DOPPLER EFFECT – ADVANCED READING
Waves
Shock wave
Taylor polynomial
Taylor expansion is very useful mathematical procedure which substitute a mathematical function by the infinite polynomial series. In practice we don't use infinite series but we use the series to the certain degree. If we limit series to the first degree polynomial, we talk about linear approximation , the function replaces the selected tangent point. If we limit series to the second-degree polynomial, we replace the function at the selected point with parabola. A higher degree polynomial gives a better approximation to the function. The method was formulated by English mathematician Sir Brook Taylor (1685-1731). The general formula for the Taylor expansion is
| f(x) = a0 + a1(x−c) + a2(x−c)2 + a3(x−c)3 + ··· ; ak = f (k)(c)/k! | (1) |
Point where we make an expansion is marked as c, expansion coefficients ak are given by ratio of k-th derivative of the function in the point c and factorial k. Basic condition for expansion convergency to the function on certain interval is that the function must have all derivatives on that interval.
Expansions of some elementary functions
In a lot of cases the xpansion only around the initial pointc = 0 is sufficient. Look on the expansions below:
| exp(x) = 1 + x + x2/2! + x3/3!+ x4/4! + ··· ; | (2) |
| sin(x) = x − x3/3!+ x5/5! − x7/7! + ··· ; | (3) |
| cos(x) = 1 − x2/2! + x4/4! − x6/6! + ··· ; | (4) |
| sinh(x) = x + x3/3!+ x5/5! + x7/7! + ··· ; | (5) |
| cosh(x) = 1 + x2/2! + x4/4! + x6/6! + ··· ; | (6) |
| ln(1+x) = x − x2/2 + x3/3 − x4/4 + ··· ; | (7) |
| 1/(1−x) = 1 + x + x2 + x3 + x4 + x5 + ··· . | (8) |
Try
In the applet below you can try Taylor expansion of different functions. Applet is a part of whole bunch of calculus applets, made by David Eck and Thomas Downey. Applet are distributable under free license Creative Commons.