Mathematics 0C2 (MATH19832)

Spring 2020, University of Manchester


Welcome to the Mathematics 0c2 course in the University of Manchester! The aim of the course is to provide an elementary second-semester course in calculus and algebra to students entering the university with no post-GCSE mathematics in the Foundation Year. Syllabus can be found from the university website: MATH19832 (This course follows directly on from Mathematics 0C1.)

We will continue where Mathematics 0C1 ended, in particular, we will give further techniques in differentiation and integration, which can be useful in applications. Moreover, we will introduce notions like sequences and series, which appear commonly in mathematics and nature. We will also define the notion of a complex number, which extend our notion of real numbers and have some beautiful geometric properties and appears commonly in problems in physics.

Succeeding in the Course Unit

What do I need to do to succeed in this course unit? There are 13 Intended Learning Outcomes (ILOs) for this course, which are listed as follows. On completion of this unit successful students will be able to:

  • Define complex numbers and sketch them using the Argand Diagram
  • Perform arithmetic operations on complex numbers and compute their moduli, arguments and conjugates
  • Express complex numbers in their polar and exponential forms and perform computations using these expressions
  • Define arithmetic, geometric and binomial sequences, evaluate their sums and compute convergent series
  • Define binomial coefficients, write binomial formula and apply it in integration exercises
  • Write Taylor and Maclaurin Series and apply them to compute limits
  • Apply implicit, logarithmic and parametric differentiation in differentiation exercises
  • Write integration by parts and integration by substitution formulae and apply them in integration exercises
  • Compute examples of improper integrals
  • Express improper rational functions as proper rational functions
  • Find partial fraction coefficients for proper rational functions
  • Apply the algorithms of simplifying improper rational functions to compute their integrals
It is helpful to follow them as the assessment (class tests and final exam) will be based on them and you can come back to this list to check what is expected and if there is something to be improved. Good luck with the course!


  • Written exam; Weighting within unit 80%. Date/time: TBA. You can use the Mathematical formula tables, which are provided by the examinations office, you can view them from here (pdf).
    See past exam papers: Note that last year we introduced complex numbers to the syllabus and some further integration topics. Thus some previous years (until 2018) exams do not exactly reflect the content of the course.

  • Class test 1; Weighting within unit 10%.
    Date/time: Tuesday 18th February, 2020, at 11 am. Held in the lecture room Renold E7. See the rules of the test.

  • Class test 2; Weighting within unit 10%
    Date/time: Tuesday 21nd April, 2020, at 11 am. Held in the lecture room Renold E7. See the rules of the test


  • Workbook (pdf), which we will use in the course. This includes exercises and examples, which we will fill during the classes. I highly recommend printing a copy of this workbook and keeping it with you during the lectures.


The lectures will take place on Tuesdays at 11:00 in Renold building room E7 and Fridays at 12:00 in the Sackville Street building room F41. In the lectures we will go through the workbook in visualiser and sometimes on blackboard, filling some of the exercises and examples. I highly recommend to attend the lectures as they provide excellent support for the studies.

In case you miss a lecture, there are still lecture podcasts available, which can be accessed from here (requires sign in). However, sometimes the podcast does not record properly or a blackboard is used so it may not be able to see all the content gone through lectures from the podcasts.


There are two possible tutorial sessions for this course: either Tuesday 10-11 am in George Begg C002, or Friday 11-12 am in George Begg C002.

In these sessions we go through some of the exercises carefully linked below, with emphasis on how to come up with the solution in the actual exam situation. Moreover, the sessions provide an excellent opportunity to ask questions about the course and we can go through together some harder parts of the course.

I recommend to attempt the exercises yourself before looking at the solutions!

Content gone through during the lectures

  • Week 1: Quadratic equation, definition of complex numbers, arithmetic operations, complex plane
  • Week 2: Arguments of complex numbers, polar and exponential form
  • Week 3: Examples of using exponential form
  • Week 4: Class test (Tuesday lecture) and arithmetic progressions
  • Week 5: More on arithmetic progressions and geometric progressions
  • Week 6: Binomial theorem and revision of differentiation, implicit differentiation
  • Week 7: Logarithmic and parametric differentiation, Taylor series
  • Week 8: Integration by substitution and parts, improper integrals
  • Week 9: Continuing integration examples
  • Easter break
  • Week 10: Class test (Tuesday lecture) and starting rational functions
  • Week 11: Continuing reducing partial fractions, integrating partial fractions
  • Week 12: Revision
Remember to fill up the unit survey for the course. Good luck with the exam!


Course leader Dr Tuomas Sahlsten (Email: tuomas.sahlsten 'at'