Jim Lambers MAT 280 Fall Semester Practice Final Exam Solution

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Jim Lambers MAT 280 Fall Semester Practice Final Exam Solution"

Transcription

1 Jim Lambers MAT 8 Fall emester 6-7 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 of the distance from (x, ) to (, 5), as the objective function that is to be minimized. The constraints are given b g(x, ) x + 4. We then have f x λg x, f λg, where λ is the Lagrange multiplier. This gives (x ) λx, ( 5) λ, x + 4. From the first two equations, we obtain two distinct expressions for λ. Equating them ields x x 5 which simplifies to 5x. ubstituting this into the constraint ields 6x 4, or x ±.9. We then have the points (x, ) ±(.9,.966). ubstituting these into f(x, ) gives the minimizer (.9,.966).. Evaluate the iterated integral olution We have π π sin x dz d dx sin x dz d dx. π sin x dx cos x π ( ( ) ( )) u/. z d dz d d u / du, u, du d. Evaluate + x da, where is the region bounded b x, and x.

2 olution We have + x da 4 x + x d dx + x x + x 4 ln u ln. 4 x x + x dx d dx dx u du, u + x, du x dx 4. Evaluate (x + ) / da, where is the region in the first quadrant bounded b the lines and x, and the circle x + 9. olution Let f(x, ) (x + ) /. onverting to polar coordinates, we obtain (x + ) / da π/ π/ π/ π/ π r 5 5 8π 5. f(r cos θ, r sin θ)r dr (r ) / r dr r 4 dr r 4 dr The upper limit of θ, π/, is obtained using the fact that the slope of the line x,, is equal to tan π. 5. Evaluate H z x + + z dv, where H is the solid hemisphere that lies above the x-plane and has center at the origin with radius.

3 olution Using spherical coordinates, we obtain H z x + + z dv π/ π π u π 9. π/ u du ρ6 6 6 (ρ cos φ) ρ(ρ sin φ) dρ dφ cos φ sin φ dφ ρ 5 dρ, u cos φ, du sin φ dφ 6. Find the volume of the solid bounded b the clinder x + 4 and the planes z and + z. olution Using clindrical coordinates, we obtain 7. Use the transformation to evaluate V r sin θ ( r r dz dr r( r sin θ) dr ) r sin θ 6 8 sin θ (6θ + 8 ) π cos θ π. u x, R v x + x x + da, where R is the square with vertices (, ), (, ), (, ), and (, ). olution olving the above equations x and, we obtain x (u + v), (v u). This ields (x, ) (u, v) x u v u x v ( ).

4 ubstituting the vertices of the square into the change of variable ields the transformed vertices in uv-space: (, ), (, ), (, 4), and (, 4). This ields the integral R x x + da 4 4 u v v dv ln v 4 (x, ) (u, v) du dv u u du (ln 4 ln )( ) ln. 8. Evaluate the line integral x dx + e d + xz dz, where is given b r(t) t 4, t, t, t. olution From r (t) 4t, t, t, we obtain x dx + e d + xz dz 9. how that the vector field x(t)(t) dx dt + e(t) d dt + x(t)z(z)dz dt dt t (4t ) + e t (t) + t 7 (t ) dt 4t 6 + te t + t 9 dt ( ) 4t et + t ( e + ). F(x,, z) e, xe + e z, e z is conservative, and use this fact to evaluate F dr where is the line segment from (,, ) to (4,, ). olution A vector field F P, Q, R is conservative if From R Q z, P z R x, Q x P. R e z Q z, P z R x, Q x e P, we find that F is in fact conservative. To evaluate the line integral efficientl, we need to find a function f such that f F. To that end, we obtain f(x,, z) P dx xe + g(, z).

5 The requirement that f Q ields the equation g (, z) Q(x,, z) (xe ) e z. olving the equation for g ields g(, z) e z + h(z). The requirement that f z R ields the equation h (z) R(x,, z) (xe + e z ) z. It follows that h(z) K where K is an arbitrar constant, and therefore f(x,, z) xe + e z + K. From the Fundamental Theorem of Line Integrals, we obtain F dr f dr f(4,, ) f(,, ) (4e + e ) (e + e ).. Use Green s Theorem to evaluate x dx x d, where is the circle x + 4 with counterclockwise orientation. olution Let { (x, ) x + 4 } be the interior of. Then, b Green s Theorem, x dx x d ( x ) x (x ) da x da. onverting to polar coordinates, we obtain x dx x d π 8π. ( r4 4 ( r )r dr ) r dr. If f(x,, z) and g(x,, z) are twice differentiable functions, show that (fg) f g + g f + f g, where f ( f).

