Computer Algebra Systems Approach to Teaching Taylor Polynomials

Transcription

1 Computer Algebra Systems Approach to Teaching Taylor Polynomials John H. Mathews Department of Mathematics, California State University, Fullerton, CA Computer algebra systems (CAS's) such as DERIVE, MACSYMA, MAPLE, mumath and Mathematica can be used as powerful assistants for performing symbol manipulations in algebra and calculus. It has been suggested that these systems will benefit undergraduates in mathematics by keeping track of the details in complicated manipulations (see Cromer, 1987 ; Freese, Lounesto & Stegenga, 1986 ; Small & Hosack, 1986 ; Woof & Hodgkinson, 1987 ; and Zorn, 1986). This article shows howthe symbolic calculus and graphics capabilities of the two CAS systems MAPLE and Mathematica can be used to enrich the understanding of both the theoretical and computational aspects oftaylor series. Both systems are available on Macintosh computers, and to fully utilize their capabilities requires several megabytes of RAM memory. The Mathematics Department at CSUF uses a satellite computer laboratory in our School of Natural Science and Mathematics. It currently includes 19 students work stations. The Taylor series expansion of a function f(x) is a familiar topic of sophomore calculus. It is the foundation of much theory in analysis and applied mathematics. Unfortunately, it is often viewed by students to be difficult to understand. Using assistance from CAS to construct the coefficients of the series, draw graphs and manipulate polynomials of high degree, gives us more time to focus our attention on the underlying principles. Using MAPLE and Mathematica has generated enthusiasm for both students and teachers. THEOREM [Taylor]. Let f be infinitely differentiable on an interval containing the numbersxo andx. Then, for eachpositive integer n, f(x) can be represented as the sum (1) f(x) = Pn (x) + Rn(x), where n f(k) (XO)(X -XO)k (2) Pn(X) = ~ kt k=0 John earned his PhD in mathematics from Michigan State University under the direction of Peter Lappan. He has been a member of the mathematics faculty of California State University, Fullerton, for twenty years, and keeps active in the areas of complex analysis, numerical analysis and computer algebra. f(n+l)(c )(X- XO)n+l (3) Rn(X ) (n+l)t where c = c(x) lies somewhere between x0 andx. The MAPLE syntax for initiating the computation of a Taylor series is taylor(f, x =x0, n + 1) ; which returns the n-th degree Taylor series for f with respect to the variable x expanded about the value x0. For example, the Maclaurin series of order n=7 for sin(x), cos(x), exp(x) and ln(1 +x) are found byissuing the commands : taylor(sin(x),x =0,8) ; x - 1/6x3 + 1/120 xs - 1/5040 x7 + O(x$) taylor(cos(x),x =0, 8) ; 1-1/2x2 + 1/24x4-1/720x6 + O(X8) taylor(exp(x), x =0, 8) ; 1 +x +1/2x2 +1/6x3 +1/24x x5 +1/720X 6 + 1/5040 x7 +0(X8) taylor(log(l+x), x =0, 8) ; X-1/2x2+1/3x3-1/4x4+1/5x5-1/6x6+1/7X7+0(X8) MAPLE has included the terms O(x8), which means that the coefficients of the Taylor series have been computed up to x7 and that an error term must be included and it involves several factors, one of which isx8. In order to do computations with the Taylor series, we must use MAPLE's command to convert them to polynomial form, i.e. we must dig out the polynomial part. For example, suppose that we want to view the polynomial approximations of degree n=3,5,7,9 to sin(x) over the interval [- 2ar,2n]. Then we issue the MAPLE commands : p3 := convert(taylor(sin(x),x=0,4),polynom) ; P3 : = x - 1/6x3 61

