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Operating Machines

Age 16 to 18 Challenge Level:

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

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 $n$.

Key questions

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 $f(x)=x$?

Possible extension

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.

Possible support

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