Course

# Introduction to Mathematical Analysis

Faculty
Science & Technology
Department
Mathematics
Course Code
MATH 2245
Credits
3.00
Semester Length
15 weeks
Max Class Size
35
Method Of Instruction
Lecture
Tutorial
Typically Offered
Summer

## Overview

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
Methods Of Instruction

Lectures, Tutorials

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%

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
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)

## Requisites

### Prerequisites

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

### Corequisites

No corequisite courses.

### Equivalencies

No equivalent courses.

## Course Guidelines

Course Guidelines for previous years are viewable by selecting the version desired. If you took this course and do not see a listing for the starting semester / year of the course, consider the previous version as the applicable version.

## Course Transfers

These are for current course guidelines only. For a full list of archived courses please see https://www.bctransferguide.ca

Institution Transfer Details for MATH 2245
Camosun College (CAMO) CAMO MATH 2XX (3)
Kwantlen Polytechnic University (KPU) KPU MATH 2331 (3)
Langara College (LANG) LANG MATH 2373 (3)
Simon Fraser University (SFU) SFU MATH 242 (3)
Thompson Rivers University (TRU) TRU MATH 2120 (3)
University Canada West (UCW) UCW MATH 2XX (3)
University of British Columbia - Okanagan (UBCO) UBCO MATH 220 (3)
University of British Columbia - Vancouver (UBCV) UBCV MATH 2nd (3)
University of Northern BC (UNBC) UNBC MATH 2XX (3)
University of the Fraser Valley (UFV) UFV MATH 265 (3)
University of Victoria (UVIC) UVIC Math 1XX (1.5)
Vancouver Island University (VIU) VIU MATH 2nd (3)

## Course Offerings

### Summer 2022

CRN
Days
Dates
Start Date
End Date
Instructor
Status
22827
09-May-2022
- 10-Aug-2022
09-May-2022
10-Aug-2022
Sinclair
Peter
Full
MATH 2245-039 – This course is offered as a guided study. Please contact the instructor to register.
Max
Enrolled
Remaining
Waitlist
0
8
-8
0