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
Contact Hours
Lecture: 4 hrs/week Tutorial: 1 hr/week
Method(s) Of Instruction
Lecture
Tutorial
Learning Activities

Lectures, Tutorials

Course Description
A one-semester introduction to analysis for students who have successfully completed the first year of calculus (six credits). 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
  • notion of completeness

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

The student who successfully completes this course will:

  • use the rules of logic to study the way in which mathematical arguments are constructed
  • analyze the structure of mathematical proofs and illustrate proof techniques by means of examples
  • use set theory to construct mathematical proofs
  • 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 work with theorems relating to properties of convergent sequences
  • define the limit of a function and continuity of a function 
  • prove and work with 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

Evaluation will be carried out in accordance with Douglas College 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 criteria:

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 current textbook. Examples of textbooks under consideration include:

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

Prerequisites

MATH 1220  (with a grade of C+ or better)