# Warwick MA143 Calculus 2: Syllabus, Topics and Practice

> A study guide for Warwick MA143 Calculus 2: differentiability, Taylor series, the Riemann integral, integration methods and improper integrals, with practice.

Source: https://practicecalculus.com/courses/university-of-warwick-ma143
Updated: 2026-11-08

## At a glance

- **Module code:** MA143-10
- **Credits:** 10
- **Level:** Undergraduate Level 1 (Year 1)
- **Duration:** 10 weeks
- **Prerequisites:** Not stated in the module catalogue (it follows MA142 Calculus 1 in the same series)
- **Assessment:** Assignments 15%, on-campus exam 85%
- **Covers:** Differentiability, Taylor's theorem, Taylor series, the Riemann integral, methods of integration, the Fundamental Theorem of Calculus, improper integrals
- **Next module:** MA259 Multivariable Calculus (Year 2 option)
- **Practice here:** [Calculus 1](/learn/calculus-1) and [Calculus 2](/learn/calculus-2)

## Key takeaways

- MA143 Calculus 2 is a 10-credit, 10-week Year 1 module and the second in Warwick's calculation-focused analysis series.
- It covers differentiability, Taylor's theorem and Taylor series, the Riemann integral, methods of integration and improper integrals.
- It builds the integral carefully and proves facts such as the integrability of continuous functions and the Fundamental Theorem of Calculus.
- Assessment is 15% assignments and 85% on-campus exam.

**MA143 Calculus 2** is the second Year 1 calculus module at the University of Warwick. Like MA142 before it, it is worth 10 credits, runs for 10 weeks and develops one-variable analysis "with a focus on calculations". Where MA142 dealt with limits, series and continuity, MA143 moves on to derivatives, Taylor series and the integral. This guide maps each MA143 topic to free practice with instant feedback.

## What Warwick MA143 covers

The official module catalogue lists this indicative outline syllabus: differentiability, Taylor's theorem, Taylor's series, the Riemann integral, methods of integration, the Fundamental Theorem of Calculus and improper integrals.

Its learning outcomes say you should be able to:

- work with differentiability, higher derivatives and the properties of differentiable functions,
- use Taylor's theorem and Taylor series, and understand when a function can be represented by a power series,
- understand how the Riemann integral is constructed,
- apply the key properties of the integral, such as the integrability of continuous functions on bounded intervals and the Fundamental Theorem of Calculus.

In US terms, MA143 mixes the second half of **Calculus 1** (derivatives and integrals) with parts of **Calculus 2** (Taylor series, integration techniques and improper integrals), taught with more attention to definitions and proofs. The catalogue does not say which term it runs in.

## MA143 topic map

| MA143 topic | Practice | Read |
| --- | --- | --- |
| Differentiability | [Differentiability](/learn/calculus-1/the-derivative/differentiability) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Rules for derivatives | [Chain rule](/learn/calculus-1/derivative-rules/chain-rule-general) | [Chain rule](/rules/chain-rule), [product rule](/rules/product-rule) |
| Mean Value Theorem | [Linear approximation and the MVT](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Mean Value Theorem](/theorems/mean-value-theorem), [Rolle's theorem](/theorems/rolles-theorem) |
| Taylor's theorem | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor's theorem](/theorems/taylors-theorem) |
| Taylor series | [Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) | [Taylor series](/blog/taylor-series), [power series](/blog/power-series) |
| The Riemann integral | [Riemann sums](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums) |
| Fundamental Theorem of Calculus | [FTC part 1](/learn/calculus-1/integrals/fundamental-theorem-part-1) | [FTC part 1](/theorems/fundamental-theorem-of-calculus-part-1), [FTC part 2](/theorems/fundamental-theorem-of-calculus-part-2) |
| Methods of integration | [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) |

## Where MA143 sits

| Module | Year | Focus |
| --- | --- | --- |
| [MA142 Calculus 1](/courses/university-of-warwick-ma142) | Year 1 | Real numbers, limits, series, continuity |
| MA143 Calculus 2 | Year 1 | Derivatives, Taylor series, the integral |
| [MA259 Multivariable Calculus](/courses/university-of-warwick-ma259) | Year 2 option | Several variables and vector calculus |

## Example problems at the MA143 level

**1. A Taylor polynomial.** The degree 3 Maclaurin polynomial of $\sin x$ is

$$\sin x \approx x - \frac{x^3}{6}$$

and Taylor's theorem bounds the error by $\frac{|x|^5}{120}$ for every real $x$, because all derivatives of $\sin x$ are bounded by $1$.

**2. Integration by parts.** With $u = \ln x$ and $dv = dx$,

$$\int \ln x\,dx = x\ln x - \int x \cdot \frac{1}{x}\,dx = x\ln x - x + C$$

**3. An improper integral.**

$$\int_0^{\infty} e^{-x}\,dx = \lim_{t \to \infty}\left[-e^{-x}\right]_0^t = \lim_{t \to \infty}\left(1 - e^{-t}\right) = 1$$

## How to do well in MA143

- **Get the techniques automatic.** Substitution, integration by parts and partial fractions should take seconds, so exam time goes on the reasoning. Our [integration by parts problems](/practice-problems/integration-by-parts) help.
- **Learn the key Maclaurin series.** $e^x$, $\sin x$, $\cos x$ and $\frac{1}{1-x}$ come up constantly.
- **Connect the integral to sums.** Drawing the Riemann sum behind each definite integral makes the Fundamental Theorem much easier to understand.
- **Do the assignments carefully.** They are only 15% of the mark, but they are the best rehearsal for the 85% exam.

> **Tip:** before using a Taylor series, check where it converges. The [ratio test](/rules/ratio-test) usually gives the radius of convergence in one line.

## Frequently asked questions

### What is MA143 at Warwick?

MA143 Calculus 2 is a Year 1 module from the Warwick Mathematics Institute. It develops one-variable analysis with a focus on calculations: differentiability, Taylor's theorem and series, the Riemann integral, methods of integration, the Fundamental Theorem of Calculus and improper integrals.

### Do I need MA142 before MA143?

The catalogue does not list a formal prerequisite, but it describes MA143 as the second module in the series that starts with MA142 Calculus 1, and it relies on the limits, series and continuity covered there.

### Which degrees take MA143?

The 2026/27 catalogue lists it as core in Year 1 of Data Science, Discrete Mathematics, and Mathematics, Operational Research, Statistics and Economics.

### How is MA143 assessed?

Assignments count for 15% and a centrally timetabled on-campus exam counts for 85%. The resit is a single in-person exam worth 100%.

### Is MA143 harder than A level calculus?

The techniques will look familiar, but the module asks why they work, for example how the Riemann integral is built and when a function equals its Taylor series. Regular practice of the calculations frees up time to focus on the ideas.