1 Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you need to remember ll tht other clculus junk tht you ve probbly forgotten! Don t fret, though: This review should help you out! Unless otherwise noted, f, g, h : R R will denote rel-vlued functions on R. Limits nd Continuity Even though most of us cn t define it precisely, we ll sort of know wht it mens to tke the limit of the vlues of function f, sy, s x pproches : It mens we nlyze the vlues of f(x) s x gets close to, nd most of the time, we ccomplish this either by () plugging x = into f directly, or (b) doing lgebr on f (e.g. fctoring,...) until we cn plug in x =. We my not lwys get vlue this wy, but if we do get vlue (sy, L), we write lim f(x) = L x for the vlue of the limit. As it hppens, this pproch lmost lwys works for elementry stuff, nd lter (red: fter we know derivtives), we hve dditionl tools such s L Hôpitl s rule to help us evlute even more limits. Tht s fine nd good, nd in Clc 3, we ll lmost lwys use the techniques we lredy hve to evlute higher-dimensionl nlogues of this concept. Even so, here s the rel definition of limit, just so you cn sy you ve seen it before. Definition: The rel number L is sid to be the limit of the function f s x if for ll ε > 0, there exists δ > 0 such tht f(x) L < ε whenever x < δ. (1) In this cse, we write L = lim x f(x). The condition in (1) is esoteric to be certin, but s the figure 1 shows, it properly chrcterizes the notion most of us hve in our minds. Recll tht we lso hve notions of limits from the left / below nd limits from the right / bove which we shouldn t forget. Without beting ded horse, you should recll wht these one-sided limits men, intuitively, nd you should lwys remember tht L is the limit of f s x in the sense of (1) if nd only if L is the limit from the left nd from the right: lim x f(x) = L if nd only if lim f(x) = L nd lim x f(x) = L. x + As it hppens, limits distribute over sums, differences, sclr multipliction, division, nd powers; in ddition, common results such s the squeeze theorem lso hold. In short: Limits provide the frmework needed to llow us the privilege of tlking bout the vlues of functions close to bd spots like holes, jumps, nd verticl symptotes in wys tht we couldn t do in lgebr, for exmple. Of course, not ll functions hve bd spots like holes, jumps, nd verticl symptotes, nd s wy to differentite those from the rest, we hve specil word. 1
2 L+ε L-ε L -δ Figure 1 Even though f() L, the limit lim x f(x) does equl L becuse for every ε > 0 (no mtter how smll), there is n intervl (x δ, x + δ) on the x-xis which mps to the intervl (L ε, L + ε) on the y-xis. Note tht smller ε vlues (i.e. smller intervls on the y-xis) will require smller δ vlues (i.e. smller intervls on the x-xis). +δ Definition: The function f : R R is sid to be continuous t x = if: () f() exists (s rel number); (b) lim x f(x) exists (s rel number); nd (c) lim x f(x) = f(). Intuitively, we imgine tht function is continuous if nd only if we re ble to drw the entirety of its grph without hving to lift our pencil (so tht it hs no holes, no jumps, no verticl symptotes...), nd s figure 2 shows, ech of these conditions is necessry for our definition to mtch our intuition. Figure 2 Discontinuity my men violting ny of the three bove conditions: f() my fil to exist (left), lim x f(x) my fil to exist (middle), or both my exist nd stisfy lim x f(x) f() (right). Tking limits of continuous functions is esier overll becuse of property (c): To tke the limit s 2
3 x of function which is continuous t x =, it suffices to just plug in. In ddition, if f nd g re defined nd continuous t the point x = nd if c is ny constnt, the functions f ± g, fg, f/g (if g() 0), cf, f g, nd g f re ll continuous t x = (ssuming the compositions meet the pproprite domin/rnge restrictions). Significnt results such s the intermedite vlue theorem lso hold. Derivtives & Differentibility First, the definitions. Definition: The derivtive of function f t x = (if it exists) is the number In this cse, we sy tht f is differentible t x =. f f( + h) f() () = lim. (2) h 0 h Definition: The derivtive of function f (if it exists) is the function f (x) defined by f f(x + h) f(x) (x) = lim. (3) h 0 h Of course, equtions (2) nd (3) hve ll the quntittive nd qulittive properties we know nd love from Clc 1: f () in (2) describes the slope of the tngent line to the curve y = f(x) t x =, for exmple, nd if r(x) is differentible function describing the position of prticle (or whtever), the functions v(x) = r (x) nd (x) = v (x) = r (x) describe the velocity nd ccelertion of the prticle, respectively. As we know from Clc 1, not ll functions hve derivtives: In prticulr, function which fils to be continuous t x = will clerly not hve derivtive there, though some continuous functions my lso not be differentible everywhere (e.g. f(x) = x fils to hve derivtive t x = 0). In this wy, being differentible is stronger thn being continuous, nd