MATH 409501/503
Fall 2013
Sample problems for Test 1
Any problem may be altered or replaced by a dierent one!
Problem 1 (15 pts.) Prove that for any n N,
13 + 23 + 33 + + n3 =
n2 (n + 1)2
.
4
Problem 2 (30 pts.) Let cfw_Fn be the sequence of Fibonacci nu
MATH 409501/503
Fall 2013
Sample problems for Test 2
Any problem may be altered or replaced by a dierent one!
Problem 1 (20 pts.) Prove the Chain Rule: if a function f is dierentiable at a point c
and a function g is dierentiable at f (c), then the compos
MATH 409501/503
Fall 2013
Sample problems for the nal exam
Any problem may be altered or replaced by a dierent one!
Problem 1 (20 pts.) Suppose E1 , E2 , E3 , . . . are countable sets. Prove that their union
E1 E2 E3 . . . is also a countable set.
Problem
MATH 409
Advanced Calculus I
Lecture 1:
Axioms of an ordered eld.
Real line
Systematic study of the calculus of functions of one
variable begins with the study of the domain of such
functions, the set of real numbers R (real line).
The real line is a math
MATH 409501
October 10, 2013
Test 1: Solutions
Problem 1 (20 pts.) Prove that for any x (0, 1) and any natural number n,
(1 x)n 1 nx +
n(n 1) 2
x.
2
The proof is by induction on n. First we consider the case n = 1. In this case the inequality
reduces to (
MATH 409
Advanced Calculus I
Lecture 6:
Limits of sequences.
Limit theorems.
Convergence of a sequence
A sequence of elements of a set X is a function f : N X .
Notation: x1 , x2 , . . . , where xn = f (n), or cfw_xn nN , or cfw_xn .
Denition. Sequence cf
MATH 409
Advanced Calculus I
Lecture 3:
Metric spaces.
Completeness axiom.
Existence of square roots.
Absolute value
Denition. The absolute value (or modulus) of a
real number a, denoted |a|, is dened as follows:
|a| =
a if a 0,
a if a < 0.
Properties of
MATH 409
Advanced Calculus I
Lecture 4:
Intervals.
Principle of mathematical induction.
Inverse function.
Problem. Construct a strict linear order on the
set C of complex numbers such that a b implies
a + c b + c for all a, b, c C.
Solution. Given complex
MATH 409
Advanced Calculus I
Lecture 5:
Binomial formula.
Inverse function and inverse images.
Countable and uncountable sets.
Well-ordering and induction
Principle of well-ordering:
The set N is well-ordered, that is, any nonempty subset of N
has a least
MATH 409501/503
Fall 2013
Sample problems for Test 2: Solutions
Any problem may be altered or replaced by a dierent one!
Problem 1 (20 pts.) Prove the Chain Rule: if a function f is dierentiable at a point c
and a function g is dierentiable at f (c), then
MATH 409
Advanced Calculus I
Lecture 7:
Monotone sequences.
The Bolzano-Weierstrass theorem.
Limit of a sequence
Denition. Sequence cfw_xn of real numbers is said to
converge to a real number a if for any > 0 there exists
N N such that |xn a| < for all n
MATH 409
Advanced Calculus I
Lecture 9:
Limit supremum and inmum.
Limits of functions.
Limit points
Denition. A limit point of a sequence cfw_xn is the
limit of any convergent subsequence of cfw_xn .
Properties of limit points.
A convergent sequence has
MATH 409501
November 14, 2013
Test 2: Solutions
Problem 1 (20 pts.) Find min x2x .
x>0
Solution: min x2x = e2/e .
x>0
2x
The function f (x) = x2x is well dened and positive on (0, ). Hence f (x) = elog f (x) = elog x =
e2x log x for all x > 0. Now it foll
MATH 409
Advanced Calculus I
Lecture 2:
Properties of an ordered eld.
Absolute value.
Supremum and inmum.
Real line
The real line is a mathematical object rich with
structure. This includes:
algebraic structure (4 arithmetic operations);
ordering (for a
MATH 409501/503
Fall 2013
Sample problems for Test 1: Solutions
Any problem may be altered or replaced by a dierent one!
Problem 1 (15 pts.) Prove that for any n N,
13 + 23 + 33 + + n3 =
n2 (n + 1)2
.
4
The proof is by induction on n. First we consider th
MATH 409501/503
Fall 2013
Sample problems for the nal exam: Solutions
Any problem may be altered or replaced by a dierent one!
Problem 1 (20 pts.) Suppose E1 , E2 , E3 , . . . are countable sets. Prove that their union
E1 E2 E3 . . . is also a countable s
MATH 409
Advanced Calculus I
Lecture 8:
Monotone sequences (continued).
Cauchy sequences.
Limit points.
Monotone sequences
Denition. A sequence cfw_xn is called increasing
(or nondecreasing) if xn xn+1 for all n N.
It is called strictly increasing if xn