math 240, final exam · 2006-03-21 · math 240, final exam may 6, 2004 name: instructor: lynnell...
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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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, "' ""'" ~Q
3. For what values of >.is every non-zero solution of the differential equation"
y + )..y'+ y = 0
unbounded ?
-.-.------........-.................--.
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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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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