CALCULUS WITHOUT LIMITS

Transcription

1 CALCULUS WITHOUT LIMITS The current stndrd for the clculus curriculum is, in my opinion, filure in mny spects. We try to present it with the modern stndrd of mthemticl rigor nd comprehensiveness but of course hve to leve out mny crucil detils t the expense of much of the wonderful intuition behind Leibniz s nottions, nd mny clssicl problems tht it helped solve. I believe in the revivl, t lest in first introduction, of the oldfshioned clculus, 1 s ws in use with gret success for well over hundred yers before the definition of limits ws mde precise by Cuchy. Its use remins in prctice by mny who rediscovered it on their own, thnks to Leibniz s nottions (sdly the vst mjority of the students miss out on it). In brief, the concept of differentils, the mysterious infinitesiml quntities, tkes the center stge. [For wht it s worth, I hve not consulted ny pre-cuchy clculus textbooks, so this my only represent my own interprettions.] 1. Differentils The scenrio is bit different from the modern emphsis on functions. We strt with two vribles, typiclly clled x nd y, tht re relted in some fshion not just in the form of y = f(x), i.e., y is given by n (explicit) expression of x, but more generlly ny reltion involving x nd y such s x + y = 1 which describes (the points on) the unit circle. The differentils will be quntities tht re written s dx nd dy, which re ment to represent infinitesiml (i.e. infinitely smll) chnges in x nd y, respectively, subject to the constrint of the given reltion. As x nd y re relted vi mthemticl formul, dx nd dy will likewise be relted in precise wy, nd the gme is to find this reltion. Wht we normlly cll the derivtive (of y with respect to x) is, not surprisingly, the quotient dy/dx of the two differentils. The procedure to compute differentils is strightforwrd: You put in x + dx in plce of x, nd y + dy in plce of y, nd mnipulte ccording to the fmilir rules of lgebr. For the reltion bove, we hve (x + dx) + (y + dy) = 1 Using the fct tht x + y = 1, we hve x + x dx + dx + y + y dy + dy = 1 x dx + dx + y dy + dy = 0 Dte: Jnury 8, The word, by the wy, simply mens set of rules or procedures (often simple nd mechnicl) for clculting something, not so much the theory (if ny) behind it. It hs lmost exclusively come to stnd for clculus of differentils nd integrls. 1

2 CALCULUS WITHOUT LIMITS Now, s dx is infinitely smll, dx will be fr smller in comprison so we cn sfely neglect it; the sme goes with dy. Therefore, we hve x dx + y dy = 0 which tells us how dx nd dy re relted, t ech point (x, y) on the circle. If you like, we cn write it in the form of derivtive dy dx = x y = x ± 1 x where the ± depends on which hlf of the circle you re on. Exmple : y = x, so tht y = x (y + dy) = x + dx y + y dy + dy = x + dx y dy + dy = dx nd dropping the higher differentil s before, we get or if you wish, Exmple 3: y = x, or Dropping the dx, nd expnd nd then dropping the dx dy, or In fct, it s hndy to remember y dy = dx dy dx = 1 y = 1 x x y = 1 (x + dx) (y + dy) = 1 (x + x dx + dx )(y + dy) = 1 x y + x dy + xy dx + x dx dy = 1 x dy + xy dx = 0 dy dx = y x = x 3 d(x n ) = nx n 1 dx even when n is frction or negtive (nd indeed ny rel number, but tht s rre in pplictions). So, we my directly pply d on both sides of ny (lgebric) reltion of x nd y, which is I suppose the word differentition originlly ment, with the help of few rules tht I ll come to shortly.

3 Exmple 4: y = sin θ. We compute s follows: CALCULUS WITHOUT LIMITS 3 y + dy = sin(θ + dθ) = sin θ cos(dθ) + cos θ sin(dθ) ccording to the trigonometric identity. Here s some hnd-wving: s dθ is infinitely smll, cos(dθ) is very close to 1 while sin(dθ) is prcticlly just dθ. So we get so y + dy = sin θ + cos θ dθ dy = cos θ dθ s we expect. Similrly, if x = cos θ, we hve dx = sin θ dθ. The picture below illustrtes (the reltions of) these differentils geometriclly, nd my help convince you the vlidity of the hnd-wving rgument. The only thing tht s bit troublesome is d(e x ) = e x dx, but the stndrd pproch vi limits is not esy (nd often omitted) nywys. So tke tht for grnted. These re ALL you need for ll prcticl clcultions involving differentils, with the help of the following rules which re ll very strightforwrd (here u nd v re ny expressions in x, or ny quntities relted to x like y ws bove): 1. Sum rule: d(u + v) = du + dv. Product (or Leibniz s) rule: d(uv) = u dv + v du (gin du dv drops out becuse it s of higher order ). 3. The quotient rule is consequence of the product rule: so we get du = d( u v v) = u v dv + vd(u v ) d( u v du u dv ) = v v 4. Chin rule: just follow your nose. For exmple: d sin(x ) = cos(x ) d(x ) = cos(x ) x dx 5. Inverse functions nd implicit functions re esy (s shown lredy). Exmple 5: y = tn 1 (x), or x = tn y dx = d( sin y cos y ) cos y d sin y sin y d cos y = cos y = cos y dy + sin y dy cos y = (1 + tn y) dy = sec y dy

