EXAM, FINAL MATH 240: CALCULUS III
APRIL 29, 2005 No books, calculators or papers may be used, other than a hand-written note card at most 5" x 7" in size.
This examination consists of eight (8) long-answer questions and four (4) multiple-choice questions. Each problem is worth ten points. Partial credits will be given only for long-answer questions, when a substantial part of a problem has been worked out. Merely displaying some formulas is not sufficient ground for receiving partial credits.
. . .
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YOUR
NAME,
PRINTED:
YOUR PENN ID (LAST 4 OF THE MIDDLE 8 DIGITS):
YOUR SIGNATURE:
YOUR
LECTURE
SECTION CHAI
(CIRCLE
ONE): CALDARARU
1
2
3
4
5
6
7
8
9-12
Total
Name: Part 1. Long-answer Questions. 1. Compute det(A3), where A is the matrix . ~ 27 ]
1
--
Name: 2. Let C be the oriented curve
2
from (3,0) to (0,2). Compute the line integral 1 (x + 1) dy + Y dx .
Name: 3. Let D be the cube
3
D={(x,y,Z)EIR31 O::;x,y,z::;l},
and let S = aD be the boundary surface of D, oriented by the unit normal vector field ii on S pointing away from D. Compute the oriented surface integral
J l (x2i + xy z; + Z3k) . ii dS .
Name: 4. Let S be the surface
4
the upper half of the unit sphere centered at the origin, oriented by the unit normal vector field ii = xi + y J + z k on s. Compute the surface integral
J 1 (x i - y J + z k) . ii dS .
Name:
5
5. Let C be the boundary of the rectangle with vertices (3,2), (-5,2), (-5, -7) and (3, -7), oriented counter-clockwise. Compute the line integral 1 y dx - x dy fc x2 + y2 .
Name: 6. Let y(t) be a function which satisfies the differential equation
6
y"(t) + (1 + t)y'(t) + (1 + t + e)y(t) = 0
y(O) = 1, y'(O) = O. Determine the values of y"(O) and y"'(O).
Name:
7
7. Suppose that a vector-valued function x(t) satisfies the following system of ordinary differential equations
~
1 2
X'(t) - A. x(t) = 0, where A = [ 0 1 ] '
and X(O)= [ -;1 ]. Determine the function x(t) explicitly.
Name: 8. Find one ("particular") of solution the system of differential equations
dx+y dt
8
{
~ - 2x dt
=
= t
0
In other words, find a pair of real-valued functions (x(t), y(t)) satisfying the above system of equations. (There are many such solutions.) [Hint: Try to replace the above system by a single differential equation, then try to find a particular solution of that equation.]
Name:
9
Part II. Multiple Choice Questions. Please circle your answer. 9. Consider the following matrices
a -1
Al
=
[ 1
a
] ' A2 =
1 -1 , A3 = [ 1 3 ]
-1
a [ a
2 1 1 1 5 , A4 = 1 2 1 a 2] [ 1 1 2]
3 4
Which ones can be diagonalized over the real numbers? (In other words, there exists an invertible matrix P with coefficients in real numbers such that p-I . Ai . P is a diagonal matrix. )
A. Al and A3 only B. A2, A3 and A4 only C. A3 and A4 only D. A2 and A4 only E. AI, A3 and A4 only F. Al and A4 only G. AI, A2, A3 and A4
Name:
10
10. Let A be a symmetric 4 x 4 matrix with real entries. Consider the following statements. I. A must have four distinct eigenvalues. II. There exists an invertible matrix 0 with real entries such that 0 . A . 0-1 is a diagonal matrix. III. The four roots of the characteristic polyno...