You may also like

problem icon

What Do Functions Do for Tiny X?

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

problem icon

Taking Trigonometry Series-ly

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

problem icon

Bessel's Equation

Get further into power series using the fascinating Bessel's equation.

Towards Maclaurin

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
(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?