6 olution We have (fg) ( (fg)) (fg) x, (fg), (fg) z f x g + fg x, f g + fg, f z g + fg z (f x g + fg x ) x + (f g + fg ) + (f z g + fg z ) z f xx g + f x g x + fg xx + f g + f g + fg + f zz g + f z g z + fg zz f(g xx + g + g zz ) + g(f xx + f + f zz ) + (f x g x + f g + f z g z ) f ( g) + g ( f) + f g f g + g f + f g. In the last steps, we have used the fact that ( f) f xx + f + f xx.. Evaluate the surface integral (x z + z) d, where is the part of the plane z 4 + x + that lies inside the clinder x + 4. olution We use the parametric equations for which x u cos v, u sin v, z 4 + u cos v + u sin v, u, v π, r u cos v, sin v, cos v + sin v, r v u sin v, u cos v, u sin v + u cos v, This ields (x z + z) d r u r v u, u, u, r u r v u. [(u cos v) + (u sin v) ](4 + u cos v + u sin v) u du dv 4u + u 4 cos v + u 4 sin v du dv [u 4 + u5 (cos v + sin v) (cos v + sin v) dv 5 ] dv 6 (π) + 5 π. (sin v cos v) π. Evaluate the surface integral F d, where F(x,, z) xz,, x

7 and is the sphere x + + z 4 with outward orientation. olution We use spherical coordinates which ields x sin φ cos θ, sin φ sin θ, z cos φ, θ π, φ π, r φ cos φ cos θ, cos φ sin θ, sin φ, r θ sin φ sin θ, sin φ cos θ,, r φ r θ 4 sin φ sin φ cos θ, sin φ sin θ, cos φ, which has outward orientation since 4 sin φ for φ π. We then have F d 8 6 π π 4 8 sin φ cos θ cos φ, sin θ, cos θ 4 sin φ sin φ cos θ, sin φ sin θ, cos φ dφ sin φ( cos θ sin φ cos φ sin θ sin φ + cos θ cos φ) dφ cos θ cos θ π π + cos θ 8 cos θ 6π sin4 π φ 4 + 6π ( 6π cos φ cos φ ( 6π + ) 64π. π sin φ cos φ dφ 6 sin θ sin φ dφ + sin φ cos φ dφ π ) π u du Here, we have used the trigonometric identities (u sin φ, du cos φ dφ) ( cos φ) sin φ dφ + 4 sin θ π π ( v ) dv (v cos φ, dv sin φ dφ) sin φ cos φ dφ cos θ + cos θ, sin θ cos θ. 4. Use tokes Theorem to evaluate F dr, where F(x,, z) x, z, xz and is the triangle with vertices (,, ), (,, ) and (,, ), oriented counterclockwise as viewed from above. Hint: to obtain an equation for the surface enclosed b, compute the equation of a plane containing the vertices. olution Using the given vertices, it can be determined that is contained within the plane x + + z. We therefore describe using the parametric equations x u, v, z u v, u, v u,

8 which ields and the normal vector r u,,, r v,,, r u r v,,, which is consistent with the counterclockwise orientation of (that is, when traversing such that this normal vector, which points upward, is visible, then the region is on the left. Appling tokes Theorem ields From We then have and thus We conclude that F dr u curl F(u, v) (r u r v ) dv du. curl F R Q z, P z R x, Q x P (xz) (z) z, (x) z (xz) x, (z) x (x), z, x. curl F (r u r v ), z, x,, (x + + z), curl F(u, v) (r u r v ) (u + v + u v). F dr u dv du A(), where is the triangle { (u, v) u, v u }. This triangle has base and height, which ields F dr. 5. Use the ivergence Theorem to evaluate the surface integral F d, where F(x,, z) x,, z and is the surface of the solid E bounded b the clinder x + and the planes z and z. olution Using clindrical coordinates, we obtain F d div F dv E [(x ) x + ( ) + (z ) z ] dv E (x + + z ) dv E (r + z )r dz dr (r z + r z ) dr

