Math 309 C - Autumn 2011
Mid-Term Exam
November 2, 2011
Name: _:7/_i
Student Number: _._._._
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Math 309
Autumn 2015
Practice Final
December 2015
Time Limit: 1 hour, 50 minutes
Name (Print):
ID Number:
This exam contains 9 pages (including this cover page) and 8 problems. Check to see if any pages
are missing. Enter all requested information on the
Math 309 C Winter 2015 Final
March 16, 2015
Name:
Student ID Number:
1
15
2
15
3
15
4
15
5
10
Total
70
You have until 4:20 pm to complete the exam. There are five problems. Read all of the
problems carefully before starting to work on them.
Turn off and
Math 309 G Spring 2015 Final
June 11, 2015
Name:
Student ID Number:
1
10
2
10
3
10
4
10
5
10
Total
50
You have until 4:20 pm to complete the exam. There are five problems. Read all of the
problems carefully before starting to work on them.
Turn off and
Fourier's Heat Conduction Equation:
History, Influences and Connections*
T N NARASIMHAN
Department of Materials Science and Mineral Engineering, Department of Environmental Science, Policy, and
Management, Lawrence Berkeley National Laboratory, University
Robert Osserman
Robert Osserman
Mathematical Sciences Research
Institute
17 Gauss Way
Berkeley, CA 94720 USA
[email protected]
Keywords: Eero Saarinen, Gateway
Arch, catenary, parabola, weighted
catenary, Robert Hooke, Galileo
Galilei, history of mechanics
Rese
Evolution of the Function Concept:
A Brief Survey
Israel Kleiner
Israel Kleiner received his Ph.D. in ring theory at McGill
University, and has been at York University for over twenty
years. He has been involved in teacher education at the
undergraduate a
Summer 2011, Math 309 C
Quiz 7
1. Suppose h : R2 R is a harmonic function. In polar coordinates you
found that h(1, ) = sin .
a) Is it possible that h is bounded (give a reason)?
b) What is h(0, 0)?
c) What is the maximum value of h over all of R = cfw_(x
Summer 2011, Math 309 C
Quiz 4
1. Find the positive eigenvalues and the corresponding eigenfunctions
associated to the boundary value problem y + y = 0 with y (0) = 0 and
y (L) = 0. Show all work.
1
Summer 2011, Math 309 C
Quiz 5
1. Find the Fourier sine series for the function
f (x) =
1 :0x1
2 :1<x2
You do NOT need to write down the explicit odd extension.
1
Summer 2011, Math 309 C
Quiz 6
1. Consider a wave of length L = 20 that satises the wave equation
2u
u
9 2 = 2 on 0 < x < 20 and for t > 0. Assume the ends of the wave are
x
t
xed and that the wave is set in motion with no initial velocity from the initi
Summer 2011, Math 309 C
Quiz 2
1. Let A be a (2 2) matrix with eigenvalues r1 = a + bi and r2 = c + di.
Fill in the following chart about the system of dierential equations x = Ax.
Assume A has two linearly independent eigenvectors.
Eigenvalue type
a > 0
Summer 2011, Math 309 C
Quiz 1
011
1. a) Find the characteristic polynomial of A = 1 0 1 and check
110
that 1 and 2 are the roots to the polynomial.
b) Find a basis for the eigenspace corresponding to the eigenvalue 1.
1
Math 309C (Ward)
Final (60 Minutes)
August 19, 2011
DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO BEGIN!
Instructions.
(1) Write your name on the line provided below these instructions.
(2) Your solutions must be NEAT and LEGIBLE.
(3) Show all work you want g
Math 309C (Ward)
Final (60 Minutes)
August 19, 2011
DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO BEGIN!
Instructions.
(1) Write your name on the line provided below these instructions.
(2) Your solutions must be NEAT and LEGIBLE.
(3) Show all work you want g
Math 309C (Ward)
Midterm (60 Minutes)
July 18, 2011
DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO BEGIN!
Instructions.
(1) Write your name on the line provided below these instructions.
(2) Your solutions must be NEAT and LEGIBLE.
(3) Show all work you want g
Summer 2011, Math 309 C
Quiz 7
1. Suppose h : R2 R is a harmonic function. In polar coordinates you
found that h(1, ) = sin .
a) Is it possible that h is bounded (give a reason)?
No. Every bounded harmonic function is constant.
b) What is h(0, 0)?
The Mea
Math 309C (Ward)
Midterm (60 Minutes)
July 18, 2011
DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO BEGIN!
Instructions.
(1) Write your name on the line provided below these instructions.
(2) Your solutions must be NEAT and LEGIBLE.
(3) Show all work you want g