Taylor Series
Text Taylor series are polynomials that approximate functions.
The existence of Taylor series is based on the fact that one can use the function value and first derivative, second derivative, third derivative, etc. at some x value x_{0} to predict what the values of a function f(x) should be for different x values some distance from the point x_{0}.
If we know up to the nth derivative of f(x) as well as the function value at x_{0}, then we can construct a Taylor polynomial of degree n.
The firstorder Taylor approximation P_{1}(x) is simply the tangent line at x_{0},
P_{1}(x) = f(x_{0})+(xx_{0})f '(x_{0}).
The secondorder Taylor approximation P_{2}(x) is the parabola that has the same function value and first and second derivatives as f(x) at the point x_{0}. The equation for this approximation is as follows: T_{2}(x) = f(x_{0})+(xx_{0})f '(x_{0})+1/2(xx_{0})^{2}f ''(x_{0}).
The nth order Taylor approximation is
Σ_{i = 0}^{n}[f^{(i)}(x_{0})/i!]*(x  {x_0})^{i}
Where f^{(i)}(x) indicates the ith derivative of f(x).
Demos
Taylor Series
 
In this demo you can see first, second, and third order Taylor approximations of functions f(x) at different values x_{0} of x. Note that accuracy increases as order increases.

Exercises 1. For a polynomial function of degree n, what condition is needed for a Taylor approximation to be exactly the same as the function itself?
2. Describe the behavior of the second order Taylor approximation for the function f(x) = x^2  1.