9 ( ) r 4 6π + 4r π. r + 8 r dr

Jim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt

Jim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt Jim Lambers MAT 28 ummer emester 212-1 Practice Final Exam olution 1. Evaluate the line integral xy dx + e y dy + xz dz, where is given by r(t) t 4, t 2, t, t 1. olution From r (t) 4t, 2t, t 2, we obtain

More information

1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r (t) = 3 cos t, 0, 3 sin t, r ( 3π

1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r (t) = 3 cos t, 0, 3 sin t, r ( 3π 1. a) 5 points) Find the unit tangent and unit normal vectors T and N to the curve at the point P 3, 3π, r t) 3 cos t, 4t, 3 sin t 3 ). b) 5 points) Find curvature of the curve at the point P. olution:

More information

Solutions to Sample Questions for Final Exam

Solutions to Sample Questions for Final Exam olutions to ample Questions for Final Exam Find the points on the surface xy z 3 that are closest to the origin. We use the method of Lagrange Multipliers, with f(x, y, z) x + y + z for the square of the

More information

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

SOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours) SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please

More information

MATH 52 FINAL EXAM SOLUTIONS

MATH 52 FINAL EXAM SOLUTIONS MAH 5 FINAL EXAM OLUION. (a) ketch the region R of integration in the following double integral. x xe y5 dy dx R = {(x, y) x, x y }. (b) Express the region R as an x-simple region. R = {(x, y) y, x y }

More information

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

One side of each sheet is blank and may be used as scratch paper. Math 244 Spring 2017 (Practice) Final 5/11/2017 Time Limit: 2 hours Name: No calculators or notes are allowed. One side of each sheet is blank and may be used as scratch paper. heck your answers whenever

More information

7a3 2. (c) πa 3 (d) πa 3 (e) πa3

7a3 2. (c) πa 3 (d) πa 3 (e) πa3 1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin

More information

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

Calculus III. Math 233 Spring Final exam May 3rd. Suggested solutions alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.

More information

Practice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr.

Practice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr. 1. Let F(x, y) xyi+(y 3x)j, and let be the curve r(t) ti+(3t t 2 )j for t 2. ompute F dr. Solution. F dr b a 2 2 F(r(t)) r (t) dt t(3t t 2 ), 3t t 2 3t 1, 3 2t dt t 3 dt 1 2 4 t4 4. 2. Evaluate the line

More information

1. If the line l has symmetric equations. = y 3 = z+2 find a vector equation for the line l that contains the point (2, 1, 3) and is parallel to l.

1. If the line l has symmetric equations. = y 3 = z+2 find a vector equation for the line l that contains the point (2, 1, 3) and is parallel to l. . If the line l has symmetric equations MA 6 PRACTICE PROBLEMS x = y = z+ 7, find a vector equation for the line l that contains the point (,, ) and is parallel to l. r = ( + t) i t j + ( + 7t) k B. r

More information

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

MAT 211 Final Exam. Spring Jennings. Show your work! MAT 211 Final Exam. pring 215. Jennings. how your work! Hessian D = f xx f yy (f xy ) 2 (for optimization). Polar coordinates x = r cos(θ), y = r sin(θ), da = r dr dθ. ylindrical coordinates x = r cos(θ),

More information

Math 234 Final Exam (with answers) Spring 2017

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

More information

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

Name: Date: 12/06/2018. M20550 Calculus III Tutorial Worksheet 11 1. ompute the surface integral M255 alculus III Tutorial Worksheet 11 x + y + z) d, where is a surface given by ru, v) u + v, u v, 1 + 2u + v and u 2, v 1. olution: First, we know x + y + z) d [ ] u +

More information

MATHS 267 Answers to Stokes Practice Dr. Jones

MATHS 267 Answers to Stokes Practice Dr. Jones MATH 267 Answers to tokes Practice Dr. Jones 1. Calculate the flux F d where is the hemisphere x2 + y 2 + z 2 1, z > and F (xz + e y2, yz, z 2 + 1). Note: the surface is open (doesn t include any of the

