# UNSW MATH1231 Mathematics 1B: Calculus Study Guide and Practice

> A study guide for UNSW MATH1231 Mathematics 1B: partial derivatives, integration techniques, ODEs, sequences, series and Taylor series, with practice.

Source: https://practicecalculus.com/courses/unsw-math1231
Updated: 2026-11-03

## At a glance

- **Course code:** MATH1231
- **Units of credit:** 6
- **Offered:** Term 1, Term 2 and Term 3
- **Prerequisites:** MATH1131 or MATH1141 or DPST1013
- **Covers:** Calculus half: partial derivatives and multivariable chain rules, integration techniques and applications, ODEs, sequences, series and Taylor series. Also linear algebra and probability distributions
- **Higher version:** MATH1241
- **Next course:** MATH2011 (Several Variable Calculus) or MATH2111 (Higher Several Variable Calculus)
- **Practice here:** [Calculus 2](/learn/calculus-2) plus the partial derivatives unit of [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH1231 Mathematics 1B is a 6 unit of credit course at UNSW Sydney that follows MATH1131 or MATH1141, offered in all three terms.
- Its calculus half covers partial derivatives, multivariable chain rules, integration techniques, ODEs, sequences, series and Taylor series.
- The rest of the course covers linear algebra (vector spaces, linear maps, eigenvalues) and a topic on probability distributions.
- MATH1241 is the higher version, and MATH1231 is a prerequisite for MATH2011 and MATH2111.

**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 topic | Practice | Read |
| --- | --- | --- |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives](/blog/partial-derivatives) |
| Multivariable chain rules | [Multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Chain rule](/rules/chain-rule) |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts](/blog/integration-by-parts) |
| Trig integrals and substitutions | [Trig substitution](/learn/calculus-2/integration-techniques/trig-substitution) | [Trig substitution](/blog/trig-substitution) |
| Partial fractions | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Partial fractions](/blog/partial-fractions) |
| Applications of integration | [Arc length and surface area](/learn/calculus-2/improper-integrals-and-applications/arc-length-and-surface-area) | [Volumes of revolution](/blog/volumes-of-revolution) |
| Ordinary differential equations | [Separable differential equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Natural exponential function](/functions/natural-exponential-function) |
| Sequences | [Sequences and their limits](/learn/calculus-2/sequences-and-series/sequences) | [Limits at infinity](/blog/limits-at-infinity) |
| Series and convergence | [Choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) | [Convergence tests](/blog/convergence-tests), [ratio test](/rules/ratio-test) |
| Taylor series | [Common Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) | [Taylor series](/blog/taylor-series), [Taylor's theorem](/theorems/taylors-theorem) |

## MATH1231 in the UNSW sequence

| Course | Calculus content | Prerequisite |
| --- | --- | --- |
| MATH1131 Mathematics 1A | Continuity, differentiability, the Riemann integral | None listed (assumes HSC Extension 1 and 2) |
| MATH1231 Mathematics 1B | Partials, integration techniques, ODEs, series | MATH1131, MATH1141 or DPST1013 |
| MATH2011 Several Variable Calculus | Multivariable and vector calculus | MATH1231, MATH1241, MATH1251 or DPST1014 |
| MATH2111 Higher Several Variable Calculus | The higher version of MATH2011 | One of those courses with a mark of at least 70 |

See the [MATH1131 guide](/courses/unsw-math1131) for a refresher on the theorems from Mathematics 1A, and the [MATH2011 guide](/courses/unsw-math2011) 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

$$\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt} = 2xy(-\sin t) + x^2\cos t = -2\cos t\sin^2 t + \cos^3 t$$

**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 $y(0) = 0$, $C = -1$:

$$y = 1 - e^{-x}$$

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

$$e^{-x^2} = 1 - x^2 + \frac{x^4}{2!} - \frac{x^6}{3!} + \cdots$$

## How to do well in MATH1231

- **Train technique selection.** Exam integrals do not tell you which method to use. The [choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) lesson builds that habit.
- **Use a fixed order for series tests.** Check the [nth-term test](/rules/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](/formulas).
- **Avoid common traps** in [common calculus mistakes](/blog/common-calculus-mistakes), and practice under time limits with [practice](/practice) sets.

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

## Frequently asked questions

### What is MATH1231 at UNSW?

MATH1231 Mathematics 1B is the second first-year mathematics course for STEM students at UNSW Sydney. Its calculus half covers partial derivatives and multivariable chain rules, further integration techniques and applications, ordinary differential equations, sequences, series and Taylor series.

### What do I need before MATH1231?

The Handbook lists MATH1131, MATH1141 or DPST1013 as the prerequisite.

### What is the difference between MATH1231 and MATH1241?

MATH1241 is the higher version of MATH1231, with greater depth and a greater emphasis on proof and rigorous argument. The two courses exclude each other.

### Is MATH1231 harder than MATH1131?

Many students find it harder, because series and integration techniques are new ideas rather than extensions of school work, and the linear algebra becomes abstract. Regular practice with mixed problems is the best preparation.

### What comes after MATH1231?

For calculus, the second-year options include MATH2011 Several Variable Calculus and MATH2111 Higher Several Variable Calculus, which needs a mark of at least 70 in MATH1231 or an equivalent course.