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
201720
PLAR
No
Semester length
15 weeks
Max class size
35
Contact hours
Lecture: 4 hrs/week Tutorial / Lab: 1 hr/week
Method(s) of instruction
Lecture
Lab
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

(Approximate lecture/class time in brackets.)

1. Logic and Proof:  [1 week]

  • elements of logic
  • various proof techniques

2. Sets and Functions:  [1 week]

  • set algebra
  • relations and functions
  • introduction to cardinality

3. The Real Numbers:  [2 weeks]

  • natural numbers
  • induction
  • definition of field
  • notion of completeness

4. Sequences:  [2 weeks]

  • subsequences
  • convergence
  • monotonicity
  • Cauchy sequences

5. Limits and Continuity: [2 weeks]

  • function limits
  • continuity and its properties
  • uniform continuity

6. Differentiation:  [2 weeks]

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

7. Integration:  [2 weeks]

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

8. Infinite series:  [2 weeks]

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

The student who successfully completes this course will: (1) have a deeper grasp of the nature of proof and have a more complete understanding of the structure of the real number system; (2) be more fluent with sequences; (3) have an enhanced appreciation of continuity and its crucial extensions beyond elementary calculus; (4) broaden his/her sense of the theoretical basis of the key calculus notions of the derivative and the Riemann integral; and (5) have a more rigorous exposure to series and power series.  The student will then be prepared to advance to the theoretical concepts in advanced calculus, differential equations, and more formal analysis courses.  As well, the student will have a better foundation for modern applied mathematics and pure science courses.

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 / labs 0-10%
Mid-term tests 20-60%
Final exam 30-40%

 

Textbook materials

Textbooks and Materials to be Purchased by Students:

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

Prerequisites

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