Math 313 - Analysis I
Spring 2009
PRACTICE QUESTIONS FOR FINAL - WORKED
SOLUTIONS
These questions are intended to represent approximately how dicult and
long the Final Exam will be, and also to indicate some of the types of questions
that might arise. The
Math 313 - Analysis I
Spring 2009
PRACTICE QUESTIONS FOR FINAL
These questions are intended to represent approximately how dicult and
long the Final Exam will be, and also to indicate some of the types of questions
that might arise. They should not be con
Math 313 - Analysis I
Spring 2009
MIDTERM #2
SOLUTIONS
(1) Suppose that f : (1, 1) is a continuous function. Suppose also that for
1
1
each natural number n N we have |f ( n )| < n . Prove that f (0) = 0.
Solution:
We know that limxt0 f (x) = f (0), since
Math 313 - Analysis I
Spring 2009
MIDTERM #2 - PRACTICE
WORKED SOLUTIONS
(1) Let f : [0, 1] R be dened by f (x) = 2x(1 x).
(a) Using only the denition involving limits, prove that f is continuous
on [0, 1].
(b) Using only the denition involving limits, pr
Math 313 - Analysis I
Spring 2009
MIDTERM #2 - PRACTICE
NOON, APRIL 3, 2009
(1) Let f : [0, 1] R be dened by f (x) = 2x(1 x).
(a) Using only the denition involving limits, prove that f is continuous
on [0, 1].
(b) Using only the denition involving limits,
Math 313 - Analysis I
Spring 2009
WORKED SOLUTIONS TO MIDTERM #1
(1) Suppose that a and b are real numbers so that b > 3.
a
3
(a) Prove that
3
b
<
|ab9|
.
9
(b) Now suppose that 3 < b < 9, and prove that
a
3
3
b
< |a 3| + |b3| .
3
Solution:
NOTE: I missed
Math 313 - Analysis I
Spring 2009
HOMEWORK #11
SOLUTIONS
(1) Suppose that f, g : [a, b] R, and suppose that f is integrable on [a, b].
Suppose also that there are nitely many points c1 , . . . , ck [a, b] so that
for all y [a, b] cfw_c1 , . . . , ck we h
Math 313 - Analysis I
Spring 2009
HOMEWORK #10
SOLUTIONS
(1) Prove that the function f (x) = x3 is (Riemann) integrable on [0, 1] and
show that
1
1
x3 dx = .
4
0
(Without using formulae for integration that you learnt in previous calculus classes.)
n
You
Math 313 - Analysis I
Spring 2009
HOMEWORK #9
WORKED SOLUTIONS
(1) Let I be an open interval, and suppose p : I I is dierentiable on
I . Suppose a I is such that p(a) = a. Let pn : I I be the n-fold
composition p p . . . p. Show that pn (a) = (p (a)n .
So
Math 313 - Analysis I
Spring 2009
HOMEWORK #8
WORKED SOLUTIONS
(1) Let A be a nonempty subset of R. Dene a function fA : R R by
fA (x) = inf cfw_|x a| | a A .
(a) Prove that for any x R the inmum of the set cfw_|x a| | a A
exists, so the function fA is we
Math 313 - Analysis I
Spring 2009
HOMEWORK #7
WORKED SOLUTIONS
(1) Suppose that f : [a, b] R is a function, and that c (a, b). Prove that
if the limit
f (x) f (c)
lim
xc (x c)2
exists then f is continuous at c.
Solution:
Let lim
xc
f (x)f (c)
(xc)2
= L.
S
Math 313 - Analysis I
Spring 2009
HOMEWORK #6
WORKED SOLUTIONS
(1) Let f be a function such that |f (u) f (v )| |u v | for all points u and
v in an interval [a, b]. Prove that f is continuous at each point of [a, b].
(This includes continuous from the rig
Math 313 - Analysis I
Spring 2009
HOMEWORK #5
SOLUTIONS
(1) Let f : R R be dened by f (x) = x , so f is the function which takes a
real number x to the greatest integer n so that n x. Let g : [1, ) R
1
1
be dened by g (x) = f (x) = x . Find lim g (x) and