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Calculus: An Introduction to Parametric Equations


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About this Lesson

  • Type: Video Tutorial
  • Length: 11:47
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 126 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Parametric Equations, Polar Coordinates (18 lessons, $27.72)
Calculus: Understanding Parametric Equations (3 lessons, $5.94)

Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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Parametric Equations and Polar Coordinates
Calculus and Parametric Equations
Introduction to Parametric Equations Page [1 of 2]
Back in Calculus I, we saw a lot of problems that involved bugs. I do not know why we find it necessary to look at bugs, but somehow, for some reason, we do. The question usually looks something like this: we are told that a bug was walking along a horizontal line, and we are given its position. Maybe its position was given by, here is an example: f(t) = t^2 - t. So there is an example. So, what that means is that t is time here, so if you give me what time you have, I can tell you where the bug is located. Now, if you wanted to find velocity, for example, of the bug you would just take the derivative of this, and so on and so on and there was the birth of Calculus.
So, there was always a little bug that was charted. For example, in this case, the bug would go backwards for a little while, then would start to go forwards, and would always travel along the line. That is what the bug does for this particular example. That is great if you happen to have a bug and it just wants to live on a line, but the world is more interesting than lines. In fact, the world has a lot of space, just for example, consider the plane. So, the poor bug is being confined in early Calculus to sort of living along this line. That is all the bug can do just live along the line. What if the bug wanted to explore? How can we describe the bug's path now? Now it is more complicated because now it is not just the bug going on a line back and forth, but now, in fact, I will need to know a couple of pieces of information. I will need to know not only the x direction, but now I will need to know the y direction. In fact, if you think about this for a second, if we consider the bug walking on this plane, but just living on this line, then to describe that I would not only have to tell you that it is going this way, but I will have to tell you where it is going this way.
So, in this simple example, I would have to give you another formula, another equation, which gives you the y direction. The y direction for this simple example would be: g(t) = 0 because the bug is not going up or down, it is only moving right or left. So, if I think of this as the y, and if I think of this as the x then this tells me the x position of the bug and the y position of bug at any time. Suppose you say, what is going on at time equals two seconds? So, two seconds after the bug starts, where is the bug? I would just plug in t (time) equals 2. If I put in 2 here, I see 2^2 which is 4 - 2 which is 2. So, I know that I am 2 units over in the x direction, and in the y direction I am still at 0. I do not move up or down; I stay at zero, and I stay at the x-axis.
So, these two equations together tell me where the bug is located on the plane. The bug, again, is still not moving in any interesting way, it is just staying along the x axis because this is constant. But if you wanted the bug to move off, then the y values would have to change as well. That would actually involve having two equations, one to give me the x location at any time and one to give me the y location at any time. These are called parametric equations, parametric because, para (meaning two) and metric (because we are no longer in the regular system of measuring feet, we are now using meters). Parametric equations, because we have two equations to describe the position, the independent variable is, time. So, it's basically a means of being able to track the position of moving objects. Give me, time and I'll tell you how far over I'm going over. There is going to be an, x equation that tells me where I am in the, x direction at any time and a corresponding, y equation to tell me where I am in the up and down direction at that same time. I can then pinpoint the point that I am located at.
That is sort of a simple-minded example. Let us take a look at a more exotic example. How about this; suppose that in the x direction, sometimes I can write it like this, x of t because x just depends upon t - I go cosine of t and then in the y direction I will move sine of t. The question is: what is the bug's path? Well, again, this is an example of a pair of parametric equations. Parametric equations because we have two equations - one is for x and one is for y and they both depend upon time. Ask the question, where is the bug located after four minutes? What I could do is I could plug in 4 in for t and I found out what the x is, I plug in 4 in for t and I find out what the y is, and that will pinpoint the point on the lane where the bug is at exactly that time. So, these are examples of parametric equations.
Now, how would I find out what the path of the bug is? Well, let's try to plot some points. So, let's see where the bug is at in the beginning when time = 0. So what is x(0)? X(0) would be the cosine of 0, and the cosine of 0 is known to be 1. What is y(0)? I plug in 0 here, I see sine of zero, and sine of 0 is 0. So, at the beginning of the day, we are right here at x=1, y=0 so that is a start. Since these are trigametric functions, I will actually plug in time that actually have radians sounding thing so that sounds like an angle, but it is just a numeric value and those are the time intervals that I will decide to look at. It might seem a little bit strange on your stopwatch, your stopwatch does not have radians there, but pi is just 3.14 something, and there is only 3.14 something on your watch. This is not a big deal, but it will make it more convenient for the calculations. Let us now plug in something else. How about we plug in pi over 4 for time; so that is a little under a minute if these are representing minutes. Where is the bug then? Well, if I plug in pi over 4 for cosine, and I remember that the cosine of pi over 4 and pi over 4 is like 45 degrees. So, the cosine of pi over 4 is the square root of 2 over 2. Similarly, when I plug that into the y, I see sine of pi over 4, which also is square root of 2 over 2. It is one of the few times in life when they actually agree. So that means that at pi over 4 times pi over 4, the bug is located at the square root of 2 over and the square root of 2 up.
So, it is now located right around here somewhere. It moved from here to here. How about at pi? If I plug in pi. What is the cosine of pi? That is -1, and what is the sign of pi? That is 0. So, at time pi, which is 3.1 minutes then we are at -1/0 so, we are now over here. So, what is the path that is being transversed? It looks like it may be going in a circle. It is a wild guess. Let us see if that wild guess makes a little bit of sense. Because, if this thing was traveling along a circle, it would look like it had radius one at the center of the origin. If that were really the case, then in fact the x and the y should satisfy that formula for a circle which means x^2 + y^2 would have to equal one. Let us see what x^2 + y^2 equals in this particular case. X^2 is cosine t so that would be cosine^2 t + sine^2 t. What is cosine^2 t + sine^2 t for any t? Well, from the Pythagorean theorem, that equals 1. So, look, I see that x^2 + y^2, in fact, does equal 1, so now it is a fact that this bug is genuinely walking along the unit circle. It is going in this direction. These two parametric equations actually allowed us to describe the path of this bug. It starts out here at 1/0, and there it goes in a circle. It is these two equations that are actually describing what you see not just the circle, but this motion. These two equations are giving this bug the position. So, these two equations are actually capturing that movement along the circle and in that direction. Parametric equations allow us to find the path of moving objects in the plane, and they are really nice ways of describing movement and path.

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