Calculus and Vectors B
|Unit level:||Level 1|
|Teaching period(s):||Semester 1|
|Offered by||School of Mathematics|
|Available as a free choice unit?:||N
- MATH10111 - Foundations of Pure Mathematics B (Compulsory)
The course unit unit aims to provide an introduction to the basic elements of calculus.
This lecture course introduces the basic ideas of complex numbers relating them the standard transcendental functions of calculus. The basic ideas of the differential and integral calculus are revised and developed. Vectors in two and three dimensions are introduced and this leads on to the calculus of functions of more than one variable and the beginnings of vector calculus.
On successful completion of this module students will be able to:
- Represent complex numbers in the complex plane and use them to relate trigonometric and exponential functions.
- Sketch polynomial, rational, inverse and some standard functions of a single variable in Cartesian and polar coordinates.
- Evaluate and interpret limits and derivatives of algebraic functions, including functions expressed in implicit or parametric form.
- Select and deploy methods for evaluating integrals of functions of a single variable.
- Construct and manipulate Taylor series of scalar functions of one and two variables.
- Manipulate vectors using tools such as scalar and vector products, deploying these quantities to solve geometric problems.
- Construct, deploy and interpret derivatives of scalar functions of more than one variable.
- Construct, evaluate and interpret simple integrals of functions of two variables, using tools such as the Jacobian and Green's theorem. Construct and evaluate line integrals.
- Locate and classify extrema of functions of two variables.
- Other - 25%
- Written exam - 75%
Assessment Further Information
Supervision attendance and participation; Weighting within unit 10%
Coursework; One in-class test, weighting within unit 15%
Two and a half hours end of semester examination; Weighting within unit 75%
- 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 the chain rule; implicit functions; logarithmic differentiation; higher derivatives (use in curve sketching).
- Infinite Series. Notation, basic notions of convergence, radius of convergence; infinite Taylor's series; expansions 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 and 3 Dimensions. Representation as directed line segments (magnitude, direction); choice of axes, components, Cartesian representation; basic properties, addition, subtraction, polar representation and relationship with complex numbers in 2 dimensions; 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 and 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; Stokes' theorem and Green's theorem.
The course is based on the following text:
James Stewart, Calculus, Early Transcendentals, International Student Edition, Thomson (any recent edition).
Feedback seminars will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
- Lectures - 33 hours
- Tutorials - 11 hours
- Independent study hours - 106 hours