University of Oxford mathematician Dr Tom Crawford derives Taylor’s Theorem for approximating any function as a polynomial and explains how the expansion works with two detailed examples. Test yourself with some exercises on Taylor’s Theorem with this FREE worksheet in Maple Learn.

The video begins by introducing the idea of approximating a function by a polynomial and the condition we choose to implement that allows the coefficients to be derived. By ensuring that both the function and polynomial approximation are equal for all derivatives at a single point, Cauchy’s Mean Value Theorem can be applied to give equality.

Taylor’s Theorem is demonstrated with two fully worked examples. First, the power series expansion for cos is derived by expanding around zero. Next, a third degree polynomial approximation is calculated for small x, when expanding the function ln(1+sin(x)).