UNSW MATH1231 Mathematics 1B: Calculus Study Guide and Practice

MATH1231 Mathematics 1B is the second first-year mathematics course for STEM students at UNSW Sydney, taken after MATH1131 or MATH1141. It is worth 6 units of credit and runs in all three terms. Like Mathematics 1A, it is half calculus and half algebra; this guide focuses on the calculus half and maps each topic to free practice, so you can learn every method by solving problems with instant feedback.

What MATH1231 covers

According to the UNSW Handbook, the calculus half of MATH1231 covers partial derivatives and multivariable chain rules, further techniques and applications of integration, ordinary differential equations, and sequences, series and Taylor series. The linear algebra half moves to abstract vector spaces, linear transformations, eigenvalues and eigenvectors, and there is also a topic on probability distributions. Maple and a typeset assignment are part of the course.

Compared with a standard Calculus 2 course, MATH1231 has the integration techniques, series and Taylor series you would expect. It adds a first look at partial derivatives and the multivariable chain rule, which many universities save for Calculus 3, and a solid block on differential equations.

MATH1231 calculus topic map

MATH1231 in the UNSW sequence

See the MATH1131 guide for a refresher on the theorems from Mathematics 1A, and the MATH2011 guide for what follows.

Example problems at the MATH1231 level

1. A multivariable chain rule. If z = x^2 y with x = \cos t and y = \sin t, then

2. A first-order linear ODE. Solve y' + y = 1 with y(0) = 0. The integrating factor is e^x, so (e^x y)' = e^x, giving e^x y = e^x + C. From y(0) = 0, C = -1:

3. A Taylor series. Replacing x with -x^2 in e^x = \sum \frac{x^n}{n!} gives

How to do well in MATH1231

• Train technique selection. Exam integrals do not tell you which method to use. The choosing a strategy lesson builds that habit.
• Use a fixed order for series tests. Check the nth-term test, then geometric and p-series, then comparison and ratio. Practice the order until it is automatic.
• Build new Taylor series from old ones. Substituting into known series, as in example 3, is faster than differentiating repeatedly. Keep the core series on your formula sheet.
• Avoid common traps in common calculus mistakes, and practice under time limits with practice sets.

Tip: check every ODE solution by substituting it back in. It takes a few seconds and catches most sign and constant errors.