Mathematics 217
Winter 2008
Homework 1.
Due January 25.
1. 3.4.4
2. 3.5.8
3. 3.5.10
4. 6.2.4
5. 6.2.8
6. 6.2.11
7. Carefully negate the denition of uniform convergence and state a possible Criterion for nonuniform convergence. Use your criterion to prove
Mathematics 306b
Final Examination
13 April 2003
1. Let I be an open interval and f : I R be a dierentiable function such that
f (x) = 0 for all x I. Show that the image f (I) of f is an open interval.
2. Let the functions fn , n N, and f have continuous
Mathematics 217
Winter 2008
Homework 6.
Due April 9.
1. Recall that the integral of the step function f (x), which equals wk on the subinterval (xk1 , xk ),
is dened as
N
b
wk |xk xk1 |.
f = I(f ) =
a
(1)
k=1
Prove that this denition agrees with the denit
Math 217, Winter 2008
Real Variables II
Syllabus
Instructors: Rasul Shakov, MC 112, [email protected] (emails will be answered within 48 hours).
Textbook: Understanding Analysis by Stephen Abbott, Springer 2002, ISBN 0387950605. The same
book was used in Ma
Mathematics 217b
Winter 2007
Sample Midterm.
Note: Sample exam length may be dierent from the actual midterm.
1. Let
f (x) = x3 3x2 + 6x 1.
f : R R,
Prove that f is a bijection, that f 1 is dierentiable on R, and calculate (f 1 ) (1).
2. Suppose that a fu
Mathematics 217
Winter 2008
Homework 4.
Due March 14.
1. 7.2.6
2. 7.3.5
3. 7.4.4
4. 7.4.7
5. Suppose that f (x) g(x) h(x) for all x [a, b], and that
b
b
b
Prove that a g exists and equals a f = a h.
1
b
a f
and
b
a h
exist and are equal.
Mathematics 217
Winter 2008
Homework 3.
Due February 20.
1. 6.6.5
2. The Taylor series expansion of the function dened by f (x) = (1 + x) about x = 0 for any xed
real number is called the binomial series.
(A) Show that
(1 + x) 1 +
n=1
( 1)( 2) ( n + 1) n
Mathematics 217
Winter 2008
Homework 2.
Due February 8.
1. 6.2.15
2. 6.2.16
3. 6.4.8
4. 6.5.7
5. 6.5.9
6. Prove that the function
f (x) =
n=1
1
(n x)2
is continuous on R except the points x = n, n N.
7. Prove that the function
f (x) =
n=1
has a continuous
Mathematics 217
Winter 2008
Homework 5.
Due March 28.
1. 7.5.6
2. 7.5.7
3. 7.5.10
4. 7.6.2
5. 7.6.4
6. Determine c > 0 so that the area under the graph of f (x) =
1
1
750+x3
from c to 2c is maximized.
Mathematics 217b
Final Examination
23 April 2004
1. Let f : R R be dierentiable with f (x) 0 for every x R (so f is
nondecreasing). Show that f is strictly increasing provided f is not identically zero
on any nonempty open interval.
2. Let f be a nonnegat
Mathematics 217b, Winter 2008
Practice Midterm Problems.
1. Show that if f is an integrable function on a closed and bounded interval, then so is the function
ef .
2. Find the set of all x R such that the power series
5n x3n
n=1
converges.
2
3. For n N, l
Mathematics 306a
Final Examination
14 December 2000
1. Show that if f is an integrable function on a closed and bounded interval, then
so is the function ef .
2. Find all continuously dierentiable functions f : R R such that f (0) = 0 and
x
0
tf (t) dt =
Mathematics 217b, Winter 2008
Practice Final Problems.
1. Show that if f : [a, b] [c, d] is an integrable function, and g is a continuous function on [c, d],
then g f is integrable on [a, b]. What if we only assume that g is integrable on [c, d]?
2. Let f