What do functions do for tiny x?
Archimedes, Bernoulli, Copernicus and de Moivre each try to think of a function which will be small near to the origin. Their choices are:
$$ A(x) = \sin(x) \quad B(x) = 1 - \cos(x) \quad C(x) = \log(1+x) \quad D(x) = 1 - \frac{1}{1-x} $$
All of these functions equal zero when x is exactly zero, but the friends want to investigate how small their functions are when x is small but not exactly zero.
Use a spreadsheet to investigate these curves graphically for smaller and smaller values of x (you may like to try graphs with x ranging from -1 to 1, -0.1 to 0.1, -0.01 to 0.01 to begin with). Before you start, you might like to try to guess what will happen close to the origin in each case.
As you zoom in the scale what do you notice happening to the graphs in each case? What similarities do the functions have as we zoom in? What differences do they exhibit?
Can you predict the shapes of the graphs for x between -0.000001 and 0.000001? Test your predictions.
For each function can you write a simple polynomial approximation with 1 or 2 terms for the shape of each curve for very small values of x?
You may like to use the example spreadsheet.
Visually what do you notice when you increase the magnification on the origin?
Congratulations Andrei for another very good solution.
The 4 functions are: $$A(x) = \sin x,\quad B(x) = 1 - \cos x,\quad C(x) = \log (1+x), \quad D(x) = 1- {1\over (1-x)}.$$ I consider the logarithm in base e. First I plotted the 4 functions using Graphmatica. In all figures $A(x)$ is violet, $B(x)$ is white, $C(x)$ is red and $D(x)$ is cyan. For $[-1, 1]$ I obtain:
For [-0.1, 0.1]
And for [-0.01, 0.01] I have:
This problem begins to motivate the existence of the Maclaurin's series in which any function which is well-behaved at the origin can be written as an 'infinite polynomial' or 'power series'. This iterative method is a numerical way of finding the coefficients of the polynomial, although calculus can be used in cases where the derivatives of the function needing to be approximated are known.