UNSW MATH1131 Mathematics 1A: Calculus Study Guide and Practice

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 vs MATH1141, and what comes next

MATH1131 and MATH1141 exclude each other, and both lead to MATH1231 or its higher version MATH1241. See the MATH1231 guide and the MATH2011 guide 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

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

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 and Extreme Value Theorem pages with that in mind.
• Keep computation sharp. Proof questions sit alongside routine derivatives and integrals, so drill the derivative rules until they are automatic.
• Watch for classic errors listed in common calculus mistakes.
• Use one formula sheet while studying: the calculus formula sheet.
• Test yourself under exam conditions with practice sets, following 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.