Math 240, Final Exam

May 6, 2004

Name:

Instructor: Lynnell Matthews

Teaching Assistant:

This exam consists of 16 pages. In order to receive full credit you need

to show all your work .

Score

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1 5

2 5

3 5

4 5

5 10

6 5

7 5

8 5 ,

9 5

10 5

11 5

12 5

13 5

14 5

Total 75

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1. Show that the differential equation (2x+y3)dx+(3xy2_e-2Y)dy = 0 is exactand find the particular solution with y( -1) = O.

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2. Find the general solution to the differentialequation"

y - 2y' + 2y = sin (x)

-- ---

, "' ""'" ~Q

3. For what values of >.is every non-zero solution of the differential equation"

y + )..y'+ y = 0

unbounded ?

4. For what values of .Adoes the matrix

[

1 2 3

]

o .A 1212

have rank 3 ?

- -

-.-.------........-.................--.

5. Determine whether each statement is true or false. If true, givea justification,if false, give a counterexample.

Let A be an n x n matrix with distinct positive eigenvalues. Then

1. det(A) > 0

2. A is diagonalizable

3. A is orthogonal

4. A is symmetric

5. Eigenvectors of A corresponding to distinct eigenvalues are orthogonal.

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6. Find a basis of eigenvectors for the matrix

[

101

]

013002

7. Let T be the followingmatrix

Use row operations to show that det T = (b - a)(c - a)(c- b).

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8. While subject to forceF(x, y) = y3i+ (x3 + 3xy2)j, a particle travels coun-terclockwiseonce around the circleof radius 3. UseGreen's Theorem to findthe work done by F.

9. Evaluate

L (1 - ye-X)dx + e-Xdy

where C is the curve r(t) = (eCos1Tt,1/(t1 + 1)),0:::; t :::;1.

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10. Use Stokes' Theorem to evaluate Ie F. dr. where

F(x, y, z) = 2yi + 3zj + xk

and C is the triangle with vertices (0,0,0), (0,2,0) and (1,1,1) oriented sothat the vertices are traversed in that order.

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11. Let Q be the solid bounded by the cylinder x2 + y2 = 4,the plane x + z = 6and the xy-plane. Find ffsF. ndS where S is the surface of Q oriented bythe outside normal, and F(x, y, z) = (x2 + sinz)i + (xy + cosz)j + eYk.

12. Prove that the given property hold for vector fields F and G and scalarfunction f. (assume the required partial derivatives are continuous)

div(F x G) = (curlF)G - F(curlG)

. .~ ,~

13. Use Laplace transforms to solve y' - 5y = f(t) where

{t2,

f(t) = 0,

y(O)= 1.

if 0 ::; t < 1;

if1::;t

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14. Solve

x' +y = t4x+y' = 0

x(O) = 0, y(O) = 2

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