Review of Calculus, cont d


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1 Jim Lmbers MAT 460 Fll Semester Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some quntity is defined to be the product of two other quntities. For exmple, rectngle of width w hs uniform height h, nd the re A of the rectngle is given by the formul A = wh. Unfortuntely, in mny pplictions, we cnnot necessrily ssume tht certin quntities such s height re constnt, nd therefore formuls such s A = wh cnnot be used directly. However, they cn be used indirectly to solve more generl problems by employing the nottion known s integrl clculus. Suppose we wish to compute the re of shpe tht is not rectngle. To simplify the discussion, we ssume tht the shpe is bounded by the verticl lines x = nd x = b, the xxis, nd the curve defined by some continuous function y = f(x), where f(x) 0 for x b. Then, we cn pproximte this shpe by n rectngles tht hve width x = (b )/n nd height f(x i ), where x i = + i x, for i = 0,..., n. We obtin the pproximtion A A n = n f(x i ) x. i=1 Intuitively, we cn conclude tht s n, the pproximte re A n will converge to the exct re of the given region. This cn be seen by observing tht s n increses, the n rectngles defined bove comprise more ccurte pproximtion of the region. More generlly, suppose tht for ech n = 1, 2,..., we define the quntity R n by choosing points = x 0 < x 1 < < x n = b, nd computing the sum R n = n f(x i ) x i, x i = x i x i 1, x i 1 x i x i. i=1 The sum tht defines R n is known s Riemnn sum. Note tht the intervl [, b] need not be divided into subintervls of equl width, nd tht f(x) cn be evluted t rbitrry points belonging to ech subintervl. If f(x) 0 on [, b], then R n converges to the re under the curve y = f(x) s n, provided tht the widths of ll of the subintervls [x i 1, x i ], for i = 1,..., n, pproch zero. This 1
2 behvior is ensured if we require tht lim δ(n) = 0, where δ(n) = mx x i. n 1 i n This condition is necessry becuse if it does not hold, then, s n, the region formed by the n rectngles will not converge to the region whose re we wish to compute. If f ssumes negtive vlues on [, b], then, under the sme conditions on the widths of the subintervls, R n converges to the net re between the grph of f nd the xxis, where re below the xxis is counted negtively. We define the definite integrl of f(x) from to b by f(x) dx = lim n R n, where the sequence of Riemnn sums {R n } n=1 is defined so tht δ(n) 0 s n, s in the previous discussion. The function f(x) is clled the integrnd, nd the vlues nd b re the lower nd upper limits of integrtion, respectively. The process of computing n integrl is clled integrtion. In this course, we will study the problem of computing n pproximtion to the definite integrl of given function f(x) over n intervl [, b]. We will lern number of techniques for computing such n pproximtion, nd ll of these techniques involve the computtion of n pproprite Riemnn sum. Extreme Vlues In mny pplictions, it is necessry to determine where given function ttins its minimum or mximum vlue. For exmple, business wishes to mximize profit, so it cn construct function tht reltes its profit to vribles such s pyroll or mintennce costs. We now consider the bsic problem of finding mximum or minimum vlue of generl function f(x) tht depends on single independent vrible x. First, we must precisely define wht it mens for function to hve mximum or minimum vlue. Definition (Absolute extrem) A function f hs bsolute mximum or globl mximum t c if f(c) f(x) for ll x in the domin of f. The number f(c) is clled the mximum vlue of f on its domin. Similrly, f hs bsolute minimum or globl minimum t c if f(c) f(x) for ll x in the domin of f. The number f(c) is then clled the minimum vlue of f on its domin. The mximum nd minimum vlues of f re clled the extreme vlues of f, nd the bsolute mximum nd minimum re ech clled n extremum of f. Before computing the mximum or minimum vlue of function, it is nturl to sk whether it is possible to determine in dvnce whether function even hs mximum or minimum, so tht 2
