Warwick MA143 Calculus 2: Syllabus, Topics and Practice

MA143 Calculus 2 is the second Year 1 calculus module at the University of Warwick. Like MA142 before it, it is worth 10 credits, runs for 10 weeks and develops one-variable analysis "with a focus on calculations". Where MA142 dealt with limits, series and continuity, MA143 moves on to derivatives, Taylor series and the integral. This guide maps each MA143 topic to free practice with instant feedback.

What Warwick MA143 covers

The official module catalogue lists this indicative outline syllabus: differentiability, Taylor's theorem, Taylor's series, the Riemann integral, methods of integration, the Fundamental Theorem of Calculus and improper integrals.

Its learning outcomes say you should be able to:

• work with differentiability, higher derivatives and the properties of differentiable functions,
• use Taylor's theorem and Taylor series, and understand when a function can be represented by a power series,
• understand how the Riemann integral is constructed,
• apply the key properties of the integral, such as the integrability of continuous functions on bounded intervals and the Fundamental Theorem of Calculus.

In US terms, MA143 mixes the second half of Calculus 1 (derivatives and integrals) with parts of Calculus 2 (Taylor series, integration techniques and improper integrals), taught with more attention to definitions and proofs. The catalogue does not say which term it runs in.

MA143 topic map

Where MA143 sits

Example problems at the MA143 level

1. A Taylor polynomial. The degree 3 Maclaurin polynomial of \sin x is

and Taylor's theorem bounds the error by \frac{|x|^5}{120} for every real x, because all derivatives of \sin x are bounded by 1.

2. Integration by parts. With u = \ln x and dv = dx,

3. An improper integral.

How to do well in MA143

• Get the techniques automatic. Substitution, integration by parts and partial fractions should take seconds, so exam time goes on the reasoning. Our integration by parts problems help.
• Learn the key Maclaurin series. e^x, \sin x, \cos x and \frac{1}{1-x} come up constantly.
• Connect the integral to sums. Drawing the Riemann sum behind each definite integral makes the Fundamental Theorem much easier to understand.
• Do the assignments carefully. They are only 15% of the mark, but they are the best rehearsal for the 85% exam.

Tip: before using a Taylor series, check where it converges. The ratio test usually gives the radius of convergence in one line.