MT1131 Calculus and Vectors

Course Description

Rationale

The course is intended to serve as an introduction to the basic elements of calculus.

Learning Outcomes

On successful completion of the course students will have acquired active knowledge and understanding of some basic concepts and results in calculus.

Syllabus

• Numbers and Functions: basic algebra of real and complex numbers; real line and complex plane; graphs and curve sketching; functions, domain and range, inverse functions; standard functions and inverse functions; basic algebra of real and complex numbers.
• Limits and Differentiation: basic notion of limit and continuity; discontinuities, left and right limits; finding some limits; definition of derivative; derivatives of standard functions and their inverses; sums, products, quotients and chain rule; implicit functions; logarithmic differentiation; higher derivatives (use in curve sketching).
• Infinite series: notation, basic notion of convergence, radius of convergence; infinite Taylor's series; expan sions for standard functions.
• More on Complex Numbers: Euler’s theorem and de Moivre’s theorem; polar form of complex numbers (polar representation of the plane); roots of unity; complex forms of sin and cos, relationship to trigonometric identities.
• Integration: definite and indefinite integrals; fundamental theorem of calculus; techniques: linearity, integration by parts, partial fractions, substitution; lengths of curves, surfaces and volumes of revolution.
• Vectors in 2-D and 3-D: representation as directed line segments (magnitude, direction) choice of axes, components, Carteisan representation; basic properties, addition, subtraction. polar representation and relation to complex numbers in 2-D; scalar and vector product; representation of lines, planes, curves and surfaces.
• Functions of more than one variable: partial derivative, chain-rule, Taylor expansion; turning points (maxima, minima, saddle-points); grad, div, curl and some useful identities in vector calculus; integration in the plane, change of order of integration; Jacobians and change of variable; line integrals in the plane, path-dependence, path independence.

Textbooks

• James Stewart. Calculus, Early Transcendentals, 5th Edition, International Student Edition. (Blue,Floor 2: 515/S27) [This text covers almost every aspect of what you will be learning, with many examples. It is recommended that you purchase this book, or ensure that you can have easy access to a copy.]