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|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 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 , where is some jumble of 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 with the -coordinate and the s with the -coordinate and make sure the two sides give you the same answer.
A curve also (as far as you’re concerned) has a derivative, , which you get by differentiating the jumble of s. This tells you how steep the curve is at any given point: you just throw in the value of and see what comes out.
Curves are objects that often have names (silly names like ) — I find it helpful to think of them like Top Trumps cards with categories like “Equation of curve”, “Equation of derivative”, “Name”, “ -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|