Tylor Polynomil Inequlities Ben Glin September 17, 24 Abstrct There re instnces where we my wish to pproximte the vlue of complicted function round given point by constructing simpler function such s polynomil which serves s n pproximtion to the originl function. In this pper we will show how such function cn be composed. In the first exmple, we will drw reltion between polynomil nd n exponentil function. In the second one, we will nlyze trigonometric function. We will then see tht these polynomils re members of fmily clled Tylor Polynomils. We will not explore to depth these pproximtions but merely outline, with the mens of bsic clculus, wy to derive them. Inequlities between polynomil nd e x In this section we will construct generl-form inequlity reltion between e x nd polynomils using the Construction Theorem of Clculus (lso known s the Second Fundmentl Theorem of Clculus). We will strt with specific cse nd then generlize it using the Induction Principl. For first-degree polynomils, we know tht e x 1 + x (1) for ll x R. We will, however, need to constrin ourselves to the domin x. Though not in the scope of this pper, the vlidity of this inequlity cn be proven 1 by showing tht (i) e x = 1 + x for x = ; nd (ii) e x > (1 + x) for ll x >. If f(x) g(x) on some intervl [, b], it follows tht f(x) dx g(x) dx. (2) Drft.9. All rights reserved c 24. Source code with limited rights cn be found t http://www.bens.ws/professionl.html. With the supervision of to be determined. 1 For the domin x. 1
Once gin, the proof for this theorem is outside the scope of this pper. The generl ide behind tht proof is showing tht the intervl [, b] cn be divided into n subintervls, in ech of which f(x) is lower-bounded by some constnt m i nd g(x) is upper-bounded by nother constnt M i, nd tht m i M i. We then let n. If we let f(x) = e x nd g(x) = 1 + x in theorem (2), it implies tht e x dx 1 + x dx. (3) Since we re interested in non-negtive vlues of x, we will let = nd introduce nother vrible, t, tht rnges from to x, thus letting b = x. Inequlity (3) is now rewritten s 1 + t dt. Using the Construction Theorem of Clculus for the integrnds e t nd 1 + t yields tht, in turn, is rerrnged s e x 1 x + x2 2, e x 1 + x + x2 2. Let s proceed with second itertion. Once gin we integrte both sides with respect to t where t x. Nmely: Solving the integrls yields 1 + t + t2 2 dt. e x 1 x + x2 2 + x3 6. Tht result might suggest the following pttern: e x 1 + x1 1! + x2 2! + x3 3! + + xn (4) for ll n N, x. In order to prove this theorem we will use the Induction Principle. Recll tht the Induction Principle sttes tht n infinite sequence of propositions P i for i = 1,..., is estbished if (i) P 1 is true, nd (ii) P k implies P k+1 for ll k [Weisstein, 24]. We will let inequlity (1) be P 1 nd inequlity (4) be P k with the subscript k substituting n. 2
We lredy showed the vlidity of proposition P 1. We will proceed, then, with the induction step. As with inequlity (1), we tke the integrl of P k with respect to t, where t x: Since we know tht in generl k 1 + t1 1! + t2 2! + t3 3! + + tk dt. (5) k! m dx = xk+1 m(k + 1) + C for k 1, integrting both sides of (5) yields e x 1 x1 1 1 + x2 1! 2 + x3 2! 3 + x4 3! 4 + + xk+1 k!(k + 1), tht consequently rewritten s Theorem (4) is now proven. e x 1 + x1 1! + x2 2! + x3 3! + x4 4! + xk+1 (k + 1)!. Inequlities between polynomil nd cos x In similr fshion, we will construct n inequlity reltion between cos x nd polynomils, strting with constnt nd building up to generl reltion. We know tht cos x 1 (6) for ll x R. Once gin, we use theorem (2) to preform repeted steps of integrtion from to x on both the right- nd left-hnd sides of the inequlity. The first step results in tht solves to cos t dt sin x x. A second itertion produces the following set of inequlities (note the lbeling chnge from x to t): nd sin t dt 1 dt t dt cos x + 1 x2 2. 3
The process repets twice more, resulting with the following four 2 inequlities: t 2 cos t + 1 dt 2 dt, nd sin x + x x3 6, sin t + t dt t 3 6 dt, cos x + x2 2 1 x4 24. (7) Inequlity (7) is then rerrnged: which seems to suggest generl form: cos x 1 x2 2! + x4 4! cos x 1 + ( x2 ) 1 (2 1)! + ( x2 ) 2 (2 2)! + + ( x2 ) n (2n)! (8) where n = 2(m 1), m N. Proving this theorem using the Induction Principle is not complicted but involves n dditionl four-itertion process nd thus will not be outlined here. Generliztion We my wonder whether there is reltion between the two conclusions in the previous sections. It turns out tht such reltion, or more precisely, generlized formul, exists. In order to find this generlized formul, we must first look t ech of the polynomils in these inequlities s n pproximtion. Ech inequlity pproximtes the vlue of function on the left-hnd side, e x in the first cse, nd cos x in the other, to the vlue of polynomil of degree n on the right-hnd side, for x tht is close to. 3 Note tht the reson tht x is close to nd not to some other constnt lies in our choice for in inequlity (3). We could hve chosen different, but then the process of integrting would hve been more complicted. Tht constrint, however, is not substntil, s functions cn be shifted long the x-xis. Recll tht function f(x) cn be pproximted by polynomil of degree n P n (x) ner x = iff the function itself nd ech of its first n derivtives 2 Note tht while e x remins intct in integrtion, cos x, on the other hnd, requires four integrtions to return to its originl form. 3 We weren t completely ccurte in tht lst sttement; in order to prove tht one such polynomil is indeed n pproximtions we need show tht the difference between the function nd the polynomil pproches s n. 4
gree with those of the polynomil P n (x) [Hughes-Hllett et l., 22]. We cn represented P n (x) s the series 4 P n (x) = C + C 1 x + C 2 x 2 + + C n x n. (9) If f(x) P n (x) nd x = it follows tht C = f(). By differentition nd substitution, we find tht C 1 = f () nd tht C 2 = 1 2 f (). In similr mnner we cn conclude tht C n = f (n) (), (1) where f (n) denotes the nth derivtive of f. Substituting eqution (1) for different n s in eqution (9) yields P n (x) = f() + f () x + f () x 2 or 5 P n (x) = 2! + f () x 3 3! n f (i) () x i. i! i= + + f (n) () x n, (11) The polynomil in eqution (11) pproximtes the function f(x) for x tht is close to. We cll this pproximtion Tylor Polynomil of Degree n nd the polynomils we found in inequlities (4) nd (8) were specil cses of it. References [Hughes-Hllett et l., 22] Hughes-Hllett, Gleson, McCllum, et l. (22). Clculus, Single nd Multivrible, chpter 1.1 Tylor Polynomils. John Wiley nd Sons, 3rd edition. [Weisstein, 24] Weisstein, E. W. (24). Principle of Mthemticl Induction. http://mthworld.wolfrm.com/principleofmthemticlinduction. html. Accessed 8/22/4. 4 The following is n dpttion from [Hughes-Hllett et l., 22]. 5 Recll tht! = 1 by definition. We lso let f () (x) f(x). 5