3 effort is not wsted in trying to solve problem tht hs no solution. The following result is very helpful in nswering this question. Theorem (Extreme Vlue Theorem) If f is continuous on [, b], then f hs n bsolute mximum nd n bsolute minimum on [, b]. Now tht we cn esily determine whether function hs mximum or minimum on closed intervl [, b], we cn develop n method for ctully finding them. It turns out tht it is esier to find points t which f ttins mximum or minimum vlue in locl sense, rther thn globl sense. In other words, we cn best find the bsolute mximum or minimum of f by finding points t which f chieves mximum or minimum with respect to nerby points, nd then determine which of these points is the bsolute mximum or minimum. The following definition mkes this notion precise. Definition (Locl extrem) A function f hs locl mximum t c if f(c) f(x) for ll x in n open intervl contining c. Similrly, f hs locl minimum t c if f(c) f(x) for ll x in n open intervl contining c. A locl mximum or locl minimum is lso clled locl extremum. At ech point t which f hs locl mximum, the function either hs horizontl tngent line, or no tngent line due to not being differentible. It turns out tht this is true in generl, nd similr sttement pplies to locl minim. To stte the forml result, we first introduce the following definition, which will lso be useful when describing method for finding locl extrem. Definition(Criticl Number) A number c in the domin of function f is criticl number of f if f (c) = 0 or f (c) does not exist. The following result describes the reltionship between criticl numbers nd locl extrem. Theorem (Fermt s Theorem) If f hs locl minimum or locl mximum t c, then c is criticl number of f; tht is, either f (c) = 0 or f (c) does not exist. This theorem suggests tht the mximum or minimum vlue of function f(x) cn be found by solving the eqution f (x) = 0. As mentioned previously, we will be lerning techniques for solving such equtions in this course. These techniques ply n essentil role in the solution of problems in which one must compute the mximum or minimum vlue of function, subject to constrints on its vribles. Such problems re clled optimiztion problems. Although we will not discuss optimiztion problems in this course, we will lern bout some of the building blocks of methods for solving these very importnt problems. The Men Vlue Theorem While the derivtive describes the behvior of function t point, we often need to understnd how the derivtive influences function s behvior on n intervl. This understnding is essentil in numericl nlysis becuse, it is often necessry to pproximte function f(x) by function 3
4 g(x) using knowledge of f(x) nd its derivtives t vrious points. It is therefore nturl to sk how well g(x) pproximtes f(x) wy from these points. The following result, consequence of Fermt s Theorem, gives limited insight into the reltionship between the behvior of function on n intervl nd the vlue of its derivtive t point. Theorem (Rolle s Theorem) If f is continuous on closed intervl [, b] nd is differentible on the open intervl (, b), nd if f() = f(b), then f (c) = 0 for some number c in (, b). By pplying Rolle s Theorem to function f, then to its derivtive f, its second derivtive f, nd so on, we obtin the following more generl result, which will be useful in nlyzing the ccurcy of methods for pproximting functions by polynomils. Theorem (Generlized Rolle s Theorem) Let x 0, x 1, x 2,..., x n be distinct points in n intervl [, b]. If f is n times differentible on (, b), nd if f(x i ) = 0 for i = 0, 1, 2,..., n, then f (n) (c) = 0 for some number c in (, b). A more fundmentl consequence of Rolle s Theorem is the Men Vlue Theorem itself, which we now stte. Theorem (Men Vlue Theorem) If f is continuous on closed intervl [, b] nd is differentible on the open intervl (, b), then f(b) f() = f (c) b for some number c in (, b). Remrk The expression f(b) f() b is the slope of the secnt line pssing through the points (, f()) nd (b, f(b)). The Men Vlue Theorem therefore sttes tht under the given ssumptions, the slope of this secnt line is equl to the slope of the tngent line of f t the point (c, f(c)), where c (, b). The Men Vlue Theorem hs the following prcticl interprettion: the verge rte of chnge of y = f(x) with respect to x on n intervl [, b] is equl to the instntneous rte of chnge y with respect to x t some point in (, b). The Men Vlue Theorem for Integrls Suppose tht f(x) is continuous function on n intervl [, b]. Then, by the Fundmentl Theorem of Clculus, f(x) hs n ntiderivtive F (x) defined on [, b] such tht F (x) = f(x). If we pply the Men Vlue Theorem to F (x), we obtin the following reltionship between the integrl of f over [, b] nd the vlue of f t point in (, b). 4
5 Theorem (Men Vlue Theorem for Integrls) If f is continuous on [, b], then for some c in (, b). f(x) dx = f(c)(b ) In other words, f ssumes its verge vlue over [, b], defined by f ve = 1 b f(x) dx, t some point in [, b], just s the Men Vlue Theorem sttes tht the derivtive of function ssumes its verge vlue over n intervl t some point in the intervl. The Men Vlue Theorem for Integrls is lso specil cse of the following more generl result. Theorem (Weighted Men Vlue Theorem for Integrls) If f is continuous on [, b], nd g is function tht is integrble on [, b] nd does not chnge sign on [, b], then for some c in (, b). f(x)g(x) dx = f(c) g(x) dx In the cse where g(x) is function tht is esy to ntidifferentite nd f(x) is not, this theorem cn be used to obtin n estimte of the integrl of f(x)g(x) over n intervl. Exmple Let f(x) be continuous on the intervl [, b]. Then, for ny x [, b], by the Weighted Men Vlue Theorem for Integrls, we hve x x x (s )2 f(s)(s ) ds = f(c) (s ) ds = f(c) (x )2 2 = f(c), 2 where < c < x. It is importnt to note tht we cn pply the Weighted Men Vlue Theorem becuse the function g(x) = (x ) does not chnge sign on [, b]. Tylor s Theorem In mny cses, it is useful to pproximte given function f(x) by polynomil, becuse one cn work much more esily with polynomils thn with other types of functions. As such, it is necessry to hve some insight into the ccurcy of such n pproximtion. The following theorem, which is consequence of the Weighted Men Vlue Theorem for Integrls, provides this insight. Theorem (Tylor s Theorem) Let f be n times continuously differentible on n intervl [, b], nd suppose tht f (n+1) exists on [, b]. Let x 0 [, b]. Then, for ny point x [, b], f(x) = P n (x) + R n (x), 5
6 where nd P n (x) = n j=0 f (j) (x 0 ) (x x 0 ) j j! = f(x 0 ) + f (x 0 )(x x 0 ) f (x 0 )(x x 0 ) f (n) (x 0 ) (x x 0 ) n n! R n (x) = x x 0 where ξ(x) is between x 0 nd x. f (n+1) (s) n! (x s) n ds = f (n+1) (ξ(x)) (x x 0 ) n+1, (n + 1)! The polynomil P n (x) is the nth Tylor polynomil of f with center x 0, nd the expression R n (x) is clled the Tylor reminder of P n (x). When the center x 0 is zero, the nth Tylor polynomil is lso known s the nth Mclurin polynomil. The finl form of the reminder is obtined by pplying the Men Vlue Theorem for Integrls to the integrl form. As P n (x) cn be used to pproximte f(x), the reminder R n (x) is lso referred to s the trunction error of P n (x). The ccurcy of the pproximtion on n intervl cn be nlyzed by using techniques for finding the extreme vlues of functions to bound the (n + 1)st derivtive on the intervl. Becuse pproximtion of functions by polynomils is employed in the development nd nlysis of mny techniques in numericl nlysis, the usefulness of Tylor s Theorem cnnot be overstted. In fct, it cn be sid tht Tylor s Theorem is the Fundmentl Theorem of Numericl Anlysis, just s the theorem describing inverse reltionship between derivtives nd integrls is clled the Fundmentl Theorem of Clculus. 6
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