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mth.122.handout.21

# mth.122.handout.21 - MTH 122 Calculus II Essex County...

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MTH 122 — Calculus II Essex County College — Division of Mathematics and Physics 1 Lecture Notes #21 — Sakai Web Project Material 1 Representations of Functions as Power Series If you were to take the function f ( x ) = 1 1 - x and carry out long division (we’ll do this is class) you’ll see that it goes on forever and has a very nice pattern, namely f ( x ) = 1 + x + x 2 + x 3 + x 4 + · · · = X n =0 x n . This, of course is the geometric series and it converges for | x | < 1. You should note that f ( x ) = 1 1 - x is defined for all x 6 = 1, but for - 1 < x < 1, f ( x ) can be written as a power series, X n =0 x n . Alhough f ( - 2) = 1 / 3 and is easy to compute, we could not use the power series to compute this, because when x = - 2 the series is not convergent. And even if we wanted to know f (1 / 2), which incidentally is 2, we wouldn’t use the power series to do the computation. Rewriting functions as power series does have uses, but for now I am more concerned with finding a power series representation of a given function. Okay to better understand what’s going on here, let’s return to a problem from introductory calculus. The tangent line approximation L ( x ) is the best linear approximation to f ( x ) near x = a because f ( x ) and L ( x ) have the same rate of change (derivative) at a . For a better approximation than a linear one, let’s try a second-degree (quadratic) approximation P 2 ( x ). In other words, we approximate a curve by a parabola instead of by a straight line. To make sure the approximation is a good one, we stipulate the following: P 2 ( a ) = f ( a ) P 2 and f should have the same value at a . P 0 ( a ) = f 0 ( a ) P 0 2 and f 0 should have the same value at a . P 00 2 ( a ) = f 00 ( a ) P 00 2 and f 00 should have the same value at a . This can, of course, go on ad infinitum . Let’s take a concrete example. 1 This document was prepared by Ron Bannon ( [email protected] ) using L A T E X 2 ε . Last revised January 10, 2009. 1

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1. Find the quadratic approximation P 2 ( x ) = A + Bx + Cx 2 to the function f ( x ) = e x that satisfies the above three conditions with a = 0. Graph P 2 and f on the same axis. Does P 2 fit f better than a tangent line in the region where a = 0? Work and Solution: We are given f ( x ) = e x and P 2 ( x ) = A + Bx + Cx 2 , and we need their derivatives. f ( x ) = e x f 0 ( x ) = e x f 00 ( x ) = e x P 2 ( x ) = A + Bx + Cx 2 P 0 2 ( x ) = B + 2 Cx P 00 2 ( x ) = 2 C Now, using a = 0 we can determine the constants A , B and C . In the functions first. f (0) = e 0 = 1 = P 2 (0) = A A = 1 Now in the first derivative. f 0 (1) = e 0 = 1 = P 0 2 (0) = B B = 1 Now in the second derivative. f 00 (0) = e 0 = 1 = P 00 2 (0) = 2 C C = 1 2 So, the function P 2 ( x ) = 1 + x + x 2 / 2. Here’s the graph. -2 -1 0 1 2 1 2 3 Figure 1: Partial graphs of the tangent line [red] to e x at a = 0, P 2 ( x ) [blue] and f ( x ) [black]. Yes, the quadratic is a better fit than is the linear function. 2
2. If we repeat this method for higher and higher-degree polynomials, we find that f ( x ) can be better approximated. Now repeat this method until you get a forth degree polynomial, P 4 ( x ) = A + Bx + Cx 2 + Dx 3 + Ex 4 , that approximates f ( x ) when a = 0. As before, graph P 4 and f on the same axis. Does the forth degree P 4 fit f better second degree P 2 in the region where a = 0?

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mth.122.handout.21 - MTH 122 Calculus II Essex County...

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