More information

Review problems for the final exam Calculus III Fall 2003

Review 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 information

Math Review for Exam 3

Math Review for Exam 3 1. ompute oln: (8x + 36xy)ds = Math 235 - Review for Exam 3 (8x + 36xy)ds, where c(t) = (t, t 2, t 3 ) on the interval t 1. 1 (8t + 36t 3 ) 1 + 4t 2 + 9t 4 dt = 2 3 (1 + 4t2 + 9t 4 ) 3 2 1 = 2 3 ((14)

More information

McGill University April 16, Advanced Calculus for Engineers

McGill University April 16, Advanced Calculus for Engineers McGill University April 16, 2014 Faculty of cience Final examination Advanced Calculus for Engineers Math 264 April 16, 2014 Time: 6PM-9PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer

More information

M273Q Multivariable Calculus Spring 2017 Review Problems for Exam 3

M273Q 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 information

Math 233. Practice Problems Chapter 15. i j k

Math 233. Practice Problems Chapter 15. i j k Math 233. Practice Problems hapter 15 1. ompute the curl and divergence of the vector field F given by F (4 cos(x 2 ) 2y)i + (4 sin(y 2 ) + 6x)j + (6x 2 y 6x + 4e 3z )k olution: The curl of F is computed

More information

MATH 332: Vector Analysis Summer 2005 Homework

MATH 332: Vector Analysis Summer 2005 Homework MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,

More information

MAT 211 Final Exam. Fall Jennings.

MAT 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 information

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

Note: Each problem is worth 14 points except numbers 5 and 6 which are 15 points. = 3 2 Math Prelim II Solutions Spring Note: Each problem is worth points except numbers 5 and 6 which are 5 points. x. Compute x da where is the region in the second quadrant between the + y circles x + y and

More information

Math 11 Fall 2016 Final Practice Problem Solutions

Math 11 Fall 2016 Final Practice Problem Solutions Math 11 Fall 216 Final Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,

More information

1. 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).

1. 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 information

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

Sections minutes. 5 to 10 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed. MTH 34 Review for Exam 4 ections 16.1-16.8. 5 minutes. 5 to 1 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed. Review for Exam 4 (16.1) Line

More information

Solutions to old Exam 3 problems

Solutions to old Exam 3 problems Solutions to old Exam 3 problems Hi students! I am putting this version of my review for the Final exam review here on the web site, place and time to be announced. Enjoy!! Best, Bill Meeks PS. There are

More information

In general, the formula is S f ds = D f(φ(u, v)) Φ u Φ v da. To compute surface area, we choose f = 1. We compute

In general, the formula is S f ds = D f(φ(u, v)) Φ u Φ v da. To compute surface area, we choose f = 1. We compute alculus III Test 3 ample Problem Answers/olutions 1. Express the area of the surface Φ(u, v) u cosv, u sinv, 2v, with domain u 1, v 2π, as a double integral in u and v. o not evaluate the integral. In

More information

D = 2(2) 3 2 = 4 9 = 5 < 0

D = 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 information

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

1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 1. The value of the double integral (a) 15 26 (b) 15 8 (c) 75 (d) 105 26 5 4 0 1 1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 2. What is the value of the double integral interchange the order

More information

HOMEWORK 8 SOLUTIONS

HOMEWORK 8 SOLUTIONS HOMEWOK 8 OLUTION. Let and φ = xdy dz + ydz dx + zdx dy. let be the disk at height given by: : x + y, z =, let X be the region in 3 bounded by the cone and the disk. We orient X via dx dy dz, then by definition

More information

Math 221 Examination 2 Several Variable Calculus

Math 221 Examination 2 Several Variable Calculus Math Examination Spring Instructions These problems should be viewed as essa questions. Before making a calculation, ou should explain in words what our strateg is. Please write our solutions on our own

More information

Solutions to the Final Exam, Math 53, Summer 2012

Solutions to the Final Exam, Math 53, Summer 2012 olutions to the Final Exam, Math 5, ummer. (a) ( points) Let be the boundary of the region enclosedby the parabola y = x and the line y = with counterclockwise orientation. alculate (y + e x )dx + xdy.

