SE 115. Real Calculus.
For year 1 CSSE students.
Semesters I and II.
ECTS credits: 10
Maths Dept sh: 4
48 lecture hours + 48 hours tutorials + 64 hours independent study.
Instructor (2000-2001): Anthony G. O'Farrell
This course aims to introduce students to the fundamentals of differential
and integral calculus in one real variable. We aim to develop a good
understanding of fundamental principles, and facility with
paper-and-pencil skills.
TEXT:
H. Anton. Calculus. 6th Edition. Wiley 1999. Chapters 1-12.
We will concentrate on those topics and applications relevant to
CS+SE.
You will also find this book useful with other courses in the
programme, and as a source of reference in the future. Buy it, and
keep it.
CONTENT:
A. Differential Calculus:.
Functions of a real variable and their graphs.
Limit at a point. Continuous function.
The derivative, as the slope of the graph. Alternative interpretations
of the derivative, as a rate of change, a margin, a speed, and acceleration,
in various contexts.
Rules for differentiating sums, products, quotients, compositions and inverses.
Higher derivatives. The relation of the second derivative to curvature.
Elementary functions: polynomials, rational functions, sin, cos, exp,
and their inverses arcsin, arccos, log. Their derivatives.
Calculating tangents, speeds, related rates, and other applications
of derivatives.
Finding stationary values (max-min theory)
Taylor polynomials. Taylor's theorem.
B. Integral Calculus:.
The concept of area. The definite integral, as an area.
The Fundamental Theorem of Calculus.
Other interpretations of the integral, and applications.
Integration of elementary functions. Techniques for integration:
substitution, parts, recursion, partial fractions.
Use of tables. Brief demonstration of computer-based tools.
C. Infinite Series:.
Series. Sequences. Convergence of sequences. Theorems on limits of sums,
products, quotients.
Convergence of series. Tests: Ratio test, Integral test.
Taylor series of elementary functions, including general binomial,
trigonometric functions, exp and log.
Radius of convergence of power series. Finding the radius of
convergence of simple power series.
Recommended Reading:
F. Giannasi and R. Low. Maths for Computing and Information Technology.
Longman. 1995.
Chapter 9 covers some of the material of the course.
Mathematics Department Notes on Calculus.
These have less detail and fewer examples
than the required text, but you will find them
useful.
A.C. Bajpai, I.M. Calus and J.A. Fairley. Mathematics for Engineers and
Scientists. Vol I & II. Library: 510 BAI.
Covers the whole of the course, and a good deal more. Also covers parts
of courses 108 and 217.
Examination: Three hour final and continuous assessment.
The final examination will have three sections, 8 questions,
6 for full marks:
Section A (Differential Calculus): 3 questions, attempt at least 2;
Section B (Integral Calculus): 3 questions, attempt at least 2;
Section C (Infinite Series): 2 questions, attempt at least 1.
updated 29-8-2000.