From the blue-and-black symmetries of a butterfly TO the mazelike grooves of a brain coral, nature reveals itself as a great artist, switching off between painting and sculpture. Nature should also ...

We can calculate the gradient of a tangent to a curve by differentiating. In order to find the equation of a tangent, we: Differentiate the equation of the curve Substitute the \(x\) value into the ...

Using high school algebra and geometry, and knowing just one rational point on a circle or elliptic curve, we can locate infinitely many others. You’re sitting at the end of a long conference table, ...

Before reading this guide, you may find it helpful to read the guide on graphs from Module 7 (M7) and the guide on indices from Module 8 (M8). In Module 7 (M7) Graphs, you learned that simultaneous ...

Express the polar equation \(r=\cos 2\theta\) in rectangular coordinates. Find the maximum height above the \(x\)-axis of the cardioid \(r=1+\cos \theta\text{.}\) Sketch the graph of the curve whose ...

Give an example of a function with one critical point which is also an inflection point. You must provide the equation of your function. A particle moves along a line with a position function \(s(t)\) ...