Prepare for University  Applied Mathematics
Age 16 to 18 Challenge Level:
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. 

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 notsosimple 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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