Ma 227 Final Exam Solutions 12/17/07

Transcription

1 Ma 7 Final Exam olutions /7/7 Name: Lecture ection: I pledge my honor that I have abided by the tevens Honor ystem. You may not use a calculator, cell phone, or computer while taking this exam. All work must be shown to obtain full credit. Credit will not be given for work not reasonably supported. When you finish, be sure to sign the pledge. Directions: Answer all questions. The point value of each problem is indicated. If you need more work space, continue the problem you are doing on the other side of the page it is on. There is a table of integrals at the end of the exam. core on Problem # # # #4 #5 #6 #7 #8 Total

2 Problem a) ( points) Find the eigenvalues and eigenvectors of A r r 4 r 4r rr 4 r Thus the eigenvalues are r,4. The system A rix is rx x x rx For r we have x x or x x so the eigenvector corresponding to r is For r 4 we have Thus the eigenvector for r 4is x x or x x. NB check, eigenvectors: b) ( points) olve the nonhomogeneous problem x t Axt, e t e t 4 where A is the matrix above in a). Let Then the DE implies x h c e t x p c e 4t ae t be t

3 or Therefore or ae t be t ae t be t ae t be t e t ae t be t e t ae t be t a a b b a b a b a b e t e t, olution is: a 5,b 4 5 x p 5 e t 4 5 e t and xt x h x p c c e 4t 5 e t 4 5 e t NB check x x x x e t x x e t, Exact solution is: x t C 5 e t C 4 e 4t,x t C 4 e 4t 4 5 e t C Problem a) ( points) Give an integral in cylindrical coordinates for the volume of the solid region bounded by the cylinder x y 4 and the planes z andy z. ketch the solid region. Do not evaluate this integral. x y 4

4 b)( points) Evaluate the surface integral Volume F V rdzdrd r sin rdzdrd nd F d,where F x,y,z e xy coszi x zj xyk and is the hemisphere x y z oriented in the direction of the positive x axis. We use tokes theorem. Thus F nd F d F dr C The boundary of is the circle y z,x. Therefore C : x, yt cost, zt sin t t so rt i costj sintk r t sintj costk F,cost,sint cossin ti j k so Problem a) ( points) Evaluate F dr dt C x cos y dydx We reverse the order of integration. Now y goes from y x to y andx goes from to. 4

5 Therefore, the region of integration is x Therefore y x cos y dydx y cos y dxdy ycos y dy sin y x sin b) ( points) Give two different integral expressions for the area of the region bounded by the parabolas x y and x 8 y. Do not evaluate these expressions. Be sure to sketch the area. x y y x

6 The curves intersect when y 8 y that is, when y. Thus at the points 4, and 4, Therefore Area da R 4 x x 8 y dxdy y 8 8 x dydx 4 8 x dydx Problem 4 a) ( points) Evaluate the line integral C zdx xdy ydz where C is given by x t,y t,z t, t. olutions: Here rt t i t j t k r t ti t j tk so F x,y,z zi xj yk F t t i t j t k C zdx xdy ydz t t t t t t dt t 5t 4 dt t4 t5 b) (5 points) Find the inverse of the matrix A and then use it solve the system AX. R R 6 R R

7 R R R R Therefore A NB check:, inverse: We note that the solution of AX Problem 5 a) (5 points) is X A Verify the divergence theorem for the vector field F x i y j zk over the sphere x y z a. We have to show that F d F N ds divf dv V We begin with the surface integral and use spherical coordinates to parametrize it. The a and r, x i y j zk a sin cos i a sin sin j a cosk, Therefore r a coscos i a cossin j a sink and r a sinsin i a sin cos j k 7

8 N r r i j k a coscos a cossin a sin a sin sin a sincos a sin cos i a sin sin j a cossin cos a cossin sin k Note that at Also so a sin cos i a sin sin j a cossink, N a j which points outward, so we use this N. F, a sincos i a sinsin j a cosk F d F N ds a sin cos a sin sin a cos sin dd b) ( points) Let a sin cos sin dd a cos 4 cos cos a 4 4 d a d 4a divf divf a dv sin ddd V a sin dd a cos d 4a D 4 d Find e Dt. e Dt e t e 4t e t 8

9 NB check: e Problem 6 4 t e t e 4t e t a)( points) Find a function x,y,z such that F x,y,z sinyi xcosyj z sin zk x siny y xcosy z z sin z Thus x siny xsiny gy,z y xcosy g y xcosy Thus g y andg hz so xsin y hz Then we have z h z z sinz so hz z cosz K. Hence xsin y z cosz K 6b)(5 points) Verify that Green s Theorem is true for the line integral ydx x y dy C where C is the ellipse 4x 9y 6 with counterclockwise orientation. Px,ydx Qx,ydy P y Q x da R To calculate the line integral around the ellipse x t. Thendx sintdt,dy costdt. C 9 y 4 6 we let xt cost and yt sint, 9

10 C ydx x y dy sint sint cost 4sin t cost dt 6sin t 6cos t 8sin tcost dt 6 cos t sin t 8sin tcost dt Also P y Q x so sint 8 sin t P y Q x da R R da Problem 7 a) ( points) Consider the two curves x y y and x y x. Give an integral or integrals in polar coordinates for the area between the two curves. Be sure to sketch the area between the two curves. Do not evaluate the integral or integrals. The two curves are given by r sin and The graphs are given below. r cos y - - x A rdrd R where R is the region common to both circles. The two circles intersect when cos sin or when tan

11 That is, at 4. r goes from to the circle r sin, for 4. Thus we need two integrals to express the area. 4 and from to r cos for A 4 sin rdrd 4 cos rdrd b) ( points) Evaluate zdv E where E lies between the spheres x y z andx y z 4inthefirstoctant. We use spherical coordinates. The equation of the unit sphere is whereas the equation of the other sphere is. ince E is in the first octant, then and. Thus Problem 8 zdv cos sinddd E sin cossindd cossin dd 5 6 a)( points) It can be shown that the Cauchy-Euler system tx t Axt t where A is a constant matrix, has a nontrivial solutions of the form xt t r u if and only if r is an eigenvalue of A and u is a corresponding eigenvector. Use this information to solve the system tx t 4 xt t We first find the eigenvectors of 4. 4 r 4 r r 4 r 5r r so the eigenvalues are r, 5. The system A rix is 4 rx x x rx

12 r yields 4x x orx x. This gives the eigenvector equation x x and hence the eigenvector.thus. r 5 leadstothe xt c t c t 5 b)( points) Rewrite the scalar equation d y dt dy y cost dt as a first order system in normal form. Express the system in the matrix form x Ax f. Let x t yt, x t y t, x t y t Then x t y t x t x t y t x t x t y t y y cost x x cost Thus x x x x x x cost

13 Table of Integrals sin xdx cosxsinx x C cos xdx cosxsin x x C sin xdx cosx 4 cosx cosx C cos xdx 4 sin x sinx C : cos x sin x dt sinx C