Mathematics 0C2 (MATH19832)

Spring 2018, 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 Dr Sean Prendiville's course, 0C1.)

Intended learning outcomes

  • Knowledge and understanding: Be familiar with partial fractions, series and functions, differentiation and integration.
  • Intellectual skills: Be able to carry out routine operations involving the topics in the syllabus.
  • Transferable skills and personal qualities: Have a set of tools and methods that can be applied in the courses given in the host department or in subsequent years.


  • Integration (4 lectures)
    Derivatives and anti-derivatives; indefinite integrals and integration using tables/inspection; definite integrals and the area under graph; integration by substitution.
  • Trigonometry (4 lectures)
    Trigonometric functions; addition formulas and further identities; inverse trigonometric functions; using trigonometric identities for integration.
  • Sequences and series (3 lectures)
    Arithmetic progressions; geometric progressions; polynomials and the binomial theorem.
  • Further calculus (6 lectures)
    Differential and derivative; Taylor series; implicit differentiation; integral and differential; integration by parts.
  • Rational functions (5 lectures)
    Proper and improper rational functions; polynomials and long division; partial fractions decomposition; integration of rational functions.



  • 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.


Tutorials are held every Friday 11-12 at the Sackville Street Building, room D39.

Content gone through during the lectures

  • 29.1.2018: Revision of differentiation, antiderivatives
  • 2.2.2018: Indefinite integral, examples
  • 5.2.2018: Definite integral, the area under graph
  • 9.2.2018: Integration by substitution
  • 12.2.2018: Trigonometric functions and identities
  • 16.2.2018: Further trigonometric identities and using them in integration
  • 19.2.2018: Class test 1
  • 23.2.2018: Trigonometric functions in integrations, inverse functions
  • 26.2.2018: Inverse trigonometric functions, solving trigonometric equations
  • 2.3.2018: Sequences, arithmetic progressions, formula for the kth element and the sum of first n terms
  • 5.3.2018: Geometric progressions, the sum of first n terms and geometric series
  • 9.3.2018: Revision on arithmetic and geometric progressions, Pascal's triangle and the binomial theorem
  • 12.3.2018: Examples on using binomial theorem, review of further calculus
  • 16.3.2018: Differentials, chain rule and product rule
  • 19.3.2018: Implicit differentiation
  • 23.3.2018: Integration by parts
  • 26.3.2018 - 15.4.2018: Easter break
  • 16.4.2018: Revision of integration by parts, Taylor series
  • 20.4.2018: More examples on Taylor series, starting rational functions
  • 23.4.2018: Class test 2
  • 27.4.2018: Reducing improper rational functions to proper rational functions using long division
  • 30.4.2018: Spitting proper rational functions to partial fractions using factorisation
  • 4.5.2018: Using what we did earlier to compute integrals of rational functions
  • 7.5.2018: May bank holiday
  • 11.5.2018: 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'