More information

Problem Points S C O R E

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

More information

Math Exam IV - Fall 2011

Math Exam IV - Fall 2011 Math 233 - Exam IV - Fall 2011 December 15, 2011 - Renato Feres NAME: STUDENT ID NUMBER: General instructions: This exam has 16 questions, each worth the same amount. Check that no pages are missing and

More information

SOME PROBLEMS YOU SHOULD BE ABLE TO DO

SOME PROBLEMS YOU SHOULD BE ABLE TO DO OME PROBLEM YOU HOULD BE ABLE TO DO I ve attempted to make a list of the main calculations you should be ready for on the exam, and included a handful of the more important formulas. There are no examples

More information

18.1. Math 1920 November 29, ) Solution: In this function P = x 2 y and Q = 0, therefore Q. Converting to polar coordinates, this gives I =

18.1. Math 1920 November 29, ) Solution: In this function P = x 2 y and Q = 0, therefore Q. Converting to polar coordinates, this gives I = Homework 1 elected olutions Math 19 November 9, 18 18.1 5) olution: In this function P = x y and Q =, therefore Q x P = x. We obtain the following integral: ( Q I = x ydx = x P ) da = x da. onverting to

More information

MATH H53 : Final exam

MATH H53 : Final exam MATH H53 : Final exam 11 May, 18 Name: You have 18 minutes to answer the questions. Use of calculators or any electronic items is not permitted. Answer the questions in the space provided. If you run out

More information

APPM 2350 Final Exam points Monday December 17, 7:30am 10am, 2018

APPM 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 information

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

x + 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 information

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

Practice problems ********************************************************** Practice problems I will not test spherical and cylindrical coordinates explicitly but these two coordinates can be used in the problems when you evaluate triple integrals. 1. Set up the integral without

More information

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

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

More information

Problem 1. Use a line integral to find the plane area enclosed by the curve C: r = a cos 3 t i + b sin 3 t j (0 t 2π). Solution: We assume a > b > 0.

Problem 1. Use a line integral to find the plane area enclosed by the curve C: r = a cos 3 t i + b sin 3 t j (0 t 2π). Solution: We assume a > b > 0. MATH 64: FINAL EXAM olutions Problem 1. Use a line integral to find the plane area enclosed by the curve C: r = a cos 3 t i + b sin 3 t j ( t π). olution: We assume a > b >. A = 1 π (xy yx )dt = 3ab π

More information

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

(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere. MATH 4 FINAL EXAM REVIEW QUESTIONS Problem. a) The points,, ) and,, 4) are the endpoints of a diameter of a sphere. i) Determine the center and radius of the sphere. ii) Find an equation for the sphere.

More information

e x3 dx dy. 0 y x 2, 0 x 1.

e 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 information

McGill University April Calculus 3. Tuesday April 29, 2014 Solutions

McGill 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 information

51. General Surface Integrals

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

More information

Multiple 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

Multiple 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 information

1. For each function, find all of its critical points and then classify each point as a local extremum or saddle point.

1. 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 information

Math 3435 Homework Set 11 Solutions 10 Points. x= 1,, is in the disk of radius 1 centered at origin

Math 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 information

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9 MATH 32B-2 (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 32B-2 (8W)

More information

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

Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work. Exam 3 Math 850-007 Fall 04 Odenthal Name: Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.. Evaluate the iterated integral

More information

Math 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 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 information

MATH 2400 Final Exam Review Solutions

MATH 2400 Final Exam Review Solutions MATH Final Eam eview olutions. Find an equation for the collection of points that are equidistant to A, 5, ) and B6,, ). AP BP + ) + y 5) + z ) 6) y ) + z + ) + + + y y + 5 + z 6z + 9 + 6 + y y + + z +

More information

Review Sheet for the Final

Review Sheet for the Final Review Sheet for the Final Math 6-4 4 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And the absence

More information

Math 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 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 information

Direction of maximum decrease = P

Direction 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 information

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

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

More information

Assignment 11 Solutions

Assignment 11 Solutions . Evaluate Math 9 Assignment olutions F n d, where F bxy,bx y,(x + y z and is the closed surface bounding the region consisting of the solid cylinder x + y a and z b. olution This is a problem for which

More information

Ying-Ying Tran 2016 May 10 Review

Ying-Ying Tran 2016 May 10 Review MATH 920 Final review Ying-Ying Tran 206 Ma 0 Review hapter 3: Vector geometr vectors dot products cross products planes quadratic surfaces clindrical and spherical coordinates hapter 4: alculus of vector-valued

