You may also like

problem icon

Towards Maclaurin

Build series for the sine and cosine functions by adding one term at a time, alternately making the approximation too big then too small but getting ever closer.

problem icon

Building Approximations for Sin(x)

Build up the concept of the Taylor series

problem icon

Taking Trigonometry Series-ly

Look at the advanced way of viewing sin and cos through their power series.

What Do Functions Do for Tiny X?

Age 16 to 18 Challenge Level:

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:

graphs for tiny x

For [-0.1, 0.1]
graphs for -.1 to .1

And for [-0.01, 0.01] I have:
graphs for -.01 to .01Each of the 4 functions could be approximated using the Taylor series expansion around 0, and the accuracy of the approximation becomes better for values of $x$ nearer to the origin. I shall use the second order polynomial: $$A(x) \approx x,\quad B(x) \approx 1- (1 - x^2/2) = x^2/2, \quad C(x)\approx x - x^2/2, \quad D(x) \approx - x - x^2.$$