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Calculus: The Remainder of a Taylor Polynomial


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About this Lesson

  • Type: Video Tutorial
  • Length: 5:48
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 62 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Sequences and Series (45 lessons, $69.30)
Calculus: Taylor and Maclaurin Polynomials (4 lessons, $6.93)

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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Sequences and Series
Taylor and Maclaurin Polynomials
The Remainder of a Taylor Polynomial Page [1 of 2]
Taylor polynomials really provide an amazing approximation, in fact a polynomial approximation, to any function that you can think of. As long as, of course, you can take derivatives of it. You might need the derivatives. The way that works is literally to make sure the derivatives all match up with the polynomial against the functions.
The actual formula for it, and we've seen this time and time again, why don't we call this P[n] for the n^th Taylor polynomial centered around x = c of some given function f(x). So we saw that it's just f(c), that's to make sure that the points agree. Now I want to slopes agree, so I make sure the derivative agrees at the point x = c, so I put this in here. And then I want to makes sure the second derivatives agree so the curvature of that point agrees, so I put in the second derivative, and that produces a square term here. I have to divide now by 2, technically 2!, which is really just 2, and then you keep doing this until you get to the n^th one, which would be .
And again, we could actually check this out by taking successive derivatives, one after the other, until we've taken n derivatives, evaluate each derivative of this polynomial at x = c, and what you're actually going to see is what? You're going to actually see the derivatives f with a case derivative evaluated at c, and that's the point of this whole thing. So all the derivatives line up. And I call this polynomial the n^th Taylor polynomial of f(x) expanded around the point x = c. Great. So there you have it.
Now we know that, in fact, f(x), the original function, does not equal this necessarily, but we have an approximation near x = c. If I plug in x = c, by the way, let's let x = c, plug in c here, look what happens by the way, if you plug in c here, nothing to plug in here, here's an x, so I put c, c - c, that's 0, c - c, that's 0, all the way out to c - c, that's 0. So everything is cancelled, and in fact, they agree perfectly. So P[n](c) = f(c). So, in fact, then they're equal at that point. But then around the point they might not actually be equal.
So there's some sort of error. The thing I want to think about now is what is the error for the approximation when you use this Taylor polynomial technique? What can you say about that error? Well, that error is something that's thought of as a remainder, is denoted by R[n](x). Now, this is just the function that is the offset, this function, this polynomial function is from f. So this is the remainder, and really all it is defined to be just what you need to be f. So f(x) will equal, not approximate, but now equal this polynomial. And then plus whatever you need to add in order to make this an equal sign. So this remainder or this error gives us how far off we are in this approximation.
So what is the answer? What is the remainder? Well, if you think about it, we can make a really good guess. Because if we look at the next polynomial, not the n^th polynomial, but the n + 1^st polynomial, then what would we see? Well, we'd see that it equals all this stuff, blah, blah, blah, blah, plus the next term, which would be the n + 1^st derivative evaluated at c times (x - c)^n + 1 all divided by n + 1!. So that would be the next term if we look at the next best approximation.
As we get higher and higher and higher in our degrees, the approximation gets better and better and better. So it would seem like the error might look like that next term. Reality is the error does look like that next term. The actual error, R[n](x) actually equals the n + 1^st derivative, just as we had predicted, evaluated at z. What's z? Well, I don't know yet, but we have the (x - c)^n + 1 all divided by n + 1!. So, in fact, it's exactly what our guess was, except that z is here.
What's z? Well, z is just some number that's between x and c. So this is where z is some number between the point we're approximating around and the value x that you're plugging in. And this z is the z that actually comes through by using the mean value theorem from early on in calculus. And if you actually write these things down precisely you can prove that such a z will exist between these two points.
So the bottom line, though, for our purpose right now is to think that if x is actually pretty close to c, which means that we're actually pretty close to where the approximation is good, then, in fact, this z is sandwiched in between these two points that are close together, so we can actually estimate this part and then we see exactly what the error term would look like.
So bottom line is if you want to figure out what the error is in Taylor's approximation for the n^th Taylor polynomial, it literally looks like the next term you would see in the n + 1^st Taylor polynomial, but instead of a c here this is just some number that's between x and c.
Up next we'll actually take a look at an application of this where we actually put this error to the test. And we'll see what we can approximate.

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