More information

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

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

More information

MATHEMATICS 200 December 2014 Final Exam Solutions

MATHEMATICS 200 December 2014 Final Exam Solutions MATHEMATICS 2 December 214 Final Eam Solutions 1. Suppose that f,, z) is a function of three variables and let u 1 6 1, 1, 2 and v 1 3 1, 1, 1 and w 1 3 1, 1, 1. Suppose that at a point a, b, c), Find

More information

Summary of various integrals

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

More information

Line and Surface Integrals. Stokes and Divergence Theorems

Line and Surface Integrals. Stokes and Divergence Theorems Math Methods 1 Lia Vas Line and urface Integrals. tokes and Divergence Theorems Review of urves. Intuitively, we think of a curve as a path traced by a moving particle in space. Thus, a curve is a function

More information

Review for the First Midterm Exam

Review 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 information

Archive of Calculus IV Questions Noel Brady Department of Mathematics University of Oklahoma

Archive of Calculus IV Questions Noel Brady Department of Mathematics University of Oklahoma Archive of Calculus IV Questions Noel Brady Department of Mathematics University of Oklahoma This is an archive of past Calculus IV exam questions. You should first attempt the questions without looking

More information

APPM 2350 FINAL EXAM FALL 2017

APPM 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 information

Without fully opening the exam, check that you have pages 1 through 10.

Without 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 information

LINE AND SURFACE INTEGRALS: A SUMMARY OF CALCULUS 3 UNIT 4

LINE AND SURFACE INTEGRALS: A SUMMARY OF CALCULUS 3 UNIT 4 LINE AN URFAE INTEGRAL: A UMMARY OF ALULU 3 UNIT 4 The final unit of material in multivariable calculus introduces many unfamiliar and non-intuitive concepts in a short amount of time. This document attempts

More information

Final Exam Review Sheet : Comments and Selected Solutions

Final Exam Review Sheet : Comments and Selected Solutions MATH 55 Applied Honors alculus III Winter Final xam Review heet : omments and elected olutions Note: The final exam will cover % among topics in chain rule, linear approximation, maximum and minimum values,

More information

MATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS

MATH 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 information

Math 31CH - Spring Final Exam

Math 31CH - Spring Final Exam Math 3H - Spring 24 - Final Exam Problem. The parabolic cylinder y = x 2 (aligned along the z-axis) is cut by the planes y =, z = and z = y. Find the volume of the solid thus obtained. Solution:We calculate

More information

Math 23b Practice Final Summer 2011

Math 23b Practice Final Summer 2011 Math 2b Practice Final Summer 211 1. (1 points) Sketch or describe the region of integration for 1 x y and interchange the order to dy dx dz. f(x, y, z) dz dy dx Solution. 1 1 x z z f(x, y, z) dy dx dz

More information

Math 263 Final. (b) The cross product is. i j k c. =< c 1, 1, 1 >

Math 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 information

Practice Problems for the Final Exam

Practice 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 information

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

e x2 dxdy, e x2 da, e x2 x 3 dx = e STS26-4 Calculus II: The fourth exam Dec 15, 214 Please show all your work! Answers without supporting work will be not given credit. Write answers in spaces provided. You have 1 hour and 2minutes to complete

More information

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

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

More information

Name: SOLUTIONS Date: 11/9/2017. M20550 Calculus III Tutorial Worksheet 8

Name: SOLUTIONS Date: 11/9/2017. M20550 Calculus III Tutorial Worksheet 8 Name: SOLUTIONS Date: /9/7 M55 alculus III Tutorial Worksheet 8. ompute R da where R is the region bounded by x + xy + y 8 using the change of variables given by x u + v and y v. Solution: We know R is

More information

Solutions for the Practice Final - Math 23B, 2016

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

More information

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

( ) ( ) ( ) ( ) Calculus III - Problem Drill 24: Stokes and Divergence Theorem alculus III - Problem Drill 4: tokes and Divergence Theorem Question No. 1 of 1 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as needed () Pick the 1. Use

More information

MATHEMATICS 317 April 2017 Final Exam Solutions

MATHEMATICS 317 April 2017 Final Exam Solutions MATHEMATI 7 April 7 Final Eam olutions. Let r be the vector field r = îı + ĵj + z ˆk and let r be the function r = r. Let a be the constant vector a = a îı + a ĵj + a ˆk. ompute and simplif the following

