Exponential and logarithmic functions

problem

What do functions do for tiny x?

Age

16 to 18

Challenge level

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

How does your function grow?

Age

16 to 18

Challenge level

Compares the size of functions f(n) for large values of n.
problem

Tuning and ratio

Age

16 to 18

Challenge level

Why is the modern piano tuned using an equal tempered scale and what has this got to do with logarithms?
problem
Favourite

Integral equation

Age

16 to 18

Challenge level

Solve this integral equation.
problem

Sierpinski triangle

Age

16 to 18

Challenge level

What is the total area of the triangles remaining in the nth stage of constructing a Sierpinski Triangle? Work out the dimension of this fractal.
problem
Favourite

Exponential trend

Age

16 to 18

Challenge level

Find all the turning points of y=x^{1/x} for x>0 and decide whether each is a maximum or minimum. Give a sketch of the graph.
problem

Complex sine

Age

16 to 18

Challenge level

Solve the equation sin z = 2 for complex z. You only need the formula you are given for sin z in terms of the exponential function, and to solve a quadratic equation and use the logarithmic function.
problem

Big, bigger, biggest

Age

16 to 18

Challenge level

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?
article
What are complex numbers?

This article introduces complex numbers, brings together into one bigger 'picture' some closely related elementary ideas like vectors and the exponential and trigonometric functions and their derivatives and proves that e^(i pi)= -1.
article
Infinite continued fractions

In this article we are going to look at infinite continued fractions - continued fractions that do not terminate.