Approximating functions

Taylor Series #

• About x=a
  • $$f(x)=f(a)+\frac{f’(a)}{1!}(x-a)^1+\frac{f’’(a)}{2!}(x-a)^2+…+\frac{f^{’n}(a)}{n!}(x-a)^n+…$$
• Approximating ‘h’ amount around ‘a’ (just another form) $$f(a+h)=f(a)+\frac{f’(a)}{1!}(h)^1+\frac{f’’(a)}{2!}(h)^2+…$$ The nth derivative of a function at a point is applicable here.

The series becomes a Maclaurin series if a=0

Maclaurin series of common functions #

$$ e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+… $$ $$ \sin(x) = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+… $$ $$ \cos(x) = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!} $$ $$ ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4} $$ $$ tan^{-1}(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7} $$