Introduction to Mathematical Analysis

Curriculum guideline

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Course
MATH 2245
Discontinued
No
Course code
MATH 2245
Descriptive
Introduction to Mathematical Analysis
Department
Mathematics
Faculty
Science & Technology
Credits
3.00
Start date
End term
Not Specified
PLAR
No
Semester length
15 weeks
Max class size
35
Course designation
None
Industry designation
None
Contact hours

Lecture: 4 hours/week
Tutorial: 1 hour/week

Method(s) of instruction
Lecture
Tutorial
Learning activities

Lectures, discussions, problem-solving practice, in-class assignments (which may include work in groups), tutorials

Course description
An introduction to analysis for students who have successfully completed the first year of calculus. This course presents foundation concepts in analysis which lay the groundwork for further study in pure and applied mathematics, in particular real analysis courses. It is normally required material for mathematics majors. Topics studied include the nature of proof, set theory and cardinality, the real numbers, limits of sequences and functions, continuity, formal coverage of the derivative and the mean value theorem, Taylor’s theorem, the Riemann integral, the fundamental theorem of calculus, and topics in infinite series.
Course content

1. Logic and Proof:

  • elements of logic
  • various proof techniques

2. Sets and Functions:

  • set algebra
  • relations and functions
  • introduction to cardinality

3. The Real Numbers: 

  • natural numbers
  • induction
  • definition of field
  • completeness of the real numbers

4. Sequences: 

  • subsequences
  • convergence
  • monotonicity
  • Cauchy sequences

5. Limits and Continuity:

  • function limits
  • continuity and its properties
  • uniform continuity

6. Differentiation: 

  • definition and properties of derivative
  • mean value theorem
  • Taylor's theorem

7. Integration: 

  • Riemann integral and its properties
  • the fundamental theorem of calculus

8. Infinite series:

  • definition of convergence
  • convergence testing
  • introduction to power series
Learning outcomes

Upon successful completion of the course, students will be able to:

  • use the vocabulary of logic and mathematics to read and write mathematical statements;
  • use the rules of logic to analyze the structure of mathematical proofs;
  • illustrate proof techniques by means of examples;
  • use set theory to construct mathematical proofs;
  • define a function and establish properties of functions acting on sets;
  • state and apply theorems relating to the cardinality of sets;
  • examine the structure and properties of the real number system;
  • use the definition of convergence of a sequence to determine the limit of a sequence;
  • prove and apply theorems relating to properties of convergent sequences;
  • define the limit of a function and continuity of a function;
  • prove and apply theorems relating to continuous functions beyond those found in elementary calculus;
  • define the derivative of a function and establish properties of differentiable functions;
  • define the Riemann integral and establish properties of integrable functions;
  • define infinite series and develop tests to determine whether an infinite series is convergent or divergent;
  • define a power series and establish basic convergence properties of power series.
Means of assessment

Assessment will be in accordance with the Douglas College Evaluation Policy. The instructor will present a written course outline with specific evaluation criteria at the beginning of the semester. Evaluation will be based on the following:

Problem sets, quizzes, assignments: 0-40%
Tutorials: 0-10%
Term tests: 20-60%
Final exam: 30-40%

 

Textbook materials

Consult the Douglas College Bookstore for the latest required textbooks and materials. Example textbooks and materials may include:

Lay, Analysis with an Introduction to Proof, Pearson, current edition

Abbott, Understanding Analysis, Springer, current edition

Chartrand, Polimeni, Zhang, Mathematical Proofs: A Transition to Advanced Mathematics, Pearson, current edition

Hammack, Book of Proof, Ingram, current edition

Dembiras, Rechnitzer, PLP: An Introduction to Mathematical Proof, current edition

Trench, Introduction to Real Analysis, current edition

Prerequisites

MATH 1220 with a minimum grade of C+