2 p5 := convert(taylor(sin (x),x=0, 6),polynom) ; p5 : = x - 1/6x3 + 1/120 x5 p7 := convert(taylor(sin(x), x=0, 8),polynom) ; P7 : = x - 1/6 x 3 + 1/120 x5-1/5040x7 p9 := convert(taylor(sin(x),x=0, 10),polynom) ; p9 : = x-1/6 x3 + 1/120 x5-1/5040 x7 + 1/ x9 it and see that it produces an approximation that is not symmetric about the origin (see Figure 2). We are now ready to look at the four polynomial approximations P3(X)1 PS (x), Nx) and P9 (x) to f(x) _ sin(x) (see Figure 1). Figure 2 Figure 1 Looking at Figure 1 reveals that the approximations of higher degree track the curve y = sin(x) closer over a wider interval. Further investigations can be made by the students. The details for expanding f in a Taylor series about xo # 0 usually requires more work than a Maclaurin series and is error prone. Let us switch to Mathematica and continue our CAS investigations. We expand sin(x) in a Taylor series of degree n =5 about x0 =,7r/3 by issuing the Mathematica command : t5 = Series [Sin [x1,fx,pi/3,5j1 S rt3 3l +x Sgrt[3] 3' +x 2-3l+x~ Sgrt[3] 4 5 -Pi -Pi + x + x 6 ( x1 Notice that Mathematica has used the notation Pi and Sgrt[3] for ac and 31/2, respectively. Notational conventions such as these are typical for CAS and it is best to stare at the output until it becomes familiar. After we convert the Taylor series to a polynomial, we can graph Students immediately find that using CAS is a powerful tool for doing homework problems. However, in order to understand the theory they must learn how the coefficients and error term are derived. This too can be assisted with Mathematica, but it is necessary to write a short program that prints out the intermediate steps. The syntax for this new Mathematica procedure is Taylor[f, x0, n] ; and the listing is given in the appendix at the end of the article. It produces more information and helps students understand the underlying principleswhen the details are messy. EXAMPLE 1. Find the Taylor polynomial of degree n = 8 and the corresponding error term for the function f(x) _ exp(-x)sin(x) expanded about the valuex0 = 0. SOLUTION. First, we can place sin(x) in the variable F by typing : F = Exp[-x] Sin[x] and the computer responds Issue the new Mathematica command : Taylor[F, 0, 8] ; Then Mathematica will print the following information : 62

3 Formulas for the Numerical values for derivatives : the derivatives : SOLUTION. First, we can place x1/3 in the variable F with the command: f<t)[x] Cosx - EY f(2) = -2 Cos [x] Sin Ex f (0)[0] = 0 ftl)[0] = 1 f( 2 )[0] = -2 and the computer's response is i x' Then issue the Mathematica command : Taylor [F, 8, 91 ; f(3) [x] = 2 Cos + 2 Sin f(3)[0] = 2 Again, a complete list of formulas and values of the Ex EX derivatives will be printed by Mathematica. For brevity, we have edited this printout. f(4)[x] = -4 Si f(5)[x] f(4 )[0] = 0 Formulas for the Numerical values for derivatives : the derivatives : - -4 Cos + 4 Sin f(5)[0] = -4 f(1) Ex Ex 1X] f (1) [8] 12 f(6) = 8 Cosx [x] f( 6)[0] = 8 _ 1. f EX (2)[8] - (144) sh 3 _ -8 Cos - 8 Sin f(7)[x] F)[0] = Ex Ex f(3) [x] =! f (3) [8] x3 (8) 16 Sin f(8)[0] = 0 The Taylor polynomial is : The function is : F[x] =Sin Ex -8 +x -8 +X) x 3 P91x] = The Taylor polynomial is: _ X)4 11(-8 + x)5 _ 77 (-8 + x) P81x1 =x - x (-8 + x)7 _ 935 (-8 + x)8 (-8 + x) The Lagrange form of the remainder term is: x9 16 Cos c - 16 Sin c R8[x] = E c E c where c lies somewhere between 0 and x. In this example, the pattern for the derivatives appears to involve multiples of the sequence {0, 1, -2, 2}, a fact that can now be proven by induction. EXAMPLE 2. Find the Taylor polynomial of degree n=9 and the corresponding error term for the function f(x) _ x1/3 expanded about the value x0 = The Lagrange form of the remainder term is : R9 [x] _ xio LIJ c 3 where c lies somewhere between 8 and x. Sincexl/3 is not symmetric aboutx o = 8, it is interesting to view this approximation. For convenience, we use the interval [0,27] (see Figure 3).

