City University London

MA2601 Calculus

The material in this page is for year 2009/10.

Teaching There will be 2 hours lectures and 1 hour tutorials per week for 10 weeks (including revision on week 10).
Notes will be provided at some stage after the lectures. Students are advised to attend lectures and take their own notes. Exercise sheets will be posted on-line and given to students during the tutorials. Solutions will be posted on-line later on.
There will be 2 courseworks (contributing 5% each towards the final mark)
The exam will take place in May 2010 together with Algebra. There will  be 4 Calculus and 4 Algebra questions, each worth 20 points. The maximum mark will be achieved for correct answers to 5 questions, with at least 2 questions answered from each part.
Past papers Solutions to past years papers can be found here.
Exercise sheets
Exercise sheet 1    and    Solutions
Exercise sheet 2    and    Solutions
Exercise sheet 3    and    Solutions
Exercise sheet 4    and    Solutions
Exercise sheet 5    and    Solutions
Exercise sheet 6    and    Solutions
Exercise sheet 7    and    Solutions
In addition to these exercise sheets, interesting exercises with solutions are available from here. This is the web page for the 2nd year Calculus module that Actuarial Science students take. It is not the same module as the one you take, but exercise sheets 1 to 4 cover partial derivatives, Lagrange multipliers and double integrals.
Coursework 1    and   Solutions
Coursework 2    and   Solutions
Module notes
Some interesting web-sites
This is a rather nice web-site which allows you to draw functions of two variables. You can also use this site to visualize the partial derivatives.
This is another web-site which allows you to visualize the effect of a change of variables in two dimensions.


This course has two main aims: first, the generalization and/or extension of many concepts you have learned in your first year calculus to functions of more than one real variable (e.g. continuity, limits, differentiability etc.) and, second, learning new methods for solving ordinary differential equations, which should also complement those you have learnt last year.
In the course we will basically study continuous and differentiable functions of several real variables (concentrating often on the example of functions of two real variables) as well as linear ordinary differential equations.
SYLLABUS: 1. Introduction: what this course is about
2. Functions of several real variables
2.1. Basic definitions and examples
    2.2. Limits and continuity
2.2.1. Neighbourhood of a point in 1, 2 and 3-dimensions
           2.2.2. Limits of functions of  two variables
           2.2.3. Continuity of functions of two variables

    2.3. Differentiation of functions of several real variables
           2.3.1. Definition of partial derivative for a function of two variables.
           2.3.2. Chain rules.
           2.3.3. Definition of differential.
           2.3.4. Transformation of partial derivatives.
           2.3.5. Extension to functions of more than two variables.
    2.4. Local properties of functions of several variables
           2.4.1. Taylor series expansions.
           2.4.2. Classification of stationary values of functions of two variables.
           2.4.3. Constraints via Lagrange multipliers.
    2.5. Integration of functions of several variables
           2.5.1. Standard coordinate systems in and R³ .
2.5.2. Integration in and R³ .
           2.5.3. Change of variables and Jacobians.
3. Differential equations
    3.1. Linear differential equations. 
           3.1.1. Second-order linear differential equations.
           3.1.2. The method of variation of parameters.
[1] Calculus, by T. Apostol (Wiley).
[2] Calculus: One and several variables, by S.N. Salas and  E. Hille  (Wiley).
[3] Calculus: A complete course, by R.A. Adams (Addison Wesley).
[4] Calculus, by H. Anton (Wiley).