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### Number and algebra

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### For younger learners

### Advanced mathematics

# Operating Machines

### Why do this problem?

This problem gives an insight into the concept of operators and
will give a great deal of practice in integration and
differentiation, reinforcing the concept that they are inverse
operations. It also reinforces ideas of closure, so important in
more advanced topics in algebra.
### Possible approach

### Key questions

### Possible extension

### Possible support

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

This is a very hands-on problem, although it will initially
require careful reading to understand the meaning of the
mathematical operations involved.

Once the meaning of the problem has been understood, students
should be encouraged to explore the possibilities generated by
repeatedly applying the operations, look for patterns and then try
to explain the meaning of the patterns.

Students should be encouraged to look for complete sequences
of functions found by repeated application of the operators, such
as the set of functions $\frac{x^n}{n!}$ where $n$ is a natural
number.

They can choose any familiar function to begin with. Some,
such as $\sin(x)$, will lead to interesting finite sequences,
whereas others, such as $\ln(x)$, will lead to problems. Others,
such as $\sqrt{x}$ will clearly lead to an infinite sequence of
possibilities, although students might find the task of writing an
algebraic expression for the general term a challenge.

To start students thinking about the more challenging
combinations of operations, first suggest looking at combining DIFF
and RECIPROCAL, starting from the function $f(x) = x$. Many
interesting properties emerge: for example, students might be able
to spot how to generate $2^n x$ for any value of $n$.

Have you carefully read the question?

What happens if you repeatedly apply DIFF to $f(x)=x$?

What happens if you repeatedly apply INT to $f(x)=x$?

What happens if you apply DIFF several times followed by INT
several times?

What functions might you try instead of $f(x)=x$?

There is plenty of extension built into the problem. The
better students should really focus on finding complete sets of
possibilities, and where possible find algebraic expressions for
the terms in such sets. They might also look at the effects of
starting with a pair of initial functions.

Focus on the functions $f(x) = x^n$ for various whole values
of $n$.

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?