More information

MATHEMATICS 200 December 2013 Final Exam Solutions

MATHEMATICS 200 December 2013 Final Exam Solutions MATHEMATICS 2 December 21 Final Eam Solutions 1. Short Answer Problems. Show our work. Not all questions are of equal difficult. Simplif our answers as much as possible in this question. (a) The line L

More information

v n ds where v = x z 2, 0,xz+1 and S is the surface that

v n ds where v = x z 2, 0,xz+1 and S is the surface that M D T P. erif the divergence theorem for d where is the surface of the sphere + + = a.. Calculate the surface integral encloses the solid region + +,. (a directl, (b b the divergence theorem. v n d where

More information

MA 441 Advanced Engineering Mathematics I Assignments - Spring 2014

MA 441 Advanced Engineering Mathematics I Assignments - Spring 2014 MA 441 Advanced Engineering Mathematics I Assignments - Spring 2014 Dr. E. Jacobs The main texts for this course are Calculus by James Stewart and Fundamentals of Differential Equations by Nagle, Saff

More information

LINE AND SURFACE INTEGRALS: A SUMMARY OF CALCULUS 3 UNIT 4

LINE AND SURFACE INTEGRALS: A SUMMARY OF CALCULUS 3 UNIT 4 LINE AN URFAE INTEGRAL: A UMMARY OF ALULU 3 UNIT 4 The final unit of material in multivariable calculus introduces many unfamiliar and non-intuitive concepts in a short amount of time. This document attempts

More information

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

Math 32B Discussion Session Week 10 Notes March 14 and March 16, 2017 Math 3B iscussion ession Week 1 Notes March 14 and March 16, 17 We ll use this week to review for the final exam. For the most part this will be driven by your questions, and I ve included a practice final

More information

Ma 1c Practical - Solutions to Homework Set 7

Ma 1c Practical - Solutions to Homework Set 7 Ma 1c Practical - olutions to omework et 7 All exercises are from the Vector Calculus text, Marsden and Tromba (Fifth Edition) Exercise 7.4.. Find the area of the portion of the unit sphere that is cut

More information

Math 11 Fall 2007 Practice Problem Solutions

Math 11 Fall 2007 Practice Problem Solutions Math 11 Fall 27 Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,

More information

Disclaimer: 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 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 information

DO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START

DO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START Math 265 Student name: KEY Final Exam Fall 23 Instructor & Section: This test is closed book and closed notes. A (graphing) calculator is allowed for this test but cannot also be a communication device

More information

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

MTH 234 Exam 2 November 21st, Without fully opening the exam, check that you have pages 1 through 12. Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 12. Show all your work on the standard

More information

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

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

More information

MLC Practice Final Exam

MLC Practice Final Exam Name: Section: Recitation/Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 13. Show all your work on the standard

More information

1. (30 points) In the x-y 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.

1. (30 points) In the x-y 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 information

MATH2321, Calculus III for Science and Engineering, Fall Name (Printed) Print your name, the date, and then sign the exam on the line

MATH2321, Calculus III for Science and Engineering, Fall Name (Printed) Print your name, the date, and then sign the exam on the line MATH2321, Calculus III for Science and Engineering, Fall 218 1 Exam 2 Name (Printed) Date Signature Instructions STOP. above. Print your name, the date, and then sign the exam on the line This exam consists

More information

Math 234 Exam 3 Review Sheet

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

More information

McGill University April 20, Advanced Calculus for Engineers

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

More information

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

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

More information

36. Double Integration over Non-Rectangular Regions of Type II

36. Double Integration over Non-Rectangular Regions of Type II 36. Double Integration over Non-Rectangular Regions of Type II When establishing the bounds of a double integral, visualize an arrow initially in the positive x direction or the positive y direction. A

More information

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

(a) 0 (b) 1/4 (c) 1/3 (d) 1/2 (e) 2/3 (f) 3/4 (g) 1 (h) 4/3 Math 114 Practice Problems for Test 3 omments: 0. urface integrals, tokes Theorem and Gauss Theorem used to be in the Math40 syllabus until last year, so we will look at some of the questions from those

More information

MATHEMATICS 200 April 2010 Final Exam Solutions

MATHEMATICS 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 information