in generl, we expect function f to be differentible t point x = if f is continuous t nd if the grph y = f(x) hs no shrp corners nd/or verticl tngents t x =. Sid differently: If f is differentible t x =, then when we zoom in towrd the point (, f()), the grph strightens out nd ppers more nd more like line. Figure 3 illustrtes this. By now, we re ll pros t derivtives: We know tht derivtives distribute over ddition/subtrction nd sclr multipliction, nd we know tht we cn use the product rule, the quotient rule, nd the chin rule to compute derivtives of products, quotients, nd compositions, respectively. Even so, however, we shouldn t forget how to compute f (x) using eqution (3), s tht will sometimes be required. Other Uses for Derivtives, Briefly: You shouldn t forget the extrs tht you re ble to do with derivtives, s mny of these will lso come up in vector-vlued functions. Some exmples: Liner pproximtions Implicit/logrithmic differentition Relted rtes (ugh!) & optimiztion 3
4 Figure 3 The function on the left is differentible t x =, s zooming in closer nd closer to (, f()) revels tht the grph becomes more nd more liner; the function on the right isn t, becuse no mtter how much you zoom in, it lwys hs point. L Hôpitl s rule for finding limits involving indeterminte forms (mny of which rise when investigting informtion bout horizontl symptotes, etc.) Men vlue theorem (!!!) Informtion bout where functions re incresing/decresing, concve up/down, etc. Suffice it to sy: If you re even little bit rusty t derivtives, you need to get unrusty yesterdy. Antiderivtives & Integrbility We ll (mostly) know the spiel: Definition: The ntiderivtive of function f (if it exists) is function F for which F (x) = f(x) (4) As it hppens, every function which is continuous on n intervl [, b] hs n ntiderivtive on the sme intervl. By the first prt of the fundmentl theorem of clculus, F (x) def = x f(t) dt (5) is one such ntiderivtive of f on the intervl [, b] (where we let x vry between nd b), nd s we find out lter, every ntiderivtive G of f is just verticl trnslte of (5): G(x) is n ntiderivtive of f(x) if nd only if G(x) = C + 4 x f(t) dt for some C.
5 One cn esily show tht such G stisfies (4), nd using the fct tht G (x) = f(x) = F (x), it follows tht G(x) F (x) is constnt (see Corollry 7 in section 3.2 of Stewrt). The result follows immeditely. In two-dimensions, the definite integrl b f(x) gives the (signed) re of the region bounded by f(x) nd the x-xis from x = to x = b. Recll tht the definite integrl is defined s the limit of Riemnn sums, nd by the second prt of the fundmentl theorem of clculus, b f(x) = F (b) F () where F is ny ntiderivtive of f. This shows in prticulr tht finding ntiderivtives/indefinite integrls is essentilly the sme s finding definite integrls. One of the most importnt properties of ntiderivtives is tht they re inverses of derivtives: ( d x ) x f(t) dt = f(x) nd f (t) dt = f(x) f(). Intuitively, this mens tht integrls cncel derivtives nd vice vers. It lso mens tht we cn go bckwrds in pplictions: For exmple, the position r(x), velocity v(x), nd ccelertion (x) of prticle (or whtever) is lso relted by the equtions r(x) = C 1 + v(x) nd v(x) = C 2 + (x). As we lerned in Clc 2, integrls lso hve lots of pplictions to rc length, surfce re, etc.; for the ske of brevity, tht won t be discussed here, but it should be noted tht those things will come up (sooner thn lter) in Clc 3. Formuls you hve to know! Here is brief summry of formuls you ll hve to know to succeed in this course. Limits: If c is constnt nd the limits lim x f(x) nd lim x g(x) both exist, then: lim x [f(x) ± g(x)] = lim x f(x) ± lim x g(x) lim x [cf(x)] = c lim x f(x) lim x [f(x)g(x)] = lim x f(x) lim x g(x) f(x) lim x g(x) = lim x f(x) lim x g(x) lim x [f(x)] n = [lim x f(x)] n (Squeeze theorem) If f(x) g(x) h(x) when x is ner nd if lim x f(x) = L = lim x h(x), then lim g(x) = L. x 5
6 Derivtives: If c is constnt nd if f nd g re both differentible, then: Also: (c) = 0 (xn ) = nx n 1 (cf) = cf (f ± g) = f ± g (fg) = fg + f g ( ) f = f g fg g g 2 (f(g(x))) = f (g(x)) g (x) (sin x) = cos x (cos x) = sin x (tn x) = sec2 x (csc x) = csc x cot x (sec x) = sec x tn x Integrls: (cot x) = csc2 x (sin 1 (x)) = 1 1 x 2 1 (cos 1 (x)) = 1 x 2 (tn 1 (x)) = x 2 (x ) = x ln (ex ) = e x (ln x) = 1 x (log b x) = 1 x ln b You should definitely know how to go bckwrds on the list of derivtives to rewrite in terms of integrls: For exmple, d (tn 1 (x)) = x x = 2 tn 1 (x) + C. You lso need to now how to do u-substitution, integrtion by prts, trig substitution, nd integrtion by prtil frctions, s well s how to integrte products of powers of trig functions (e.g. sin 2 x cos 3 x ). In ddition: If k is constnt nd if f nd g re both integrble, then: k = kx + C x n = xn+1 n+1 + C kf = k f(x) + C (f ± g) = f ± g + C Finlly: ln x = x ln x x + C tn x = ln sec x + C cot x = ln sin x + C sec x = ln sec x + tn x + C csc x = ln csc x cot x + C 6