Growing

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

What Do Functions Do for Tiny X?

Looking at small values of functions. Motivating the existence of the Taylor expansion.

Equation Attack

The equation a^x + b^x = 1 can be solved algebraically in special cases but in general it can only be solved by numerical methods.

Building Approximations for Sin(x)

Age 16 to 18
Challenge Level

In general, any function whose derivatives of all orders are finite can be written as a power series

$$f(x) = f(0) + x f'(0) + \frac{x^2}{2!}f''(0) + \dots + \frac{x^n}{n!}f^{(n)}(0)+\dots$$

This expansion is an infinite series (not a polynomial). Truncating this series at a given point provides us with a polynomial approximation to f(x).

The question of how big the errors are in this approximation is a difficult one to answer, and more details will be discovered at university in Numerical Analysis courses.

You may also like

Growing

What Do Functions Do for Tiny X?

Equation Attack

Building Approximations for Sin(x)

Age 16 to 18Challenge Level

Age 16 to 18
Challenge Level