Advanced Calculus I
Advanced Calculus I MATH 315
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This 5 page Class Notes was uploaded by Michale Kuhlman on Monday September 28, 2015. The Class Notes belongs to MATH 315 at George Mason University taught by Robert Sachs in Fall. Since its upload, it has received 19 views. For similar materials see /class/215003/math-315-george-mason-university in Mathematics (M) at George Mason University.
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Date Created: 09/28/15
October 25 2007 Three Applications of the Derivative Math 315 These notes describe the three applications of the derivative that were discussed in class Taylor polynomial approximation convergence of sequences de ned itera tively Newton s method The rst is in our text in section 74 pp 210211 and also p 214 for the Lagrange form of the remainder The second does not appear Newton s method is described in section 75 of the text pp 220222 Each of these topics is in my opinion more properly done now since they do not involve in nite sums which is the topic of chapter 7 1 Taylor polynomials and the remainder You learned in your earlier calculus course that for a given function f on an interval a b which has derivatives of order n for each point mo in a b there is a unique polynomial of degree at most 71 which approximates 1 near mg by matching the values of f and its rst 71 derivatives at 0 namely 1000950 quot0 n 2 m7x02Tm7z0 950 f90090900 which is known as the Taylor polynomial of order n for 1 near me It will be denoted Tn when the context of f and the point mo are implied The question we address is How close is f approximated by the Taylor polyno mial The most important applications come from n 1 and n 2 the linear and quadratic approximations respectively They are used for example to explain how to solve maxmin problems and as we shall see in Newton s method The case of n 0 namely x versus flt0 is the Mean Value Theorem In that case we proved Theorem 1 is continuous on 11 and d erentiable on 11 there exists at least one point c E a 1 such that 7 fa fc b 7 a If we apply that theorem to a pair of points mo and z inside some other interval where the hypotheses hold we get the case of n 0 in Taylor s theorem with remainder where the right hand side is the remainder The basic idea of generating the higher order polynomials is to use a linear ap proximation for successive derivatives which are then integrated So if f is approximated by the linear function 1quot 0 f m0m 7 mo then an integration back to 1 yields the quadratic approximation W 3 mo f mox 7 me f z 7 mo Tm A similar linear approximation of f leads to T3z and in general Tnm arises by linear approximation of f 1z at 0 In class based on the homework exercise the following limit was obtained case of n l and then extended to general 71 which we do below Lemma 2 For afunction 1 which is twice d erentiable in a neighborhood ofmo hm 90 i 950 f90090 900 f 900 2 17 x 7 m0 More generally for a function f with n l derivatives M 7 Tux fltn1900 1117120 m7x0 1 7 71 1 39 Recall that we proved this using L Hopital s rule 71 1 times This suggests one way the Taylor polynomial can arise in an iterative fashion and leads to the conjecture for the remainder form for x 7 Tnz which means NOT taking the limit which based on the mean value theorem should become Theorem 3 Taylor s theorem with Cauchy s form ofthe remainder Let n be a natural number let a andb be distinct extended real numbers let 1 11 7 R and suppose fmll exists on a b Then for each pair ofpoints z m0 6 a b there is a number 0 between m and me such that f 10 7 Tnx 7 7 0n1 We will see a total of four proofs of this two with and two without integrals The rst proof is the one in our book and it uses the generalized mean value theorem in a clever way as follows Proof 1 Assume without loss of generality that 0 lt z and use the Generalized Mean Value Theorem on the following two functions of t with mg g t g m m7 n n Glttgt7fltzgt72f War Ft k0 The second function uses the base point mo in the Taylor polynomial as a variable and is the clever choice Note that 0 while Fm0 As well Cm 0 while Cmo x 7 Tnm and each function is differentiable in 25 between the points z and mo Applying the generalized