4 4 CALCULUS WITHOUT LIMITS (which, by the wy, gives nother useful rule d(tn y) = sec y dy.) If one wishes to see dy in terms of x nd dx on the other side, we shll note tht 1 + tn y = 1 + x nd obtin dy = dx 1 + x Note tht this clculus of differentils instntly pplies to multi-vrible functions with little extr efforts. For exmple, to differentite z = e xy dz = e xy d(xy) = e xy (x dy + y dx) = y e xy dx + x e xy dy which is often clled the totl differentil nd written in terms of prtil derivtives: dz = z z dx + x y dy Leibniz s nottion leds nicely to Crtn s clculus of differentil forms.. Integrls In ddition to tking the quotient of two differentils to get t meningful quntity (the derivtive), we cn lso dd up infinitely mny infinitesimls, nd this process is clled integrtion. Agin Leibniz provides us with the mgicl nottion: f(x) dx which mens, loosely speking, to um up n infinite collection of infinitesimls of the form f(x) dx, one for ech x between nd b. For the simplest cse when f(x) = 1 (constnt), we see tht the sum of ll the infinitesiml increments dx just ccumulte nd give the totl increment in x from x = to x = b, hence dx = b The generl problem of integrtion is very strightforwrd procedure, t lest in principle: find quntity y (i.e., find reltion y = F (x)), by whtever mens you cn, such tht dy = f(x) dx, so the sum of these infinitely mny dy s would give you the totl increment in the vrible y from the point when x = to x = b. In nottion, f(x) dx = y=f (b) y=f () dy = F (b) F () A common old typogrphy for s. See, e.g., the originl copy of the Declrtion of Independence.

5 CALCULUS WITHOUT LIMITS 5 which is precisely the Fundmentl Theorem of Clculus. For exmple: 1 0 dx 1 + x = x=1 x=0 d tn 1 x = tn 1 (1) tn 1 (0) = π 4 Of course the hrd prt is to find the right y, nd ll the integrtion techniques re just tricks to move the symbol d right next to, so they could cncel out in some sense. To illustrte it, consider 1 1 x dx = cos θ d cos θ = sin θ( sin θ dθ) nd with the help of trig identity sin θ = 1 cos(θ), we cn proceed 1 cos(θ) = dθ = 1 dθ + 1 cos(θ) d(θ) = 1 dθ In other words, the quntity d sin(θ) y = 1 θ + 1 sin(θ) with x = cos θ 4 stisfies dy = 1 x dx, s one cn redily check. If we wish to evlute the definite integrl, sy from x = 0 to x = t, we simply need to find the totl increment of y from the point when x = 0 to x = t, by keeping trck of the end vlues of θ. Alterntively, one my write y purely in terms of x: y = 1 cos 1 (x) sin θ cos θ = 1 cos 1 (x) + 1 x 1 x so tht t 0 1 x dx = ( 1 cos 1 (t) + 1 ) ( t 1 t 1 ) π + 0 = 1 sin 1 (t) + 1 t 1 t This result cn be red off from this picture: so the intermedite vrible θ does hve geometric mening.

6 6 CALCULUS WITHOUT LIMITS Such geometric considertions re often behind integrtion techniques. exmple, this picture For nother illustrtes the formul y dx = xy x dy which (if we used u nd v insted) is nothing but the formul for integrtion by prts. 3. Tylor series One importnt prt of the subject of clculus is the Tylor series, nd in the process of mking precise to wht types of functions, nd over which domins, it pplies, we ve mde it one of the hrdest topics for the students. I found tht n pproch through the (originl) Men Vlue Theorem is getting t the hert of the mtter very quickly. It seems to me tht the Men Vlue Theorem originlly ment for the ssertion f(x) dx = f(c) (b ) for some c between nd b. The vlue f(c) is rightfully the men vlue of the function over the intervl. Of course this theorem my fil if f(x) tkes jump, nd skips over its men vlue, but something slightly less thn continuity would suffice, nmely tht f(x) be the derivtive of some other function, nd it is this version tht hs tken over the nme of Men Vlue Theorem. [However, it hs become pprent tht being continuously differentible is fr more useful criterion thn simply being differentible, nd hs won the nottion C 1.] The generliztion for functions of severl vribles (sy x 1, x,...), over compct nd connected region D, is eqully intuitive: f(x 1, x,...) dx 1 dx = f(c) vol(d) D

7 CALCULUS WITHOUT LIMITS 7 for some c D. Now, let s strt with the Fundmentl Theorem of Clculus f(x) = f() + nd note tht we could do the sme for f (x 1 ) Substituting this in, we get f(x) = f() + f (x 1 ) = f () + x x x1 ( x1 f () + = f() + f ()(x ) + f (x 1 ) dx 1 f (x ) dx f (x ) dx ) dx 1 x x1 f (x )dx dx 1 The ltter integrl is over tringle (in the x 1 x -plne), so by the Men Vlue Theorem it is f (c) (x ) for some c [, x]. Therefore, f(x) = f() + f ()(x ) + f (c) (x ) The process clerly itertes, so we end up with Tylor s Theorem (with Lgrnge s reminder): r f (n) () f(x) = (x ) n + f (r+1) (c) (x )r+1 n! (r + 1)! n=0 for some c [, x], where the n! comes from the volume of the n-simplex. For nice functions such s e x, sin x, nd cos x, f (r+1) (c) in the reminder is bounded (s c [, x] nd r N vry, but x is fixed) tht the reminder tends to 0 s r, for ech fixed x R. For other functions such s tn 1 (x), this is true for x in some finite intervl (which turns out to be lwys of the type from R to + R, with or without the endpoints). In these cses we spek of the Tylor series of f(x) round x = : f(x) = n=0 f (n) () (x ) n n!