Page 51 - Calculus for the Clueless, Calc II

In order to eliminate t, we solve for t in the equation for y: t = cos (1 - y/a). Therefore, x = a{cos (1 - y/a) -sin
-1
-1
[cos (1 - y/a)]}. Pretty awful, isn't it?! For all practical purposes, this form is impossible to use. We really need
-1
parameters here!!!!
We wish to take derivatives using parameters. The first derivative will give us the slope, and the second will tell
its upness and downness.

Example 4—

You might think we came up with d y/dx by taking dy/dx and taking the derivative of the top over the
2
2
derivative of the bottom. You'd be wrong. You say, "Of course. You must use the quotient rule!" Again you'd
be wrong. The second derivative is the derivative of the first derivative. So...

Again, in this particular example, you could eliminate the t, but in the cycloid, you really could not.

Example 5—

The a's cancel.