k=1 s k (x) (3) and that the corresponding infinite series may also converge; moreover, if it converges, then it defines a function S through its sum

Transcription

1 0. L Hôpital s rule You alreay kow from Lecture 0 that ay sequece {s k } iuces a sequece of fiite sums {S } through S = s k, a that if s k 0 as k the {S } may coverge to the it k= S = s s s 3 s 4 = s k. k= The the right-ha sie of is calle a ifiite series, a the left-ha sie is the sum to which it coverges. You also kow from Lecture 7 what a fuctio sequece is, a that a fuctio sequece may likewise coverge. So it shoul t surprise you that ay fuctio sequece {s k } iuces a fuctio sequece of fiite sums {S } through S = s k 3 k= a that the correspoig ifiite series may also coverge; moreover, if it coverges, the it efies a fuctio S through its sum S = s s s 3 s 4 = s k. 4 k= The most importat case is where s k is a power fuctio, efie by s k = a k k 5 with a k 0 as k. The the right-ha sie of 4 is calle surprise, surprise! a power series. If a 0 a a a 3 3 a 4 4 a 5 5 = a k k 6 k=0 yiels g, where g is a kow fuctio, the we call 6 the power-series represetatio of g; a otherwise, g is just the sum of the power series. So, i effect, either the series yiels a alterative represetatio of a fuctio we alreay kow, or else it efies a bra ew fuctio. For eample, recall from Lectures 7 a 9 that the epoetial fuctio is efie by e. 7 The series coverges oly if s k 0 as k because s k = S k S k, a so if {S } coverges the s k S k S k S k S k = S S = 0. k k k k

2 From the biomial theorem, however, we have a = a. a..3 a a a 5 plus a lot more terms i fact, 5 of them. 8 If we set a = / we get = = so that, o usig e = 0, we have = = ! =0 0 which is a coverget power series. Power-series represetatios of fuctios ca be etremely useful. For eample, 0 yiels two aitioal ways i which to show that the epoetial is its ow erivative. If we use the properties of the epoetial fuctio iscovere i Lecture 9 to perturb y = e to y δy = e δ, the we obtai δy = e δ e = e e δ e = e e δ. But replacig by δ i 0 yiels e δ = δ δ δ3 δ4 δ5 = δ oδ, a so {e } = y δy δ δ e e δ δ δ = e δ δ oδ δ = e. = e δ = e e δ { δ δ oδ } = e { 0} δ It coverges for all values of. I geeral, a power series will coverge for some values of a iverge for other values of ; however, the power series of three of our most importat fuctios, amely, ep, si a cos have the very ice property that they always coverge.

3 Or else we ca procee as follows : e = 3. = = = = = e. So let us co- Ay other power series ca likewise be ifferetiate term by term. sier the power series of a arbitrary fuctio g: g = a 0 a a a 3 3 a 4 4 a 5 5 = a k k 3 k=0 Differetiatig oce yiels { g = a0 a a a 3 3 a 4 4 a 5 5 } { } { a } { 3 3 a } { 4 4 a } 5 5 = {a 0} a {} a = 0 a a a 3 3 a a = a a 3a 3 4a 4 3 5a Differetiatig agai yiels g = { {g } = a a 3a 3 4a 4 3 5a 5 4 } { } { 4a } { 4 3 5a } 5 4 = {a } a {} 3a 3 = 0 a 3a 3 4a 4 3 5a = a 6a 3 a 4 0a 5 3 a so o. Settig = 0 i 3 yiels g0 = a 0 ; settig = 0 i 4 yiels g 0 = a ; settig = 0 i 5 yiels g 0 = a ; a so o. I other wors, a 0 = g0; a = g 0; a = g 0; a so o. Substitutig back ito 3 yiels 5 g = g0 g 0 g 0 6a Clearly, the ame we gave our arbitrary fuctio itself is arbitrary, a so if 6 hols the just as surely oes h = h0 h 0 h 0 6b Makig various assumptios whose valiity we have o choice but to take for grate at this stage i our stuy of the calculus. Provie the aswer coverges, which aturally we assume.

4 Hece g h = g0 g 0 g 0 h0 h 0 h 0 There is a problem with 7, however: by virtue of all those ots, it is less precise tha we like our equatios to be. We therefore efie O calle big oh of for m to mea ay ifiite power series i which is the lowest iteger epoet: ituitively, O = A B C, where A, B a C are iepeet of a hece costats i the it as 0. Thus, by ispectio, O has the followig two properties: O = 0 for, With this ew otatio we ca be more precise about 7: 7 O m = O m for m. 8 g h = g0 g 0 g 0 O 3 h0 h 0 h 0 O 3. 9 Now, suppose that you wish to calculate the it of the quotiet g as 0. By h our geeral rule about its, the it of this quotiet is the quotiet of the its, provie the aswer eists. So if g0 a h0 are both ozero the g h = g0 h0. 0 If g0 = h0 = 0, however, the 9 implies g h g 0 O h 0 O g 0 O h 0 O = g 0 h 0 after iviig both umerator a eomiator by a usig 8; a if g 0 a h 0 are also both zero, i.e., if g0 = h0 = g 0 = h 0 = 0, the 9 implies g h g 0 O 3 h 0 O 3 g 0 O h 0 O = g 0 h 0 after iviig both umerator a eomiator by a usig 8 agai. A so o. A slight geeralizatio of this result is sometimes useful. Suppose that G a H are relate to g a h by g = G a, h = H a. 3 The, from the chai rule, we have g = G a {a} = G a{0} = G a, g = {G a} = G a { a} = G a{ 0} = G a, etc., So O is a ew ki of juk term, relate to o by O = o if, but wellig o this relatioship will serve o purpose here. Provie, of course, that all the erivatives eist, which we assume.

