# Southampton MATH1064 Calculus I: Syllabus, Topics and Practice

> A study guide for MATH1064 Calculus I at Southampton: limits, epsilon-delta, derivatives, the Mean Value Theorem, integration and ODEs, with free practice.

Source: https://practicecalculus.com/courses/university-of-southampton-math1064
Updated: 2026-11-08

## At a glance

- **Module code:** MATH1064
- **Credits:** 15 CATS points, Level 4 (Year 1)
- **Semester:** Semester 1
- **Prerequisites:** Not stated on the module page; it is described as a bridge from A Level Mathematics
- **Covers:** Functions, epsilon-delta limits, continuity, the Intermediate Value Theorem, differentiation, Rolle's theorem and the Mean Value Theorem, L'Hôpital's rule, Taylor series, integration techniques, improper integrals, first and second order ODEs
- **Assessment:** 40% coursework, 60% exam
- **Next module:** MATH1065 Calculus II (Semester 2)
- **Practice here:** [Calculus 1](/learn/calculus-1) plus the first units of [Calculus 2](/learn/calculus-2)

## Key takeaways

- MATH1064 Calculus I is a 15-credit Level 4 module taught in Semester 1 at the University of Southampton.
- It bridges A Level and university mathematics, revisiting familiar topics in more depth and adding new ones such as epsilon-delta limits.
- The syllabus covers functions, limits and continuity, differentiation, integration and ordinary differential equations.
- It is assessed by 40% coursework and a 60% exam, and leads into MATH1065 Calculus II.

**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

| MATH1064 topic | Practice | Read |
| --- | --- | --- |
| Functions, domain, range and inverses | [Function notation](/learn/calculus-1/functions-warmup/function-notation) | [Inverse function rule](/rules/inverse-function-rule) |
| Limits and the algebra of limits | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [Limit laws](/rules/limit-laws), [how to evaluate limits](/blog/how-to-evaluate-limits) |
| Continuity and the Intermediate Value Theorem | [Continuity](/learn/calculus-1/limits/continuity) | [Intermediate Value Theorem](/theorems/intermediate-value-theorem) |
| Derivative as a limit, tangent lines | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Product, quotient and chain rules | [Chain rule](/learn/calculus-1/derivative-rules/chain-rule-general) | [Product rule](/rules/product-rule), [chain rule](/rules/chain-rule) |
| Trig, hyperbolic, exponential and log derivatives | [Trig derivatives](/learn/calculus-1/derivative-rules/trig-derivatives) | [Hyperbolic functions](/functions/hyperbolic-functions), [derivative of ln x](/derivatives/ln-x) |
| Inverse trig derivatives | [Inverse trig derivatives](/learn/calculus-1/derivative-rules/inverse-trig-derivatives) | [Derivative of arctan x](/derivatives/arctan-x) |
| Rolle's theorem and the Mean Value Theorem | [Linear approximation and the MVT](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Rolle's theorem](/theorems/rolles-theorem), [Mean Value Theorem](/theorems/mean-value-theorem) |
| L'Hôpital's rule | [L'Hôpital's rule](/learn/calculus-1/applications-of-derivatives/lhopitals-rule) | [L'Hôpital's rule](/rules/lhopitals-rule) |
| Maxima, minima and curve sketching | [Critical points](/learn/calculus-1/applications-of-derivatives/critical-points) | [Curve sketching](/blog/curve-sketching-with-derivatives) |
| Taylor series | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor series](/blog/taylor-series) |
| Riemann sums and the Fundamental Theorem | [Riemann sums](/learn/calculus-1/integrals/riemann-sums) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |
| Substitution, parts and partial fractions | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [u-substitution](/blog/u-substitution), [partial fractions](/blog/partial-fractions) |
| Improper integrals | [Integrating to infinity](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals](/blog/improper-integrals) |
| First order ODEs | [Separable equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Differential equation problems](/practice-problems/differential-equations) |

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 $\frac{dy}{dx} = xy$:

$$\int \frac{dy}{y} = \int x\,dx \quad \Rightarrow \quad y = Ae^{x^2/2}$$

**4. An improper integral.**

$$\int_0^{\infty} e^{-x}\,dx = \lim_{t \to \infty} \left(1 - e^{-t}\right) = 1$$

## 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](/theorems/mean-value-theorem) precisely, not just use them.
- **Keep the formulas close.** The [calculus formula sheet](/formulas) has every standard derivative and integral on one page.
- **Revise with timed problems** using [practice exams](/practice) before the January exam.

> **Tip:** after MATH1064, [MATH1065 Calculus II](/courses/university-of-southampton-math1065) starts with second order inhomogeneous ODEs, so make sure the constant coefficient method from this module is solid before Semester 2.

## Frequently asked questions

### What is MATH1064 at Southampton?

MATH1064 Calculus I is a first-year mathematics module that bridges A Level and university mathematics. Its official syllabus covers functions, limits including the epsilon-delta definition, continuity, differentiation, integration and first and second order ordinary differential equations.

### Is MATH1064 just A Level Maths again?

No. Southampton says some material will be similar to A Level Maths and Further Maths but treated in more depth, and some will be new. New topics include the formal definition of a limit, Rolle's theorem, the Mean Value Theorem and improper integrals.

### How is MATH1064 assessed?

The 2026-27 module page lists 40% coursework and a 60% exam. Teaching is 48 hours of lectures, problem classes and workshops, with 102 hours of independent study.

### What comes after MATH1064?

MATH1065 Calculus II in Semester 2, which covers second order inhomogeneous ODEs and calculus of several variables, including partial derivatives, Lagrange multipliers and double integrals.

### How can I prepare for MATH1064?

Make your A Level differentiation and integration fast and accurate, and practise working with inequalities and limits. The free Calculus 1 path on PracticeCalculus covers the computational side in the same order.