MATH 2400 LECTURE NOTES: POLYNOMIAL AND RATIONAL
FUNCTIONS
PETE L. CLARK
Contents
1. Polynomial Functions
2. Rational Functions
1
6
1. Polynomial Functions
Using the basic operations of addition, subtraction, multiplication, division and
composition of fu
MATH 2400 SECOND MIDTERM EXAM
PETE L. CLARK
Directions: You will have 60 minutes to complete the exam. Calculators are not
permitted. The problems come in three groups. You must solve exactly four problems altogether and at least one problem in each group
MATH 2400 FREEROLL THIRD MIDTERM EXAM
PETE L. CLARK
Directions: This is a freeroll exam: your performance on this exam will only be
used to increase1 your midterm exam grade. Accordingly please feel free to do as
few or as many problems as you choose, kee
MATH 2400: PRACTICE PROBLEMS FOR EXAM 2
PETE L. CLARK
1)a) State the Extreme Value Theorem.
b) Give an example of a function f : [a, b] R which is continuous except at a single point c (a, b) and such that both conclusions of the Extreme Value Theorem
fai
MATH 2400 LECTURE NOTES: COMPLETENESS
PETE L. CLARK
Contents
1. Dedekind Completeness
1.1. Introducing (LUB) and (GLB)
1.2. Calisthenics With Sup and Inf
1.3. The Extended Real Numbers
2. Intervals and the Intermediate Value Theorem
2.1. Convex subsets of
MATH 2400 LECTURE NOTES: DIFFERENTIAL MISCELLANY
PETE L. CLARK
Contents
1. LHpitals Rule
o
1.1. The Cauchy Mean Value Theorem
1.2. LHpitals Rule
o
2. Newtons Method
2.1. Introducing Newtons Method
2.2. A Babylonian Algorithm
2.3. Questioning Newtons Metho
MATH 2400 LECTURE NOTES: DIFFERENTIATION
PETE L. CLARK
Contents
1. Dierentiability Versus Continuity
2. Dierentiation Rules
3. Optimization
3.1. Intervals and interior points
3.2. Functions increasing or decreasing at a point
3.3. Extreme Values
3.4. Loca
MATH 2400 LECTURE NOTES: INTEGRATION
PETE L. CLARK
Contents
1. The Fundamental Theorem of Calculus
2. Building the Denite Integral
2.1. Upper and Lower Sums
2.2. Darboux Integrability
2.3. Verication of the Axioms
2.4. An Inductive Proof of the Integrabil
MATH 2410: PRACTICE PROBLEMS FOR EXAM II
PETE L. CLARK
1) Let cfw_an be a real innite sequence.
n=1
a) Let L R. Dene an L.
b) Dene an .
c) Let S R. Dene n=1 an = S .
2) a) State the Monotone Sequence Lemma.
b) State the Bolzano-Weierstrass Theorem for s
MATH 2410 FIRST MIDTERM EXAM
PETE L. CLARK
Directions: You will have 75 minutes. No calculators. Good luck!
1) State the denition of Darboux integrability and give two equivalent conditions
for a function to be Darboux integrable.
Solution: A function f :
MATH 2410 SECOND MIDTERM EXAM
PETE L. CLARK
Directions: You will have 75 minutes. No calculators. Good luck!
1) a) State the Bolzano-Weierstrass Theorem (for sequences).
Solution: Every bounded sequence admits a convergent subsequence.
b) Prove/disprove:
MATH 2400: PRACTICE PROBLEMS FOR EXAM 2
PETE L. CLARK
1)a) State the Extreme Value Theorem.
Solution: Let f : [a, b] R be a continuous function. Then f has both a maximum
and minimum value. Consequently, f is bounded.
b) Give an example of a function f :
MATH 2400: PRACTICE PROBLEMS FOR EXAM 1
PETE L. CLARK
1) Find all real numbers x such that x3 = x. Prove your answer!
2) a) Prove that 6 is an irrational number. You may use the fact that if an
integer x2 is divisible by 6, then also x is divisible by 6.
MATH 2400: PRACTICE PROBLEMS FOR EXAM 1
PETE L. CLARK
1) Find all real numbers x such that x3 = x. Prove your answer!
Solution: If x3 = x, then 0 = x3 x = x(x + 1)(x 1). Earlier we showed
using the eld axioms that if if a product of numbers is equal to ze
SPRING 2011 MATH 2400/2400H COURSE SYLLABUS
PETE L. CLARK
Course: Math 2400 / 2400H: Honors Calculus With Theory I
Instructor: Prof. Pete L. Clark, Ph.D.
Lectures: MWF 11:15am - 12:05pm , Th 11am-12:15pm, in Boyd 222
My Oce: Boyd 502
Oce Hours: MWF 12:15p
LECTURE NOTES ON MATHEMATICAL INDUCTION
PETE L. CLARK
Contents
1. Introduction
2. The (Pedagogically) First Induction Proof
3. The (Historically) First(?) Induction Proof
4. Closed Form Identities
5. More on Power Sums
6. Inequalities
7. Extending binary
MATH 2400 LECTURE NOTES
PETE L. CLARK
1. First Lecture
1.1. The Goal: Calculus Made Rigorous.
The goal of this course is to cover the material of single variable calculus in a
mathematically rigorous way. The latter phrase is important: in most calculus
c
MATH 2400 FIRST MIDTERM EXAM
PETE L. CLARK
Directions: You will have 75 minutes to complete the exam. Calculators are not
permitted (nor would they be helpful.). Please attempt all problems. When in
doubt, err on the side of writing a little too much rath
MATH 2400 LECTURE NOTES: CONTINUITY AND LIMITS
PETE L. CLARK
Contents
1. Introduction to Calculus
1.1. Some derivatives without formal limits
2. Some derivatives without a careful denition of limits
3. Limits in Terms of Continuity
4. Continuity Done Righ
MATH 2400 LECTURE NOTES: DIFFERENTIATION
PETE L. CLARK
Contents
1. Dierentiability Versus Continuity
2. Dierentiation Rules
3. Optimization
3.1. Intervals and interior points
3.2. Functions increasing or decreasing at a point
3.3. Extreme Values
3.4. Loca
MATH 2410: PRACTICE PROBLEMS FOR EXAM I
PETE L. CLARK
1) Let f : [a, b] R. Give careful denitions for the following:
a) f is bounded.
b) f is Darboux integrable.
c) f is Riemann integable. (Hint: this is the condition that the Riemann sums
b
converge to a