4 f = 1/(x ^ 2 + 1) Figure 3. Interval of Convergence. Another important topic about infinite series is the interval of convergence. For illustrative purposes, we shall consider the Runge function f(x) = 1(x2 + 1). The Taylor polynomials of degree n =10 and n =12 expanded about x=0 are obtained when we issue the commands : p10 = Normal [Series [f, {x, 0, X 2 + x4 _X6 + X8 -x10 Figure 4. Why do the approximations P10 and P 12 diverge away from f near x = 1 and x=-1? Because, P10( 1) = P10(-1 ) = 0, P12( 1 ) = P12( -1 ) = 1 and f(1) = f(-1) = 1/2. The error for all ofthese cases is ± 1/2. This pattern is continued with all of the partial sums of the infinite Taylor series. The Taylor series will converge to f(x) when -1 < x < 1 but will not converge at the endpoints x = ± 1. The theoretical approach is to investigate the error term E10(x) = W1) (c)x11/11! We can gain insights after we plot the function g(x) = f( 11 ) (x)/11! (see Figure 5). g = Together [ (D[f, {x, 11}])/11!] 12x - 220x X5-792x x11 plot[g, {x, -1.5,1.5}] and p12 = Normal[Series [f, {x, 0,13 }]] 1 _X2 + x4 _X6 + X8 -x10 + x12 We can visually inspect how close these approximations are to f(x) when we plot the graphs of f, P 10 (x) and P12(x) (see Figure 4). Figure 5. Looking at the graph of g(x) we can see that I g(x) I < 0.95 and since Ixll I < 1 when IxI < 1, we have established the error bound 64

5 I E1o(x) I = I g(c) I Ix 11 I < * 1 = Investigations such as these take some of the mystery out of mathematical analysis. The Calculus of Power Series The study of power series should be viewed as the extension of the study of polynomials. The connection can be madewhen we use the symbol manipulation power of Mathematica to investigate meaningful examples. cosine = Normal [Series [Cos[x],{x, 0, 13}]] x 2 x 4 X6 X8 x10 x The termwise derivative of this polynomial is easily found: D [cosine, x] x.3 xs x7 x9 x11 - x This can be compared with the series for sin x. sine = Normal [Series [Sin [x], {x, 0,12}]] x x3 xs x7 x9 x The conclusion to be drawn is that d/dz cos(x) = - sin(x). A well known identity from trigonometry can be attempted when we issue the command Normal[sine^2 + cosine^2] x14 x 16 x x The interpretation to be made is that this is a good approximation to 1 whenx is small or even when Ix I < 1. This can be contrasted with the series manipulation ap proach given in Example 3 below, but be prepared to answer some sticky questions. The Cauchy Product CAS can be usedto illustrate abstract ideals byfinding the Taylor series of the product f(x)g(x). It is known that the formula involves a linear combination of derivates and binomial coefficients. First, we can discover the pattern for the higher derivates of the product h(x) _ f(x)g(x) with the following dialogue. D[f[x] g[x],x] g[x] f' [x] + fjx] g' [x] The second derivative of the product is : D[f[x] g[xl, {x, 2}] 2 f' [x] g' [x] + g[x] f" [x] + f[x] g"[x] Similarly, the third and fourth derivatives are : D[f[x] g [x], {x, 3}] 3 g'[x] f"[x] + 3 f'[x] g" [x] + g[x] f( 3 ) [x] + f[x] g(3) [X] and D[f[x] g [x], {x, 4}] 6 f"[x]g"[x]+4g'[x]f(3)[x]+4f'[x]g(3) [x] +g[x] f(4) [x] + qx] g (4 )[x] A regrettable feature of CAS is that the answer might not appear in a format that the user expects or wants ; i.e., the previous results are illuminated when they are rewritten as fg'+f'g fg"+2f'g'+f"g f g"' +3Fg"+3f"g'+f"'g f g"" +4f'g"'+6f"g"+4f"' g'+f""g The pattern of binomial coefficients can now be recognized. Therefore, our Mathematica experimentation led us to the conjecture h(k)(x) =, I (~ ) (x)g(k-j) (x), which can be proven with mathematical induction. Therefore, the Taylor polynomial Pn (x) of degree n for the product h(x) is 65

