### What Do Functions Do for Tiny X?

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

### Building Approximations for Sin(x)

Build up the concept of the Taylor series

### Taking Trigonometry Series-ly

Look at the advanced way of viewing sin and cos through their power series.

# Towards Maclaurin

##### Age 16 to 18 Challenge Level:
(1) We know $\cos x \leq 1$ for all $x$. By considering the derivative of the function $$f(x) = x - \sin x$$ prove that $\sin x \leq x$ for $x \geq 0$.

(2) By considering the derivative of the function $$f(x) = \cos x - \left(1 - {x^2\over 2}\right)$$ prove that $\cos x \geq 1 - {x^2\over 2}$ for $x \geq 0$.

(3) By considering the derivative of the function $$f(x) = \left(x - {x^3 \over 3!}\right) - \sin x$$ prove that $\sin x \geq (x - {x^3 \over 3!})$ for $x \geq 0$.

(4) By considering the derivative of the function $$f(x) = \cos x - \left(1 - {x^2 \over 2!} + {x^4\over 4!}\right)$$ prove that $\cos x \leq \left(1 - {x^2\over 2!} + {x^4 \over 4!}\right)$ for $x \geq 0$.

(5) What can you say about continuing this process?