How You Will Be Evaluated
Grades: 330 points are possible, distributed as follows:
180 points total for assignments (12 @ 15 points each)
75 points total for discussions (3 (25 points each)
75 points total for the final (5 questions @ 15 points each; cumu
Topics to Review
Sections 3.1 and 3.2 present intro-
ductory material. The main sec-
tion of this chapter is 3.3 because
it contains the general method of
separation of variables. Sections
3.3, 3.5, and 3.8 use the basic
theory of Fourier series as pre-
s
38 Chapter 2 Fourier Series
(c) Obtain the Taylor series expansion
sincv =i(1) 1'2" (00 < x < 00)
710: "(271+ 1)!'
(d) Integrate the series in (c) term by term, and use the alternating series test to
obtain the inequalities
7r sinx
1.85 </ dx < 1.86.
Midterm 1 Applied Analysis
Name
Problem
Points:
Surname
1
2
3
4
Each Problem is worth 20 points
Good luck!
5
Problem 1:
Solve the initial value problem
ey
2
u
u
3x2
= 0,
x
y
u(x, 0) = x3 .
2
dy
Solution: Write as ux 3x
u = 0. Next solve dx
= 3x
, a separ
Geometricexplanation of the notions of elliptic, hyperbolic and parabolic for PDEs
General form:
Auxx + Buxy + Cuyy + Dux + Euy + F u + G
Translate uxx x2 , uxy xy, uyy y 2
Elliptic: AC B 2 > 0, special case A = C = 1, B = 0: uxx + uyy = 0 translates to
x
Section 2.5 Mean Square Approximation and Parsevals Identity 53
In Exercises 916, find the sine series expansion of the given function on the interval
0 < a: < 1.
9- x(1$)- 10.1m2. 11. sin7rx. 12. singx.
13. sinvrcc cosmr. 14. (1 + cos 7m) sin 7m. 15. ea.