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Even as the finite encloses an infinite series
And in the unlimited limits appear,
So the soul of immensity dwells in minutia
And in narrowest limits no limits inhere.
What joy to discern the minute in infinity!
The vast to perceive in the small, what divinity!
Today we continue with polar equations.
The material to be covered is thus
In discussing a problem with Matthew, I made the point that parametric equations are useful for representing curves that cannot be represented as ordinary functions -- because the curves fail the vertical line test.
The most obvious example of this is a circle. It's so important, yet we can't represent it as a function $y=f(x)$. We have to write, for example (and somewhat ashamedly) \[ y(x)=\pm\sqrt{r^2-x^2} \]
But if we think of it as a parametric equation, it's easy to write as a function of time -- as a motion -- just the way that you might trace it out on the board or on your paper: \[ \begin{array}{c} {x(t)=\cos(t)}\cr {y(t)=\sin(t)} \end{array} \] as $t$ varies over $[0,2\pi)$. That's if you like to start at (1,0), and draw in a counter-clockwise fashion. If you like to start at the top and go in a clockwise fashion (like time on a clock), then you might prefer \[ \begin{array}{c} {x(t)=\cos(\frac{\pi}{2}-t)}\cr {y(t)=\sin(\frac{\pi}{2}-t)} \end{array} \] as $t$ varies over $[0,2\pi)$....
In today's materials you'll see some pretty wild curves -- and even learn how to compute their lengths!
We'll also compute tangent lines to curves. In terms of motions, the tangent line is the path a particle would travel (initially) if whatever force is keeping the particle in its trajectory were to release. If we were together in class, I'd be swinging a ball attached to a string over my head, and imagining along with you what would happen if one were to cut the string....