# UNSW MATH1131 Mathematics 1A: Calculus Study Guide and Practice

> A study guide for UNSW MATH1131 Mathematics 1A: limits, continuity, derivatives, the Mean Value Theorem and the Riemann integral, mapped to practice.

Source: https://practicecalculus.com/courses/unsw-math1131
Updated: 2026-11-03

## At a glance

- **Course code:** MATH1131
- **Units of credit:** 6
- **Offered:** Term 1, Term 2 and Term 3
- **Assumed knowledge:** NSW HSC Mathematics Extension 1 and 2
- **Covers:** Calculus half: continuous and differentiable functions, the Riemann integral, the Maximum-Minimum, Intermediate Value and Mean Value Theorems. Algebra half: vectors, matrices, Gaussian elimination
- **Higher version:** MATH1141 (Higher Mathematics 1A)
- **Next course:** MATH1231 (Mathematics 1B) or MATH1241
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MATH1131 Mathematics 1A is a 6 unit of credit first-year course at UNSW Sydney, offered in all three terms.
- Half of the course is calculus: continuity, differentiability, the Riemann integral and the big theorems that justify them. The other half is algebra.
- It assumes a good grasp of NSW HSC Mathematics Extension 1 and 2.
- It is usually followed by MATH1231 Mathematics 1B, and MATH1141 is its higher version.

**MATH1131 Mathematics 1A** is the first-year mathematics course most STEM students at UNSW Sydney take in their first term. It is worth 6 units of credit and runs in Term 1, Term 2 and Term 3. The course is half calculus and half algebra; this guide focuses on the calculus half and maps every topic to free practice, so you can learn each idea by solving problems with instant feedback.

## What MATH1131 covers

The UNSW Handbook describes MATH1131 as building on high school calculus to give a solid foundation for further mathematics. The calculus half deepens your understanding of continuous and differentiable functions and introduces the Riemann integral. It uses theorems such as the Maximum-Minimum Theorem, the Intermediate Value Theorem and the Mean Value Theorems to justify properties of common functions and to solve theoretical and applied problems. The algebra half introduces vectors, matrices and solving linear systems by Gaussian elimination. Maple is used throughout.

Compared with a standard **Calculus 1** course, MATH1131 assumes you have already met derivatives and integrals at school (NSW HSC Extension 1 and 2) and spends more time on why the results are true. Theorems are not just names to memorize; they are tools you apply.

## MATH1131 calculus topic map

| MATH1131 topic | Practice | Read |
| --- | --- | --- |
| Limits of functions | [One-sided and infinite limits](/learn/calculus-1/limits/one-sided-and-infinite-limits) | [How to evaluate limits](/blog/how-to-evaluate-limits), [squeeze theorem](/theorems/squeeze-theorem) |
| Continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Continuity](/blog/continuity) |
| Intermediate Value Theorem | [Continuity](/learn/calculus-1/limits/continuity) | [Intermediate Value Theorem](/theorems/intermediate-value-theorem) |
| Differentiability | [When derivatives fail](/learn/calculus-1/the-derivative/differentiability) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Differentiation rules | [Mixed rules practice](/learn/calculus-1/derivative-rules/derivative-rules-practice) | [Chain rule](/rules/chain-rule), [inverse function rule](/rules/inverse-function-rule) |
| Maximum-Minimum Theorem | [Absolute extrema on an interval](/learn/calculus-1/applications-of-derivatives/absolute-extrema) | [Extreme Value Theorem](/theorems/extreme-value-theorem) |
| Mean Value Theorem | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Mean Value Theorem](/theorems/mean-value-theorem), [Rolle's theorem](/theorems/rolles-theorem) |
| Applications of derivatives | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) |
| Riemann integral | [Area with rectangles](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums) |
| Fundamental Theorem of Calculus | [Derivatives of integrals](/learn/calculus-1/integrals/fundamental-theorem-part-1) | [FTC part 1](/theorems/fundamental-theorem-of-calculus-part-1), [Mean Value Theorem for integrals](/theorems/mean-value-theorem-for-integrals) |

