Practice Final
Math 0420
April 19, 2014
The exam contains two parts. There are six problems in Part 1 and nine
problems in Part 2.
Problems in Part 1 are 5 points each. If you get n points for a problem
then your score will be 3 maxcfw_n 3, 0 for this pro
Midterm Exam
Math 0420
Spring 2014
Solutions
1. Let S R and x R. Assume that x is a cluster point of S. Show that there exists a convergent
sequence cfw_xn S such that xn = x and lim xn = x.
Solution:
2.
See Proposition 3.1.2.
Let f : [1, 3] R be dened b
Solutions for HW8
Math 0420
March 31, 2014
Problem (4.4.1). Compute the nth Taylor Polynomial at 0 for the exponential function.
Proof. Let f (x) = ex . Then we know that for each n N, f (n) (x) =
ex and therefore f (n) (0) = 1. So the nth Taylor polynomi
Practice Final Solutions
Math 0420
April 19, 2014
The exam contains two parts. There are six problems in Part 1 and nine
problems in Part 2.
Problems in Part 1 are 5 points each. If you get n points for a problem
then your score will be 3 maxcfw_n 3, 0 fo
Solutions for HW6
Math 0420
February 22, 2014
1
Problem (3.6.3). Which of these sets are closed: (0, 1), Z, Q, cfw_ n : n N?
Proof. The set of cluster points of (0, 1) is [0, 1] and since [0, 1]
(0, 1),
(0, 1) is not closed.
The set of cluster points of Z
Solutions for HW5
Math 0420
February 21, 2014
Problem (3.4.2). Suppose f : (a, b) R is uniformly continuous. Show
limxb f (x) exists.
Proof. Pick a sequence in (a, b), say cfw_xn that converges to b. Then cfw_xn is
Cauchy and by lemma 3.4.5 so is cfw_f