5 a similarly for h. So, i particular, g 0 = G a, g 0 = G a, etc.; a similarly, h 0 = H a, h 0 = H a, etc. Also, g0 = Ga a h0 = Ha, from 3. So 0-3 yiel the followig: Ga = Ha = 0 AND H G a 0 = a G Ga = Ha = G a = H a = 0 AND H G a 0 = a G a so o. Moreover, we eve have G = H = AND G,H FINITE = G G = G a H a = G a H a = G H 4a 4b a so o. We refer to 4 as L Hôpital s rule. We ca ofte use L Hôpital s rule to etermie the itig behavior of ietermiate quotiets, or eve after maipulatio ietermiate proucts or iffereces. It eve works for ifiite its. Nevertheless, L Hôpital s rule is also ofte uecessary, a ee ot yiel the most elegat approach, eve where it works. Cosier, e.g., L g h where g = 3 a h = e. 5 Because g = 3, g = 6, g = 6, h = e 0 = e, h = e 0 = e a h = e 0 = e, we have both g0 = g 0 = g 0 = 0 a h0 = h 0 = h 0 = 0, so that the a so o of L Hôpital s rule yiels L = g 0 h 0 g h It woul have bee simpler, however, to observe from 0 so that 5 implies L 3 e = 6 e 0 = 6. 6 e = 6 3 O 4, O 4 6 O = 6 = 6, 8 0 after ivisio by 3 a use of 8. O the other ha, there are cases where L Hôpital s rule is ueiably the most efficiet tool for the job. Cosier, e.g., L G H where G = a H = l e. 9 By supposig istea that G a H are relate to g a h by g/ = G, h/ = H a otig that G g, H h a g //h / = G /H by the chai rule, etc. Although i this course we have a strog preferece for thikig that aythig that eserves the ame of it shoul at least have the gooess to be fiite.

6 Because G G = a H H =, a because G = implyig at oce that G = a, from the chai rule, H = e { e } = e {0 e } = e e = e 30 so that H H = =, from 4b we obtai 0 L = G H G H = =. 3 Or cosier L l, 3 which is a ietermiate ifferece of type. Notig that l = l l, 33 we ca eploit 4 with a =, G = l = G = 0 = /, G = 0 { } = a H = l = H = 0 l = l /, H = / 0 = / to observe that, because G = l = 0, G = / = 0 a H = l = 0, H = l / = 0, L = G H G H = =. 34 Whether, however, L Hôpital s rule yiels the eatest metho for obtaiig this it is moot. A alterative approach is to ote that with g = l i 6 we obtai l = O Now, substitutig u = or = u i 3 a recallig that the efiitio of Ou oes ot istiguish betwee Ou a Ou, we obtai L u 0 l u u u Ou 3 u 0 u u3 Ou 4 u 0 u l u u l u Ou u 0 u Ou u 0 u {u u Ou 3 } u{u u Ou 3 } = = after ivisio by u a use of 8 with u i place of. Note that, i the seco lie, we coul have replace u3 Ou 4 by Ou 3 a uou by Ou i the eomiator, without affectig the result i ay way. I prefer the seco approach. Which o you prefer? 36

7 Eercise. Show that siδ = δ oδ a cosδ = δ oδ.. The lie y = a b is sai to be a oblique asymptote for the ratioal fuctio f as if f is of type O /O for some a {f a b} = 0 a similarly for. Fi the oblique asymptote for f efie o, by f = Suitable problems from staar calculus tets Stewart 003: pp , ## 5-6; p. 34, ## 55-6, 63 a 64 see Eercise a its solutio. Referece Stewart, J. 003 Calculus: early trasceetals. Belmot, Califoria: Brooks/Cole, 5th e. Solutios to selecte eercises. Use 6 with g = si a h = cos to obtai si = O 3 a cos = O 3 ; the replace by δ.. Here =. After simplificatio we have f a b = 5 3b 3a b a b a 3 3 This epressio approaches zero as if a oly if the egree of the polyomial i the umerator is less tha the egree of the polyomial i the eomiator. So the coefficiets of a 3 must both be zero i the umerator. That is, ab = 0 a a = 0 or a =,b =. So the oblique asymptote is y =.