Exponential and Logarithmic Functions (4.1, 4.2, 4.4, 4.6)
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1 WQ017 MAT16B Lecture : Mrch 8, 017 Aoucemets W -4p Wellm 115-4p Wellm 115 Q4 ue F T 3/1 10:30-1:30 FINAL Expoetil Logrithmic Fuctios (4.1, 4., 4.4, 4.6) Properties of Expoets Let b be positive rel umbers. 0 = 1 x y = x+y ( x ) y = xy (b) x = x b x x = 1 x Defiitio. (Nturl Expoetil Fuctio) The irrtiol umber e is efie to be The turl expoetil fuctio is f(x) = e x ( e = lim x 0 (1 + x) 1/x ( = lim 1 + x = exp(x) ) Defiitio. (Logrithmic Fuctios) The logrithm to the bse is efie by The turl logrithmic fuctio is efie s log x = b if oly if b = x. l x = b if oly if e b = x. Logrithms to other bses c be evlute usig the chge of bse formul: log x = l x l Iverse properties of Logrithms Expoets ecll tht iverse fuctios hve the property tht f(f 1 (x)) = x f 1 (f(x)) = x. ) l e x = x e l x = x Solve for x Expoetil Growth or Decy Defiitio. If y is positive qutity whose rte of chge with respect to time is proportiol to the qutity preset t y time t, the y is of the form y = Ce kt where C is the iitil vlue k is the costt of proportiolity. Expoetil growth is iicte by k > 0 expoetil ecy by k < 0. 1
2 WQ017 MAT16B Lecture : Mrch 8, 017 Properties of Logrithms l xy = l x + l y l x = l x Derivtives (4.3, 4.5, 8.6) Let u be ifferetible fuctio of x. x [ex ] = e x Let u be ifferetible fuctio of x. x [l x] = 1 x x [eu ] = e u u x x [l u] = 1 u u x Other Bses Differetitio x [x ] = (l ) x x [u ] = (l ) u u x ( ) 1 1 x [log x] = l x x [log u] = ( 1 l ) 1 u u x Implicit Differetitio Slopes Tget Lies, Locl extrem l Hôpitl s ule ule. If the limit of f(x) g(x) provie the limit exists or is ifiite. s x pproches c prouces ietermite form, the f(x) lim x c g(x) = lim f (x) x c g (x) Iefiite Itegrls (5.1, 5., 5.3, 6.1, 6.3, 8.5) = F (x) + C Itegrtio ules Costt ule: k x = k x + C Costt Multiple ule: k = k Sum/Differece ule: f(x) ± g(x) x = Note: 0 x = C ± g(x) x
3 WQ017 MAT16B Lecture : Mrch 8, 017 Techiques of Itegrtio u-substitutio Geerl Power ule: u u = u C, 1 1 u Geerl Log ule: x = l u + C u x Geerl Expoetil ule: e u u x x = eu + C Itegrtio by Prts: u v = uv v u Prtil Frctios Trigoometric Fuctios y Iitil Vlue Problem Give: Differetil Equtio: x = f (x) Iitil Coitio (Iitil Vlue) f(0) = y 0 Fi y = f(x). Defiite Itegrls (5.4, 5.5, 5.6, 5.7, 6.1) Fumetl Theorem of Clculus If f is oegtive cotiuous o the close itervl [, b], the = F (b) F () where F is y fuctio such tht F (x) = f(x) for ll x [, b]. Properties of Defiite Itegrls Let f, g be cotiuous o [, b]. k = k f(x) ± g(x) x = ± g(x) x c = + where < c < b c = 0 = If f is eve fuctio If f is o fuctio b = = 0 0 3
4 WQ017 MAT16B Lecture : Mrch 8, 017 Are of egio Defiitio. If f g re cotiuous o [, b] g(x) f(x) for ll x [, b], the the re of the regio boue by the grphs of f, g, x =, x = b is give by Volumes of solis of revolutio Are = f(x) g(x) x Defiitio (The Disk Metho). The volume of the soli forme by revolvig the regio boue by the grph of f the x-xis ( x b) bout the x-xis is Volume = π [f(x)] x Defiitio (The Wsher Metho). Let f, g be cotiuous, oegtive, g(x) f(x) for ll x o the close itervl [, b]. The volume of the soli forme by revolvig the regio boue by the regio boue by the grphs of f g ( x b) bout the x-xis is Volume = π [f(x)] [g(x)] x Here f(x) is clle the outer rius g(x) is the ier rius. Averge of fuctio i=1 f(x i ) logy of the verge of umbers i rge [, b]: Improper Itegrls (etext 6.4) is qutity kow s the verge of umbers. The cotiuous 1 b = lim b = = lim c + c Numericl Itegrtio etext 5.6, 6.3 Let f be cotiuous fuctio o itervl [, b]. Prtitio of the itervl ito equl prts is { = x 0, x 1,..., x = b}. Ech subitervl hs legth b. Mipoit ule b 1 ( ) xi + x i+1 f 4
5 WQ017 MAT16B Lecture : Mrch 8, 017 Trpezoil ule b 1 f(x i ) + f(x i+1 ) Simpso s ule (Here must be eve.) b 1 Probbility Theory (9.1, 9., 9.3) = b (f(x 0 + 1f(x 1 ) + + f(x 1 ) + f(x )) f(x i ) + 4f(x i+1 ) + f(x i+ ) 6 = b 3 (f(x 0) + 4f(x 1 )f(x ) + + 4f(x ) + f(x 1 ) + f(x )) Discrete Cotiuous Experimet Outcomes Smple Spce: S S is fiite (coutble) (ucoutbly) iifiite Evets: F om Vrible: X : S is fiite (coutble) itervl [, b], (, ) Probbility Distributio P : F [0, 1] Probbility esity fuctio: f : [0, 1] P (x) = 1 cotiuous, oegtive, = 1 Probbility: P (X I) = P (x) P (X I) = x I I Expecte Vlue: E[X] xp (x) x Vrice σ = vr(x) (x µ) P (x) (x µ) Str Devitio σ = vr(x) 5
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