Taylor Polynomial Approximations in Polar Coordinates Sheldon P. Gordon The College Mathematics Journal, Vol. 24, No. 4. (Sep., 1993), pp. 325-330. Stable URL: http://links.jstor.org/sici?sici=0746-8342%28199309%2924%3a4%3c325%3atpaipc%3e2.0.co%3b2-m The College Mathematics Journal is currently published by Mathematical Association of America. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/maa.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org. http://www.jstor.org Wed Aug 22 12:30:43 2007
Taylor Polynomial Approximations in Polar Coordinates Sheldon P. Gordon Shelly Gordon is Professor of Mathematics at Suffolk Community College. He is a member of the Harvard Calculus Reform Project and co-director of the Math Modeling/PreCalculus Reform Project. He is co-editor of Statistics for the Twenty-First Century published in the MAA Notes series. His primary interests concern the implications of technology on the entire mathematics curriculum and the place of applications in that curriculum. One of the major features of many of the calculus reform projects currently being conducted, including the Harvard Calculus Project, is a renewed emphasis on Taylor polynomial approximations [I]. Graphics software and graphing calculators allow students to explore the concept of such approximations visually and so come to a much deeper understanding of the process of approximation of one function by another. We will consider a natural extension of these ideas and attempt to approximate a curve given in polar coordinates using a Taylor polynomial. Suppose we begin naively by considering a polar function r =f(8) in the neighborhood of a point P(ro, a) and define the linear approximation to the function as the tangent line at that point. As is shown in most calculus texts, the slope of the tangent line to a polar curve is given by Therefore, the equation of the tangent line to the polar curve at the point P is given in polar coordinates by so that r sin 6' - r, sin a = m(r cos 6' - rocos a) r = r,(sin a - rn cos a) sin 8 - rn cos 8 We invite those readers who feel somewhat masochistic to continue this process to determine the polar equation for the approximating parabola and the corresponding higher degree polynomials. You will see that this naive approach is not particularly fruitful. Rather, it is necessary to re-interpret the idea of a Taylor polynomial approximation to reflect the circular symmetries that naturally arise in equations given in polar coordinates. First, the constant approximation to f(8), r =f(a) = r,, is a circle of radius r, centered at the pole. Next, the first-order approximation, ' is a translation and rotation of the standard Archimedean spiral r = a8. This VOL. 24, NO. 4, SEPTEMBER 1993 325
"linear" approximation bends along with the original curve and so should be a reasonably accurate approximation to the curve for points near 0 =a. In a similar way, the quadratic approximation, Y =f((~) +f1(~)(e -a) +flya)(e -a)2/2 is also a counter-clockwise spiral. Notice that the sign of f"(a) does not affect the counter-clockwise unrolling of the spiral as 0 increases. However, the constant and "linear" terms may affect the overall spiral behavior when 0 is close to a in the sense that the spiral may not unroll in a monotonically increasing manner. Clearly, if we continue this process, the Taylor polynomial approximation in polar coordinates is a linear combination of terms of the form (8 - a)". Each such term is a spiral opening in the counter-clockwise direction. However, since a Taylor polynomial is "centered" about 8 =a, it may be helpful to think of the approximation as consisting of two separate spirals, both initiating at the point where 0 = a and where one spiral opens counter-clockwise for 0 > a and increasing and the other opens clockwise for e < a and decreasing toward - m. Before examining how the Taylor polynomial approximations apply, we digress to consider the properties of such polynomial spirals. We immediately observe that the larger n is, the more rapidly the corresponding spiral unrolls for 8 - a > 1 and the less rapidly for 0 - a < 1. Further, if the coefficient of such a term is negative or if the values of (0 -a)" are negative, then the spiral is reflected back through the pole, but will still open in the counter-clockwise direction. See Figure 1. Figure 1 Graphs of r = 9, r = g2 and r = -9. Suppose now, for simplicity, that we consider a polar polynomial of degree n in 0, P(0) = aoen+ alen-' + a2en-2+... +an. In the large, the polar graph of such a polynomial will be a counter-clockwise spiral. On a more local basis, various types of behavior are possible. The following observations are evident after a little thought: 1.The real roots of P(0)correspond to angles at which the curve passes through the pole. 2. As 0 runs through an interval between successive real roots 8, < 8, of P(B), the point (8, P(0)) moves around a closed loop of the polar graph of P, leaving the 326 THE COLLEGE MATHEMATICS JOURNAL
