Exploring and noticing

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.

In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.

Which curve is which, and how would you plan a route to pass between them?

What can you deduce about the gradients of curves linking (0,0), (8,8) and (4,6)?

Get started with calculus by exploring the connections between the sign of a curve and the sign of its gradient.

Find the location of the point of inflection of this cubic.

Here you have an opportunity to explore the proofs of the laws of logarithms.

Which of the statements must be true?

Explore the strange geometrical properties of the Koch Snowflake.