Southampton MATH1064 Calculus I: Syllabus, Topics and Practice

MATH1064 Calculus I is the Semester 1 calculus module for Year 1 mathematics students at the University of Southampton. Southampton describes it as a bridge between A Level mathematics and university mathematics: some of the material will feel familiar, but it is treated in more depth, and some of it will be new. This guide walks through the official syllabus and links every topic to free practice so you can learn by solving problems.

What Southampton MATH1064 covers

According to the 2026-27 module page, MATH1064 studies functions, limits, continuity, differentiation, integration and ordinary differential equations. The syllabus is organised in five blocks:

• Functions: domain and range, inverse functions, and injections, surjections and bijections.
• Limits: informal and formal ideas of a limit, the epsilon-delta definition with examples, the algebra of limits, continuity of sums, products and compositions, and the Intermediate Value Theorem.
• Differentiation: slopes and tangent lines, the formal definition as a limit, the product, quotient and chain rules, derivatives of trigonometric, hyperbolic, inverse trigonometric, exponential and logarithmic functions, Rolle's theorem and the Mean Value Theorem, L'Hôpital's rule, maxima and minima, curve sketching and Taylor series.
• Integration: integration via Riemann sums, the Fundamental Theorem of Calculus, substitution, integration by parts, partial fractions and improper integrals.
• Differential equations: first order equations (separable, linear, homogeneous, Bernoulli and Clairaut) and second order constant coefficient equations with complementary functions and particular integrals.

That covers all of a typical Calculus 1 course and a good part of Calculus 2, plus a first course in differential equations.

MATH1064 topic map

Second order constant coefficient ODEs are on the syllabus but are not yet covered by a lesson here, so lean on your problem classes for those.

Example problems at the MATH1064 level

1. An epsilon-delta limit. To prove \lim_{x \to 2} (3x - 1) = 5, note |(3x - 1) - 5| = 3|x - 2|. Given \varepsilon > 0, take \delta = \frac{\varepsilon}{3}.

2. The Mean Value Theorem. For f(x) = x^3 on [0, 2], there is a c with f'(c) = \frac{8 - 0}{2 - 0} = 4, so 3c^2 = 4 and c = \frac{2}{\sqrt{3}}.

3. A separable ODE. Solve \frac{dy}{dx} = xy:

4. An improper integral.

How to do well in MATH1064

• Take the coursework seriously. It is worth 40% of the module, so steady weekly work pays off directly.
• Practise inequalities. One learning outcome is handling inequalities for real numbers accurately, and epsilon-delta proofs depend on it.
• Do not rely on A Level habits. Learn to state theorems such as the Mean Value Theorem precisely, not just use them.
• Keep the formulas close. The calculus formula sheet has every standard derivative and integral on one page.
• Revise with timed problems using practice exams before the January exam.

Tip: after MATH1064, MATH1065 Calculus II starts with second order inhomogeneous ODEs, so make sure the constant coefficient method from this module is solid before Semester 2.