18-01F07-L32

18-01F07-L32 - MIT OpenCourseWare http:/ocw.mit.edu 18.01...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Please use the following citation format: David Jerison, 18.01 Single Variable Calculus, Fall 2007 . (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Transcript – Lecture 32 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Today we're going to continue our discussion of parametric curves. I have to tell you about arc length. And let me remind me where we left off last time. This is parametric curves, continued. Last time, we talked about the parametric representation for the circle. Or one of the parametric representations for the circle. Which was this one here. And first we noted that this does parameterize, as we say, the circle. That satisfies the equation for the circle. And it's traced counterclockwise. The picture looks like this. Here's the circle. And it starts out here at t = and it gets up to here at time t = pi / 2. So now I have to talk to you about arc length. In this parametric form. And the results should be the same as arc length around this circle ordinarily. And we start out with this basic differential relationship. dx ^2 is dx ^2 + dy ^2. And then I'm going to take the square root, divide by dt, so the rate of change with respect to t of s is going to be the square root. Well, maybe I'll write it without dividing. Just write it as ds. So this would be (dx / dt)^2 + (dy / dt)^2 dt. So this is what you get formally from this equation. If you take its square roots and you divide by dt squared in the inside, the square root and you multiply by dt outside. So that those cancel. And this is the formal connection between the two. We'll be saying just a few more words in a few minutes about how to make sense of that rigorously. Alright so that's the set of formulas for the infinitesimal, the differential of arc length. And so to figure it out, I have to differentiate x with respect to t. And remember x is up here. It's defined by a cos t, so its derivative is - a sin t. And similarly, dy / dt = a cos t. And so I can plug this in. And I get the arc length element, which is the square root of )- a sin t) ^2 (+ a cos t) ^2 dt. Which just becomes the square root of a ^2 dt, or a dt. Now, I was about to divide by t. Let me do that now. We can also write the rate of change of arc length with respect to t. And that's a, in this case. And this gets
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 11

18-01F07-L32 - MIT OpenCourseWare http:/ocw.mit.edu 18.01...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online