KCL 4CCM111A Calculus I: Syllabus, Topics and Practice

4CCM111A Calculus I is the autumn-semester calculus module for Year 1 mathematics students at King's College London. It is a required module in the Mathematics BSc and sits alongside Sequences and Series and Linear Algebra and Geometry I. This guide explains what the module covers according to King's own module description and maps each topic to free practice, so you can build fluency by solving problems with instant feedback.

What KCL Calculus I covers

King's describes the aim of the module as reviewing and enhancing pre-university mathematics so that students develop genuine confidence and fluency. A solid grasp of the basics then frees you to concentrate on the new, more abstract ideas in other Year 1 modules.

The official syllabus lists: injective, surjective and bijective functions; trigonometric functions; hyperbolic functions; inverse functions; the logarithm and exponential functions; limits; differentiation from first principles; integration from first principles; series; Taylor's theorem and L'Hôpital's rule.

So the module starts from material you met at A Level and rebuilds it carefully. Most of it matches a Calculus 1 course, while series and Taylor's theorem come from the start of Calculus 2. The module page does not list integration techniques such as integration by parts in detail, so check your lecture notes for how far each topic goes.

4CCM111A topic map

Each row links to a lesson you can practise and a reference page that explains the idea.

Where 4CCM111A fits in Year 1

Calculus I and Sequences and Series run side by side. Calculus I is about fluent computation, while Sequences and Series makes the idea of a limit rigorous with proofs. In the spring, Calculus II takes everything into several variables.

Example problems at the 4CCM111A level

1. A derivative from first principles. For f(x) = x^2,

2. A hyperbolic identity. Using \cosh x = \frac{e^x + e^{-x}}{2} and \sinh x = \frac{e^x - e^{-x}}{2}, you can check that \cosh^2 x - \sinh^2 x = 1 and that \frac{d}{dx}\cosh x = \sinh x.

3. L'Hôpital's rule. The limit \lim_{x \to 0} \frac{1 - \cos x}{x^2} has the form \frac{0}{0}, so

4. A Taylor polynomial. Near x = 0, e^x \approx 1 + x + \frac{x^2}{2} + \frac{x^3}{6}, and Taylor's theorem tells you how large the error can be.

How to do well in KCL Calculus I

• Hand in every weekly sheet. King's sets exercises each week and goes through them in tutorials. They are the best preparation for the class tests and the exam.
• Do not coast on A Level. The module reviews familiar material, but it asks for proofs from first principles. Practise writing out the limit definition of the derivative in full.
• Get comfortable with hyperbolic functions early. They often feel new to students who did not take Further Maths. The hyperbolic functions page covers the definitions and identities.
• Keep the formulas close. The calculus formula sheet has the standard derivatives and integrals on one page.
• Test yourself under time pressure with practice exams before the January exam.

Tip: series and Taylor's theorem also appear in Sequences and Series, from a more rigorous angle. Practising the computations here makes the proofs there much easier to follow.