University calculus: what first-year modules cover and how to practise

What first-year calculus modules cover

Course names and order differ, but the same building blocks keep appearing. Here is what three real courses list:

In the US, this material is often split into Calculus II (integration and series) and Calculus III (multivariable calculus), though the split varies, so check your course catalogue or module handbook for the exact list and the exam format.

The core topics

• Limits and continuity: precise limits, continuity and the behaviour of functions near a point.
• Techniques of integration: substitution, integration by parts, partial fractions.
• Improper integrals: integrals over infinite ranges or with infinite integrands, and when they converge.
• Sequences and series: convergence tests, power series and Taylor and Maclaurin series.
• Functions of several variables: partial derivatives, the chain rule, gradient, directional derivatives, and stationary points.
• Multiple integrals: double and triple integrals, changing the order, and polar or other coordinates with a Jacobian.
• Differential equations: first order (separable, integrating factor) and second order linear equations with constant coefficients.
• Vector calculus in many courses: line and surface integrals and the big integral theorems.

Worked examples

Integration by parts

An improper integral

The area under 1/x² from 1 to infinity is finite, but the same integral of 1/x diverges.

A Taylor polynomial

The Maclaurin series of eˣ starts 1 + x + x²/2 + x³/6. At x = 0.1 this gives 1.105167, very close to the true value e^0.1 = 1.105171.

Partial derivatives

For f(x, y) = x²y + sin(xy), treating y as a constant gives ∂f/∂x = 2xy + y cos(xy), and treating x as a constant gives ∂f/∂y = x² + x cos(xy).

A double integral

A first order differential equation

Solve dy/dx + 2y = 0 with y(0) = 5. Separating gives dy/y = −2 dx, so ln|y| = −2x + c and y = Ae^(−2x). The condition y(0) = 5 gives A = 5, so y = 5e^(−2x).

Warm up with definite integrals in our Calculus course. Pressing Mark opens the question in the course, where you answer it and it is marked:

Use the daily paper as practice

Uni-Sine publishes a free daily University Calculus paper. It is a short timed practice paper: 8 questions worth 25 to 35 marks, with a suggested 40 to 55 minutes, marked when you finish. It is not your university's exam, so use it alongside your module's problem sheets and past papers.

• What it covers well: limits and continuity, techniques of integration, improper integrals, sequences and series, Taylor series, partial derivatives, multiple integrals, and first and second order differential equations.
• What it does not cover: vector calculus (line and surface integrals and the integral theorems) and the proofs some analysis-style courses ask for.

Common mistakes in university calculus

• Treating an improper integral as ordinary. Write it as a limit and check whether it converges.
• Forgetting the Jacobian. Changing to polar coordinates turns dx dy into r dr dθ.
• Losing the constant in ODEs. Apply the initial condition only after integrating.
• Assuming a series converges because its terms shrink. The harmonic series 1 + 1/2 + 1/3 + ... diverges.

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