Operating machines
What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?
Age
16 to 18
Challenge level
Problem
My friend has two function machines. The first, called RECIPROCAL, takes as an input any function $f(x)$ and returns another function $1/f(x)$. The second, called PRODUCT, takes two functions $f(x)$ and $g(x)$ and returns a new function which is their product $f(x)g(x)$.
I start off with the function $f(x)=x$. Which functions can I make using the function machines RECIPROCAL and PRODUCT? Can you describe an entire theoretical set of such outputs? Prove your assertions. (Note that I have several copies of this function to hand). Are there any other starting functions which yield the same set of outputs? What sets of outputs will I get by starting with different initial functions?
I have two operator machines. The first, called DIFF, takes any input function and returns another function which is its derivative. (If the derivative function does not exist, then DIFF returns the original function). The second, called INT, takes any input function and returns the implicit integral (i.e. no constant of integration) of the original function (if this integral does not exist then INT returns the original function).
If I again start off with the function $f(x) = x$, what set of outputs can I make using DIFF and INT? Are there any other starting functions which yield the same set of outputs? What sets of outputs will I get by starting with different initial functions?
Can you find any initial functions which yield a finite set of possibilities under RECIPROCAL and PRODUCT or under DIFF and INT?
What happens if you begin with $f(x) = x$ and use any combination of RECIPROCAL and DIFF?
Explore the possibilities when you can use different combinations of RECIPROCAL, PRODUCT, DIFF and INT for various choices of initial function. Can you find any interesting sets of outcomes?
See also the problem Calculus Countdown
NOTES AND BACKGROUND
In mathematics and science, the concept of differential and integral operators is very important, with applications ranging from quantum chemistry to analysis of waves. The simplest examples of these are given in this problem under the guise of DIFF and INT.
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I start off with the function $f(x)=x$. Which functions can I make using the function machines RECIPROCAL and PRODUCT? Can you describe an entire theoretical set of such outputs? Prove your assertions. (Note that I have several copies of this function to hand). Are there any other starting functions which yield the same set of outputs? What sets of outputs will I get by starting with different initial functions?
I have two operator machines. The first, called DIFF, takes any input function and returns another function which is its derivative. (If the derivative function does not exist, then DIFF returns the original function). The second, called INT, takes any input function and returns the implicit integral (i.e. no constant of integration) of the original function (if this integral does not exist then INT returns the original function).
Image
If I again start off with the function $f(x) = x$, what set of outputs can I make using DIFF and INT? Are there any other starting functions which yield the same set of outputs? What sets of outputs will I get by starting with different initial functions?
Can you find any initial functions which yield a finite set of possibilities under RECIPROCAL and PRODUCT or under DIFF and INT?
What happens if you begin with $f(x) = x$ and use any combination of RECIPROCAL and DIFF?
Explore the possibilities when you can use different combinations of RECIPROCAL, PRODUCT, DIFF and INT for various choices of initial function. Can you find any interesting sets of outcomes?
See also the problem Calculus Countdown
NOTES AND BACKGROUND
In mathematics and science, the concept of differential and integral operators is very important, with applications ranging from quantum chemistry to analysis of waves. The simplest examples of these are given in this problem under the guise of DIFF and INT.
Getting Started
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 $f(x)=x$?
Student Solutions
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.
Teachers' Resources
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 $n$.