## MATH1131 vs MATH1141, and what comes next

| Course | What it is | Terms |
| --- | --- | --- |
| MATH1131 Mathematics 1A | Standard first-year calculus and algebra | Terms 1, 2 and 3 |
| MATH1141 Higher Mathematics 1A | Same topics, more depth, more proof | Terms 1 and 3 |
| MATH1231 Mathematics 1B | Multivariable chain rules, integration techniques, ODEs, series, linear algebra | Terms 1, 2 and 3 |
| MATH2011 Several Variable Calculus | Second-year multivariable and vector calculus | Term 1 |

MATH1131 and MATH1141 exclude each other, and both lead to MATH1231 or its higher version MATH1241. See the [MATH1231 guide](/courses/unsw-math1231) and the [MATH2011 guide](/courses/unsw-math2011) for the rest of the sequence.

## Example problems at the MATH1131 level

**1. Using the Intermediate Value Theorem.** Show $x^3 + x - 1 = 0$ has a root in $[0, 1]$. Let $f(x) = x^3 + x - 1$, which is continuous. Since $f(0) = -1 < 0$ and $f(1) = 1 > 0$, the IVT gives a $c \in (0, 1)$ with $f(c) = 0$.

**2. Using the Mean Value Theorem.** Show $|\sin a - \sin b| \le |a - b|$ for all real $a, b$. By the MVT there is a $c$ between $a$ and $b$ with

$$\sin a - \sin b = \cos c\,(a - b)$$

and since $|\cos c| \le 1$, the inequality follows.

**3. Differentiating an integral.** If $F(x) = \int_0^{x^2} \cos t\,dt$, the Fundamental Theorem with the chain rule gives

$$F'(x) = \cos(x^2) \cdot 2x = 2x\cos(x^2)$$

## How to do well in MATH1131

- **Learn the theorems by their hypotheses.** Exam questions often hinge on checking continuity on a closed interval or differentiability on an open one. Read the [Mean Value Theorem](/theorems/mean-value-theorem) and [Extreme Value Theorem](/theorems/extreme-value-theorem) pages with that in mind.
- **Keep computation sharp.** Proof questions sit alongside routine derivatives and integrals, so drill the [derivative rules](/learn/calculus-1/derivative-rules/derivative-rules-practice) until they are automatic.
- **Watch for classic errors** listed in [common calculus mistakes](/blog/common-calculus-mistakes).
- **Use one formula sheet** while studying: the [calculus formula sheet](/formulas).
- **Test yourself under exam conditions** with [practice](/practice) sets, following [how to prepare for a calculus exam](/blog/how-to-prepare-for-a-calculus-exam).

> **Tip:** when a question says "show that there exists", think IVT or MVT first. Most existence questions in MATH1131 calculus come down to one of them.

## Frequently asked questions

### What is MATH1131 at UNSW?

MATH1131 Mathematics 1A is the standard first-year mathematics course for STEM students at UNSW Sydney. Its calculus half studies continuous and differentiable functions and introduces the Riemann integral, and its algebra half covers vectors, matrices and Gaussian elimination.

### What is the difference between MATH1131 and MATH1141?

MATH1141 Higher Mathematics 1A covers the same topics with greater depth and more emphasis on proof and rigorous argument. The two courses exclude each other, and MATH1141 is offered in Terms 1 and 3.

### What comes after MATH1131?

Students usually continue with MATH1231 Mathematics 1B, or its higher version MATH1241.

### Is MATH1131 hard?

It moves faster than HSC mathematics and asks you to justify results with theorems, not only compute. Students who keep up with weekly problems and practice derivatives and integrals regularly usually manage it well.

### How should I prepare for MATH1131 calculus?

Revise functions, exponentials, logs and trig from Extension 1 and 2, then practice limits and continuity. The free Calculus 1 path on PracticeCalculus starts with exactly that.