mean value theorem yields a point c where the claimed result holds Proof 2 This is similar but uses Rolle s theorem repeatedly to nd 0 after cooking up a term to make that theorem be applicable Looking at f 7 Tnm for the t1 dtn we we we we Proof4 Lagrange form of remainder text p 214 add 1 to their 71 Remainder is the following integral 1 12 7 725 W1 t dt Mmm f ltgt which is seen to be correct by showing it satis es the right differential equation and initial conditions as per Proof 3 Fans of integration will recognize that this comes from reversing the order of the nested integrations and thereby integrating the derivative of 1 last instead of rst 2 Convergence of iterative sequences revisited We now revisit our earlier discussion of iterative sequences meaning a se quence where my 9zn for some differentiable function g In Chapter 2 we learned the following two facts 11fmn 7 LthenL gL 2 If is monotone and bounded it is convergent Now we can use our understanding of derivatives to generate a useful third result Ifg is differentiable maps some interval into itself and lg l r lt l for all z in a neighborhood containing all sequence elements then the sequence converges to a unique xed point The key step is to show that the factor r in the hypothesis is a geometric factor that indicates that 9 moves points closer together by a factor of r or better namely ln2 7 mn1l l9ltn1 7 lgCnn1 7 g Tln1 7 Therefore the distances between successive points shrinks faster or at the rate of T which allows us to sum the distances by a geometric series Mathematicians call this the Banach Contracting Mapping Principle and it works in a general setting of distance metric spaces as long as Cauchy sequences are convergent complete ness More carefully observe the following The xed point if it exists is unique Suppose that L and M are xed points and distinct Then L 7 M 7 ML 7 WW M 7 M which makes lL 7 M l O which is a contradiction hence the uniqueness The sequence is Cauchy As mentioned above repeated use of the de nition yields lzn 7 znl rim 7 zn1l lm 7 mol for all n and therefore lnk 90ml S lnk n7k71l ln7k71 n7k72lln1 90ml r k71 Tnk72 r lzl 7 zol and this tends to zero as 71 tends to in nity for any k gt 0 An example revisited Problem 2 from Exam 1 used the function gz 4 7 3z with starting value mo 4 numbered from 1 there Since 9 3z2 which is less than 1 when z2 gt 3 we nd that the mapping is contracting for z 2 2 say with lg g 34 there So the sequence starting from 4 converges since 9 maps 2 4 into itself Wasn t that easier Near the xed point z 3 we expect terms to decrease their distance from 3 by a factor very close to g 3 13 This is easily observed numerically 3 Newton s method and revisited When iterations as discussed above have 9 L 0 then we need to think about the quadratic approximation in order to understand how the sequence be haves As we saw for the iterative computation of such iterations may converge extremely rapidly To solve f 0 for some x in an interval 0 1 Newton used the linear approxi mation to f at some approximate solution x xn to generate the next approximate solution xn1 More precisely solve for the x that makes xn fxnx 7 35 0 and call that solution xn1 which of course requires 1quot 7 O Graphically we nd where the tangent line crosses the level y O algebraically this says HM and observe that such an iteration has the properties n1 mn 7 1 If xn 0 then xn1 xn 21me 7LthenLL7 f so L 0 3 By Taylor s theorem x xn f xnx 7 xn f cnx 7 xn2 so we calculate fmn which shows that the nearness to 0 of f xn as output is a factor times the value of f squared fltzn1gt7 f ltgtltznl7 m2 gmme gt2 5 Using a presumed limit L with L O we nd x x 7 L z f Lx7L assuming L 7 0 Applying this with x xn1 andx xn and using property 3 enables us to analyze the convergence rate of xn to L yielding f 0nf L 2fn2 where there is some additional approximate cancellation assuming xn is near L and 1quot doesn t vary rapidly xn1 7 L z xn 7 L2 Newton s method is a very valuable tool for solving equations numerically It is also the basis for some fancy theoretical mathematics especially when the role of x is played by functions This includes some of the best works of John Nash
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