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Here we collect 10 essential applied mathematics problems to get you thinking before you embark on your degree, although many of the problems from the physics and engineering areas will also be relevant. They will be very useful to anyone intending to study mathematics, physics or engineering: they will give you a good mathematical grounding in some of the topics likely to arise in your degree course and refine your problem solving skills.

Remember, these problems are designed to make you think and there is not necessarily a 'right' answer. Approach them in a thoughtful way; they are hopefully both interesting and stimulating. What questions do they raise in you mind? Where do these questions lead you? Take them to a level that feels comfortable for you.

Finally, once you have done the problems, study the solutions. These will give you additional insights into the problems and the underlying mathematics.

Calculus Countdown	Get started with a game! Calculus is crucial at university, so the more familiar you are with it, the better.
Whose line graph is it anyway? Integration Matcher	Review your knowledge of functions, graphs and processes by thoroughly understanding these two problems.
Implicitly	Think about this implicity defined equation. The extension part will test out your calculus.
Stats statements	A good understanding of statistics and probability will set you in good stead. Here you can put your knowledge into practice in these thought provoking questions.
Curve fitter 2	This problem will reinforce your understanding of geometry; the extension provides a difficult mathematical challenge.
Making functions from their equations	Which came first: the equation or the function? In advanced applications, functions can be defined as solutions to differential equations. This problem will introduce you to these concepts.
Fix me or crush me	Vectors and matrices together form a basic foundation stone of mathematics. Here you can explore how the two work together.
Operating machines	Operators are a new way of looking at functions and equations. Here you can explore these ideas.
Bessel's equation	Equations at university become complicated rather quickly. In this problem you can get a feel for some of the ideas involved in advanced equations.
The not-so-simple pendulum 2	See how second order differential equations evolve in this challenging exercise in calculus.

The following article, interspersed with small problems for you to try, is a very useful introduction to complex numbers

An introduction to complex numbers



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