6 ( 4 ) Pn(x) = 2 kt n h(k) ( x p) (x-xo)k k=0 h(k) (xo) = E (k) f(j) (xo)g( k-j) (xv), j=0 J EXAMPLE 3. Verify that formulas (4) and (5) can be used to find the Taylor polynomial of degree n = 4 for the function h(x) = exp(x)cos(x). SOLUTION. we know that Let f(x) = exp(x) and g(x) = cos(x), then f(0) =1, f'(0) =1, f"(0) =1, f"'(0) =1 and f....(0) =1 g(0) =1, g'(0) =0, g"(0) = -1, g"' (0) = 0 and g.... (0) Therefore, h(0) = 1(1) (1) = 1 h'(0) = 1(1) (0) + 1(1) (1) = 1 h"(0) = 1(1)(-1) + 2(1) (0) + 1(1) (1) = 0 h"'(0) =1(1) (0) + 3(1) (-1) + 3(1) (0) + 1(1)(1) = -2 h"" (0) =1(1)(1)+4(1)(0)+6(1)(-1)+4(1)(0) + 1(1)(1) =-4 Hence, the desired Taylor polynomial is P4 (x) = 1 + 1/1! x + 0/2! x2-2/3! x3-4/4!x4 = 1 + x - x3/3 - x4/6 Of course, this can be verified with the Mathematica cosx = Series [Cos [x], {x, 0, 4 } ] O[x]5 expx = Series [Exp [x], {x, 0, 4 }] l+x [x] 5 coax expx l+x-3-6 +O[x]5 Notice that Mathematica has multiplied the two series together and ignored terms involving powers ofx larger than 4. After all, this is to be expected, because if you do not yet know the coefficients forx 5 in the series for cos(x) and exp(x) then you cannot compute it in their product. Conclusion We have seen how shifting the burden of computing to CAS leaves more time to explore more meaningful problems, rather than studying only a few routine exercises. Details involving error analysis can be obtained for several complicated functions. Graphs of functions and their higher derivatives are easy to obtain. The obstacle of not enough time is erased and implementing an experimental approach is feasible. It does open up the door to asking questions, many of which cannot be answered on the spot. Ifan attitude of encouragement is used then the situation makes students active participants in mathematical exploration, rather than passive recipients of a fixed body of facts. Appendix The Mathematica program listing which prints out the details for a Taylor series is : Taylor[f,x0_,n j : = {Print ["Formulas for the derivatives:"], Do[{Der = D[f, {x, k1l, Print["r,Superscript["(",k,")"], "[x] Der], Print[" {k, 0, n}l, Print["Numerical values for the derivatives:"], Do[{Der = D[f, {x, k}], Val = ReplaceAll]Der,x-+x0], Print["r,Superscript["(",k,")"], "[",x0,"] Val], Print[" {k, 0, n]], Print["The function is :","\n"," F[x] _ ",fl, Print["The Taylor polynomial is:"], S = Normal [Series [f, {x, x0, n}]], Print ["P",Subscript[n],"[x] S], m=n+1, Der = D[f, {x, m1l, Err = ReplaceAll[Der,x-c]c^m/m!, Print["The Lagrange form of the remainder term is :"], Print["R",Subscript[n],"[xl Err], Print["where c lies somewhere between ", x0," and x."] } References Cromer, T. (1987). Some applications of computer algebra. Colleagiate Microcomputer, 5(2), Freese, K, Lounesto, P., & Stegenga, D. (1986). The use of mumath in the calculus classroom. Journal of Computers in Mathematics and Science Teaching, 6, Small, D., & Hosack, J. (1986). Computer algebra systems, tools for reforming calculus instruction. MAA Notes, 6, Mathematical Association of America, Woof, C., & Hodgkinson, D. (1987). mumath. A microcomputer algebra system. New York : Academic Press. Zom,P.(1986). Coputersymbolicmanipulation inelementary calculus. MAA Notes, 6, Mathematical Association ofamerica,