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The solutions for various mathematical problems from a university-level mathematics exam, including problems related to heat conduction, complex analysis, and differential equations.
Typology: Exams
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Tuesday, January 13, 2004
Your name, printed:................................................................
Your signature:......................................................................
Your Penn student ID#:........................................................... Circle the name of your Professor: Shatz Wilf
This is a closed book exam. You may use during the exam, a single 5x7 inch card on which you may have written on both sides, if you wish. No more than one such card, and no larger size, please. No other books, papers, calculators, or other materials may be consulted during an exam. The exam consists of twenty (20) multiple choice questions, and each question offers a choice of five or six answers. Circle the answer that you choose, in each case, on the question sheet itself. The exam booklets are given to you for your convenience in working out the problems, but do not hand in these booklets when the exam is over. Hand in only the exam itself, with your answers to the questions circled.
For grader’s use only:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Do not turn the page and start work until the proctors announce the start of the exam.
∑^ ∞ n=
Ane−kn^2 π^2 t/L^2 sin nπx L. Then which one of the following is true: (a) ux(0, t) = ux(L, t) = 0 (b) ux(0, t) = ux(L, t) (c) ux(0, t) = u(0, t) = 0 (d) u(0, t) = u(L, t) = 0 (e) ux(L, t) = u(L, t) = 0
(a) 1/ (b) 2πi (c) 1/ (d) πi (e) none of the above
(a) 2πi (b) πi/ 12 (c) πi/ 3 (d) eπi
k^2
( (^) 3 + 4i 3 − 4 i
)k (z − 3)k
(a) are z 0 = 3 and R = √ 2 (b) are z 0 = 0 and R =
(c) are z 0 = 3 and R = 1 (d) are z 0 = 0 and R =
(e) are none of the above
(a) π/ 4 (b) 1/ 8 (c) π/ 8 (d) 1/ 4 (e) none of the above
(a) (^) (z−^1 1) 2 + (^) z−^11 + 1 − (z − 1) + (z − 1)^2 −... (b) (^) (z−−^1 1) 2 + (^) z−^11 + 1 − (z − 1) + (z − 1)^2 −... (c) (^) (z−^1 1) 2 − (^) z−^11 + 1 − (z − 1) + (z − 1)^2 −... (d) (^) (z−^1 1) 2 + (^) z−^11 + 1 + (z − 1) + (z − 1)^2 +... (e) (^) (z−−^1 1) 2 + (^) z−^11 + 1 + (z − 1) + (z − 1)^2 +...
(b) (eπ^ + e−π^ )/(e^2 π^ − e−^2 π) (c) (eπ^ − e−π^ )/(e^2 π^ + e−^2 π) (d) (eπ^ + e−π^ )/(e^2 π^ + e−^2 π) (e) (e^2 π^ − 1)/(e^4 π^ − 1) (f) (e^2 π^ + 1)/(e^4 π^ + 1)