pole tangent to the line 8 = +8, and returning tangent to the line 8 = + 8,. 3. If 8, is a real root of P with even multiplicity, the polar graph of P has a cusp at the pole tangent to the line 8 = i-8,,. Note that the sign of P(8) does not change as 8 passes through the value 8,. 4. Each of the relative maxima and minima of the polynomial corresponds to a point where the distance from the pole to the spiral is a relative maximum or minimum. 5. Each of the points of inflection of the polynomial corresponds to a point where the spiral has the same curvature as a "linear" spiral, r = a8, for a given value of 8. However, it is much harder to distinguish such points geometrically in the polar plane than in the rectangular plane. We next see how these properties of polynomial spirals provide the type of agreement we would like to achieve between a polar curve and the corresponding Taylor polynomial approximation in polar coordinates. The agreement can best be appreciated visually. In Figures 2a, 2b and 2c, we show the successive spiral polynomial approximations r = 0, r = 8-03/3! and r = 8 - d3/3! + e5/5! to the circle r = sin 0. By the fifth-degree term, the approximation cannot be distinguished, by eye, from the circle between 8 = -n-/2 and n/2. Figure 2a Graph of r = 6 (outer curve) as approximation to r = sin 6 on 6 E [O, T]. Figure 2b Graph of r = 6-03/3!(inner curve) as approximation to r = sin 0 on 6 E [O, TI. VOL. 24, NO.4, SEPTEMBER 1993 327
Figure 2c Graph of r = 0 - e3/3! + e5/5! as approximation to r = sin e on e E [O, TI, Figure 3a Graph of r = ;o2 (outer curve) as approximation to r = 1- cos e (inner curve) on e E [-r,rl. Figure 3b Graph of r = e2/2 - e4/4! (inner curve) as approximation to r = 1- cos e (outer curve) on 6 EI-%-, r]. 328 THE COLLEGE MATHEMATICS JOURNAL
Figure 3c Graph of r = 02/2-0"4! + 0"6! (outer curve) as approximation to r = 1- cos 0 (inner curve) on 0 E [-T,TI. In Figures 3a, 3b and 3c, we show the successive approximations of degree n = 2,4,6, respectively, to the cardioid r = 1- cos 0. Notice how the resulting polynomials all have double roots at 0 = 0 since the leading term in each case is quadratic in 8 and so each polynomial accurately approximates the cusp in the cardioid. Notice also that successive approximations are alternatively outside and inside the cardioid. Further, each successive approximating curve is clearly a better fit to the cardioid. In Figures 4a, 4b and 4c, we show three approximations to the four-leaf rose, r = sin28, corresponding to Taylor polynomials of degree n = 1, 3, and 9 respectively. (In these graphs, we use different viewing "windows" to optimize the view; one unfortunate consequence is that the dimensions of the rose appear somewhat different in each case.) Notice how, as the degree of the approximating polynomial increases and it has more real roots, successively more loops are produced to approximate the petals of the rose curve. By the last case with n = 9, the shape of the approximation mirrors the rose rather well and, in fact, it is virtually impossible to distinguish by eye the upper pair of petals of the rose from the upper pair of loops of the approximation. The above discussions can be presented effectively in the classroom using a computer and projector or graphing calculators if each student has one. Alternatively, they can be assigned to students to investigate on their own using what Figure 4a Graph of r = 20 as approximation to r = sin 20 on 0 E [ -7, T]. VOL. 24, NO. 4, SEPTEMBER 1993
Graph of r = 20 - (20)"3! Figure 4b as approximation to r = sin 20 on 0 E [-T, TI. Figure 4c Graph of ninth degree Taylor polynomial approximation to r = sin20 on 0 E [-T, TI. graphing technology is available. It is a particularly nice exercise because it extends the coverage of two important themes in calculus, Taylor approximations and polar coordinate graphs, by showing the interplay between them. That reinforcement helps students better understand both topics. Equally importantly, too many topics covered in calculus appear, to the students, to be totally unrelated since they appear in different chapters and there is no cross-reference between them. An exercise of this type serves to break down one of the walls between two important ideas and helps the students see some of the fascinating connections between them. Acknowledgment. The author gratefully acknowledges the support provided by the National Science Foundation under grants #USE-89-53923 and #USE-91-50440. The views expressed here are those of the author and do not necessarily reflect those of the Foundation, the Harvard Calculus Reform Project or the Math Modeling/PreCalculus Reform Project under which the work was conducted. Reference 1. Sheldon P. Gordon, Introducing Taylor polynomial approximations in introductory calculus, PRIMUS, 1 (1991) 305-313. THE COLLEGE MATHEMATICS JOURNAL