Module Code - Title:
MS4022
-
CALCULUS 2
Year Last Offered:
2025/6
Hours Per Week:
Grading Type:
N
Prerequisite Modules:
MS4021
Rationale and Purpose of the Module:
This module introduces the student to sequences and series, integral calculus, ordinary differential equations and functions of several variables. It develops problem solving skills in these topics.
Syllabus:
- Sequences and series: Limit of a sequence, convergence of a sequence;
series, convergence, tests for convergence, absolute and conditional convergence. Power series.
- MacLaurin and Taylor series: Order notation, big-O, little-O notation, asymptotic equivalence, Taylor's Theorem and remainders, applications.
- Indefinite Integral: Integration of standard functions, techniques including
integration by parts, substitution and partial fractions.
- Definite Integral: The limit of a Riemann sum, fundamental theorem of
calculus, Area between two curves, Volumes of revolution, Improper integrals.
- Introduction to ordinary differential equations: Definition of an ODE,linearity, first order variables separable, solution technique by integration.
- Introduction to functions of two real variables: Continuity, partial derivatives
and their geometrical interpretation, Leibniz's rule, conditions (without proof) for maximum, minimum, saddle-point.
Learning Outcomes:
Cognitive (Knowledge, Understanding, Application, Analysis, Evaluation, Synthesis)
On successful completion of this module, students should be able to:
1. Test a sequence or series for convergence using a classical method such as the ratio test or else be able to directly determine if it is convergent.
2. Calculate Taylor series, use Taylor's theorem to estimate remainders, and recognise and utilize big-O order notation.
3. Use techniques including integration by parts, substitution, and partial fractions to find indefinite integrals.
4. Calculate Riemann sums and find definite integrals as limits of Riemann sums.
5. Apply the Fundamental Theorem of Calculus, including differentiation under the integral sign.
6. Classify ordinary differential equations according to their order, linearity and homogeneity.
7. Solve first-order separable ordinary differential equations.
8. Find partial derivatives of functions of two variables and determine and classify local maxima, local minima, and saddle points.
Affective (Attitudes and Values)
none
Psychomotor (Physical Skills)
none
How the Module will be Taught and what will be the Learning Experiences of the Students:
Normal weekly lectures and tutorials. Mid-term exam(s) and final exam.
Motivating examples from research in mathematical modelling.
Graduates will demonstrate a firm grounding in the rigorous approach to calculus and have an appreciation of its many applications.
Research Findings Incorporated in to the Syllabus (If Relevant):
Prime Texts:
Adams, R. A. (2013)
Calculus: A Complete
Course, 8th ed.
, Prentice Hall
Other Relevant Texts:
Stewart J. (2012)
Calculus, 7th ed.
, Cengage Learning.
Anton, H. (2012)
Calculus: a new horizon, 10th ed.
, Wiley
Wade, W.R. (2009)
An introduction to analysis, 4th ed.
, Pearson
Programme(s) in which this Module is Offered:
BSFIMAUFA - FINANCIAL MATHEMATICS
BSECMSUFA - Economics and Mathematical Sciences
BSPHEDUFA - PHYSICAL EDUCATION
BSSCCHUFA - Science Choice
BSMSCIUFA - MATHEMATICAL SCIENCES
BSMAPHUFA - MATHEMATICS AND PHYSICS
Semester(s) Module is Offered:
Spring
Module Leader:
doireann.okiely@ul.ie