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# Operating Machines

Part 1: Using RECIPROCAL and PRODUCT

Starting with $f(x)=x$, $x^n$ is possible for all integers n, and there are no other possibilities. The same set of possibilities can be created by starting with $f(x)= \frac{1}{x}$. No other starting function will yield the same set.

Part 2: Using DIFF and INT

Starting with $f(x) = x$, it is possible to create $0$ and $\displaystyle{\frac{x^n}{n!}}$ for any non-negative integer $n$

Starting with any function from this set, it is possible to generate all others using DIFF and INT.

Part 3:

Functions yielding a finite set of possibilites under DIFF and INT are $e^x$, $\sin x$ and $\cos x$

$\displaystyle{\frac{\mathrm{d}}{\mathrm{d}x} e^x = \int e^x = e^x}$

$\displaystyle{\frac{\mathrm{d}}{\mathrm{d}x} \left( \sin x \right) = \cos x

\rightarrow \frac{\mathrm{d}}{\mathrm{d}x} \left(\cos x \right) = -\sin x

\rightarrow \frac{\mathrm{d}}{\mathrm{d}x} \left( -\sin x \right) = \cos x

\rightarrow \frac{\mathrm{d}}{\mathrm{d}x} \left( -\cos x \right) = \sin x}$

The sequence then repeats.

There are others, for example $e^{-x}$ yields two possibilities, and $\sinh x$ and $\cosh x$ also yield two possibilities.

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Part 1: Using RECIPROCAL and PRODUCT

Starting with $f(x)=x$, $x^n$ is possible for all integers n, and there are no other possibilities. The same set of possibilities can be created by starting with $f(x)= \frac{1}{x}$. No other starting function will yield the same set.

Part 2: Using DIFF and INT

Starting with $f(x) = x$, it is possible to create $0$ and $\displaystyle{\frac{x^n}{n!}}$ for any non-negative integer $n$

Starting with any function from this set, it is possible to generate all others using DIFF and INT.

Part 3:

Functions yielding a finite set of possibilites under DIFF and INT are $e^x$, $\sin x$ and $\cos x$

$\displaystyle{\frac{\mathrm{d}}{\mathrm{d}x} e^x = \int e^x = e^x}$

$\displaystyle{\frac{\mathrm{d}}{\mathrm{d}x} \left( \sin x \right) = \cos x

\rightarrow \frac{\mathrm{d}}{\mathrm{d}x} \left(\cos x \right) = -\sin x

\rightarrow \frac{\mathrm{d}}{\mathrm{d}x} \left( -\sin x \right) = \cos x

\rightarrow \frac{\mathrm{d}}{\mathrm{d}x} \left( -\cos x \right) = \sin x}$

The sequence then repeats.

There are others, for example $e^{-x}$ yields two possibilities, and $\sinh x$ and $\cosh x$ also yield two possibilities.

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?