THIRD PROBLEM SET
Math 5616H: Honors Analysis
Due W 6 March, 2013.
Total 60 points.
1. Problem #4 on p. 165. (10 points)
2. Problem #7 on p. 165. (10 points)
3. Problem #9 on p. 166. (10 points)
4. Problem #13(a) on p. 167. (10 points)
5. Problem #18) on
Math 5616H
Midterm 2, with solutions
Spring 2013
April 19, 2013
1. (25 points) Let f : R R be continuous, and periodic with period 2. Let z be a real
number. Is it true that the limit of averages
1
N N
N
f (x + nz) =
lim
n=1
1
2
f (t) dt?
(1)
For which z
Math 5616H: Introduction to Analysis
Spring Semester 2013
Lectures: M W F 2:303:20 in Vincent H 207.
Professor: Robert Gulliver, Vincent Hall 452. Phone: (612) 625-1560.
Oce hours M F 3:354:25, W 11:1512:20 or by appointment.
Email: gulliveratmath.umn.edu
Math 5616H: Introduction to Analysis II. Spring 2012
Homework 7. Problems and Solutions.
#1 (Ch. 9: #6). If f (0, 0) = 0 and
f (x, y) =
x2
xy
+ y2
if (x, y) = (0, 0),
prove that (D1 f )(x, y) and (D2 f )(x, y) exist at every point of R2 , although f is no
Math 5616H: Introduction to Analysis II. Spring 2012
Homework 1. Problems and Solutions.
#1 (Ch. 5: #15). Suppose that f is a twice-dierentiable real function on (a, ), and M0 , M1 , M2 are the
2
least upper bounds of |f (x)|, |f (x)|, |f (x)|, respective
Math 5616H: Introduction to Analysis II. Spring 2012
Homework 4. Problems and Solutions.
#1 (Ch. 7: #22.) Assume f R() on [a, b], and prove that there are polynomials Pn
such that
b
|f Pn |2 d = 0.
lim
n
a
Proof. From f R() on [a, b] it follows that |f |
Math 5616H: Introduction to Analysis II. Spring 2012
Homework 6. Problems and Solutions.
#1 (Ch. 8: #14). If f (x) = ( |x|)2 on [, ], prove that
f (x) =
and deduce that
n=1
2
4
+
cos(nx)
3
n2
n=1
1
2
= ,
n2
6
1
4
= .
n4
90
n=1
Proof. Since f (x) is an eve
Math 5616H: Introduction to Analysis II. Spring 2012
Homework 2. Problems and Solutions.
#1 (Ch. 5: #14). (i) Let f be a dierentiable real function dened in (a, b). Prove that
f is convex if and only if f is monotonically increasing.
(ii) Assume next that
Math 5616H: Introduction to Analysis II. Spring 2012
Homework 3. Problems and Solutions.
#1 (Ch. 7: #13(a). Assume that cfw_fn is a sequence of monotonically increasing functions on
R1 with 0 fn (x) 1 for all x and all n. Prove that there is a function f
Math 5616H: Introduction to Analysis II. Spring 2012
Homework 5. Problems and Solutions.
#1 (Ch. 8: #8.) For n = 0, 1, 2, . . ., and x real, prove that
| sin(nx)| n | sin x|.
Proof. Starting from the trivial case n = 0, we can proceed by induction. If thi
FIFTH PROBLEM SET
Math 5616H: Honors Analysis
Due W 10 April, 2013.
Total 70 points.
1. Problem #8 on p. 197. (10 points)
2. Problem #10 on p. 197. (10 points)
3. Problem #11 on p. 198. (10 points)
4. Problem #12 on p. 198. (15 points)
5. Problem #22 on p
SIXTH PROBLEM SET
Math 5616H: Honors Analysis
Due W 24 April, 2013.
Total 70 points.
1. Problem #6 on p. 239. (10 points)
2. Problem #7 on p. 239. (10 points)
3. Problem #8 on p. 239. (10 points)
4. Problem #9 on p. 239. (10 points)
5. Problem #13 on p. 2
SEVENTH PROBLEM SET
Math 5616H: Honors Analysis
Due W 8 May, 2013.
Total 70 points.
1. Problem #14 on p. 224. (10 points)
2. Problem #21 on p. 241. (10 points)
3. Problem #27 on p. 242. (10 points)
4. Problem #28 on p. 242. (10 points)
5. Problem #1 on p.
SECOND PROBLEM SET
Math 5616H: Honors Analysis
Due W 20 February, 2013.
Total 60 points.
1. Problem #10(a,b,c) on p. 139.(20 points)
2. Problem #16 on p. 141. (10 points)
3. Problem #1 on p. 165. (10 points)
4. Problem #2 on p. 165. (10 points)
5. Problem
FOURTH PROBLEM SET
Math 5616H: Honors Analysis
Due W 27 March, 2013.
Total 100 points.
1. Problem #10 on p. 166. (10 points)
2. Problem #11 on p. 166. (10 points)
3. Problem #21 on p. 169. (10 points)
4. Problem #23 on p. 169. (10 points)
5. Problem #1 on
Math 5616H
Midterm 1 with solutions
Spring 2013
March 8, 2013
Total 80 points
1. (15 points) Let f (x) and g(x) be real continuous functions on an interval [a, b], such that
b
b
g 2 (x) dx = 1.
f 2 (x) dx =
a
a
Prove that
b
f (x)g(x) dx 1,
a
and that
b
f