Math 404 Quiz 5.
Your name:
Let f (x) = x on [, ]. a) Compute Fourier coefficients of f 1 2
c0 = if n = 0 we have cn = 1 2
xdx = 0,

xeinx dx =

1 einx x 2 in


1 2
1

(1)n einx dx = in in
b) Apply Parseval's theorem to conclude that 2 1 = 2
Math 404 Quiz 8.
Your name:
Let F : (x, y) (sin(x + y), exy  1). represent F as a composition of two primitive mappings in a neighborhood of (0, 0).
Take G1 : (x, y) (x, exy  1) if u = x, v = exy  1 then x = u, y = u  ln(v + 1). Therefore for G2 :
Math 404 Quiz 1.
Your name:
If "yes" prove it, if "no" give a counterexample: A continuous function f : R R has the following properties: f (0) = 0 and f (x) < 0 if x > 1
(i) Is it true that f has to attain its maximum?
(ii) Is it true that f has to att
Math 404, Midterm 3.
5. Let F : R2 R2 be a C 1 mapping and F (0, 0) = I (here I denotes the
identity mapping). Prove that there is a neighborhood U of (0, 0), such that
restriction of F on U is a composition of two primitive mappings.
Let
F(x, y ) = (z (
Midterm 2.
Each problem costs 16 points. In the first two problems you will need to prove a theorem (or its part) from the list (the choice is mine): 1. 8.11, 8.12, 8.14, Fejr's theorem(prob. 15), 8.16. e 2. 9.2, 9.5, 9.7, 9.8, 9.12, 9.15, 9.17, 9.21, 9.2
Math 404 Quiz 2.
Your name:
Let be a monotonically increasing function on [a, b]. Prove that R() if and only if is continuous
IF: Assume is not continuous at y [a, b] and yi y such that (yi )(y) > 0 for any i. R(), therefore > 0 partition P such that U
Math 404 Quiz 4.
Your name:
inx .  cn e
Let f be a smooth 2periodic function. Assume f that
Prove
f

in cn einx .
Sinse f is smooth, f R on [, ]. Let f n = 1 inx fe 2

n einx . Then
1 2 1 2
f einx dx =

=


f (ineinx )dx = 0 + incn .

1
Math 404 Quiz 7. Your name:
Consider mapping R2 R2 , f : (x, y) (u, v) defined by u = x2  y 2 , v = 2xy Find [f (x, y)] in standard basis. 2x 2y Find Jacobian Jf (x, y). 2x 2y 2y = 4(x2 + y 2 ) 2x 2y 2x
Jf (x, y) = det
Conclude that f is onetoone in
Math 404 Quiz 6.
Your name:
Let A be a linear transformation from Rn to Rm and cfw_ai j is its matrix in the standard bases. Prove that A ai j .
i,j
Ax
ij
ai j xj 
ij
ai j  x
therefore A
ij
ai j .
1
Math 404, Midterm 1. Your name: 1. Function f is monotonic on [a, b], and is continuous on [a, b]. Prove that f R(). 2. Function f R on [a, b] and there is a differentiable function F on [a, b] such that F = f . Prove that
b
f (x)dx = F (b)  F (a).
a
3.