the Math of Form.

60 years of developing into a tour de force.

Sonja Kirchberger's terribly pretty legs!!!! photo by Luca Teuchmann.

So, what is a curve?

You don’t really need a technical definition, and it probably wouldn’t help you even if I could provide one. I’m going to give you a loose definition that a curve is anything you can draw. Obviously, that’s a pretty wide-ranging definition, and there’s only a limited subset of all of the possible curves you need to care about for C1.

In most of A-level, you only care about functions, which have the nice quality that they never back-track: for any value of x$x$ you can think of, if you draw a vertical line through that value, it crosses the curve once. Or nonce.

Every curve has a (possibly very complicated) equation in the form y=f(x)$y=f(x)$ , where f(x)$f(x)$ is some jumble of x$x$ s and numbers. Just like with the straight line, you can tell whether a point is on the line by checking the two sides of the equation: replace the y$y$ with the y$y$ -coordinate and the x$x$ s with the x$x$ -coordinate and make sure the two sides give you the same answer.
A curve also (as far as you’re concerned) has a derivative, dydx=f′(x)$\frac{dy}{dx}=f^{\prime }(x)$ , which you get by differentiating the jumble of x$x$ s. This tells you how steep the curve is at any given point: you just throw in the value of x$x$ and see what comes out.

Curves are objects that often have names (silly names like C$C$ ) — I find it helpful to think of them like Top Trumps cards with categories like “Equation of curve”, “Equation of derivative”, “Name”, “y$y$ -intercept”, “Solutions”, “Turning points” and so on. You can even draw out the card if it helps…
(A particularly useful thing to note: if the gradient is 0, the curve is temporarily flat; this is known as a turning point, or a stationary point, or an extremum, or a local maximum or minimum, depending on how awkward they want to be.) - Written by Colin+ in core 1, geometry.

pregnant Melanie Brown, singer Mel B., is like so many pregnant mothers shaped by an inexpressible body of knowledge outside the reach of touch, but within the grasp of concept. - Dawn Wolf, Keeper of Stories 5.22.17