered on the origin. onverting to polar coordinates then
More informationSummary for Vector Calculus and Complex Calculus (Math 321) By Lei Li
Summary for Vector alculus and omplex alculus (Math 321) By Lei Li 1 Vector alculus 1.1 Parametrization urves, surfaces, or volumes can be parametrized. Below, I ll talk about 3D case. Suppose we use e
More informationNote: Each problem is worth 14 points except numbers 5 and 6 which are 15 points. = 3 2
Math Prelim II Solutions Spring Note: Each problem is worth points except numbers 5 and 6 which are 5 points. x. Compute x da where is the region in the second quadrant between the + y circles x + y and
More informationMATHEMATICS 200 April 2010 Final Exam Solutions
MATHEMATICS April Final Eam Solutions. (a) A surface z(, y) is defined by zy y + ln(yz). (i) Compute z, z y (ii) Evaluate z and z y in terms of, y, z. at (, y, z) (,, /). (b) A surface z f(, y) has derivatives
More informationContents. MATH 32B2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9
MATH 32B2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B2 (8W)
More informationMath 11 Fall 2016 Final Practice Problem Solutions
Math 11 Fall 216 Final Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,
More informationJim Lambers MAT 280 Fall Semester Practice Final Exam Solution
Jim Lambers MAT 8 Fall emester 67 Practice Final Exam olution. Use Lagrange multipliers to find the point on the circle x + 4 closest to the point (, 5). olution We have f(x, ) (x ) + ( 5), the square
More informationMath 20C Homework 2 Partial Solutions
Math 2C Homework 2 Partial Solutions Problem 1 (12.4.14). Calculate (j k) (j + k). Solution. The basic properties of the cross product are found in Theorem 2 of Section 12.4. From these properties, we
More informationPractice problems. m zδdv. In our case, we can cancel δ and have z =
Practice problems 1. Consider a right circular cone of uniform density. The height is H. Let s say the distance of the centroid to the base is d. What is the value d/h? We can create a coordinate system
More informationAPPM 2350 FINAL EXAM FALL 2017
APPM 25 FINAL EXAM FALL 27. ( points) Determine the absolute maximum and minimum values of the function f(x, y) = 2 6x 4y + 4x 2 + y. Be sure to clearly give both the locations and values of the absolute
More informationMath 210, Final Exam, Fall 2010 Problem 1 Solution. v cosθ = u. v Since the magnitudes of the vectors are positive, the sign of the dot product will
Math, Final Exam, Fall Problem Solution. Let u,, and v,,3. (a) Is the angle between u and v acute, obtuse, or right? (b) Find an equation for the plane through (,,) containing u and v. Solution: (a) The
More informationBi. lkent Calculus II Exams
Bi. lkent Calculus II Exams 988208 Spring 208 Midterm I........ Spring 208 Midterm II....... 2 Spring 207 Midterm I........ 4 Spring 207 Midterm II....... 5 Spring 207 Final........... 7 Spring 206 Midterm
More informationPractice Problems for the Final Exam
Math 114 Spring 2017 Practice Problems for the Final Exam 1. The planes 3x + 2y + z = 6 and x + y = 2 intersect in a line l. Find the distance from the origin to l. (Answer: 24 3 ) 2. Find the area of
More information12/19/2009, FINAL PRACTICE VII Math 21a, Fall Name:
12/19/29, FINAL PRATIE VII Math 21a, Fall 29 Name: MWF 9 Jameel AlAidroos MWF 1 Andrew ottonlay MWF 1 Oliver Knill MWF 1 HT Yau MWF 11 Ana araiani MWF 11 hris Phillips MWF 11 Ethan Street MWF 12 Toby
More informationTangent Plane. Linear Approximation. The Gradient
Calculus 3 Lia Vas Tangent Plane. Linear Approximation. The Gradient The tangent plane. Let z = f(x, y) be a function of two variables with continuous partial derivatives. Recall that the vectors 1, 0,
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 informationMultiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015
Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction
More informationAnswer sheet: Final exam for Math 2339, Dec 10, 2010
Answer sheet: Final exam for Math 9, ec, Problem. Let the surface be z f(x,y) ln(y + cos(πxy) + e ). (a) Find the gradient vector of f f(x,y) y + cos(πxy) + e πy sin(πxy), y πx sin(πxy) (b) Evaluate f(,
More informationMATH 261 FINAL EXAM PRACTICE PROBLEMS
MATH 261 FINAL EXAM PRACTICE PROBLEMS These practice problems are pulled from the final exams in previous semesters. The 2hour final exam typically has 89 problems on it, with 45 coming from the postexam
More information(You may need to make a sin / costype trigonometric substitution.) Solution.
MTHE 7 Problem Set Solutions. As a reminder, a torus with radii a and b is the surface of revolution of the circle (x b) + z = a in the xzplane about the zaxis (a and b are positive real numbers, with
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 information234 Review Sheet 2 Solutions
4 Review Sheet Solutions. Find all the critical points of the following functions and apply the second derivative test. (a) f(x, y) (x y)(x + y) ( ) x f + y + (x y)x (x + y) + (x y) ( ) x + ( x)y x + x
More 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 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 informationSummary 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 informationMath 10C  Fall Final Exam
Math 1C  Fall 217  Final Exam Problem 1. Consider the function f(x, y) = 1 x 2 (y 1) 2. (i) Draw the level curve through the point P (1, 2). Find the gradient of f at the point P and draw the gradient
More informationG 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 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 informationPrint Your Name: Your Section:
Print Your Name: Your Section: Mathematics 1c. Practice Final Solutions This exam has ten questions. J. Marsden You may take four hours; there is no credit for overtime work No aids (including notes, books,
More informationDivergence Theorem December 2013
Divergence Theorem 17.3 11 December 2013 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:
More information4 Partial Differentiation
4 Partial Differentiation Many equations in engineering, physics and mathematics tie together more than two variables. For example Ohm s Law (V = IR) and the equation for an ideal gas, PV = nrt, which
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 informationDivergence Theorem Fundamental Theorem, Four Ways. 3D Fundamental Theorem. Divergence Theorem
Divergence Theorem 17.3 11 December 213 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:
More informationDEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS. MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 2018
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 208 Version A refers to the regular exam and Version B to the makeup. Version A. A particle
More information1 Functions of Several Variables Some Examples Level Curves / Contours Functions of More Variables... 6
Contents 1 Functions of Several Variables 1 1.1 Some Examples.................................. 2 1.2 Level Curves / Contours............................. 4 1.3 Functions of More Variables...........................
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 information