AP Calculus AB Cram Sheet

Transcription

1 ABCrmSheet.n AP Clcls AB Crm Sheet Definition of the Derivtive Fnction: f ' (x) = lim hø0 f Hx+hL - Å h ÅÅÅ Definition of Derivtive t Point: f H+hL- f HL f ' () = lim hø0 (note: the first efinition reslts in fnction, the secon efinition reslts in nmer. Also h f H+hL- f HL note tht the ifference qotient,, y itself, represents the verge rte of chnge of f from x = to x = + h) h Interprettions of the Derivtive: f ' () represents the instntneos rte of chnge of f t x =, the slope of the tngent line to the grph of f t x =, n the slope of the crve t x =. Derivtive Formls: (note: n k re constnts) x HkL = 0 x (k f(x))= k f ' (x) x H Ln = nh L n- x x [f(x) ± g(x)] = f ' (x) ± g ' (x) [f(x) g(x)] = f(x) g ' (x) + g(x) f ' (x) x I ÅÅÅÅ ghxl M = ghxl - g ' HxL ÅÅ ÅÅÅ Å HgHxLL 2 x x sin(f(x)) = cos (f(x)) f ' (x) cos(f(x)) = -sin(f(x)) f ' (x) x tn(f(x)) = sec2 H L ÿ x ln(f(x)) = ÅÅÅ ÿ x e = e ÿ x = ÿ ln ÿ x sin- = x cos- = - Å è!!!!!!!!!!!!!!!!!!! -H L 2 Å Å è!!!!!!!!!!!!!!!!!!! -H L 2 Å

2 ABCrmSheet.n 2 x tn- = x H f - HxLL t x = L'Hopitls's Rle: If lim xø lim xø ÅÅÅÅ Å +H L 2 = ÅÅÅÅ 0 ghxl 0 or f HL eqls ÅÅÅ = lim ghxl xø Å g ' HxL Å t x = n if lim xø Å exists then g ' HxL The sme rle pplies if yo get n ineterminte form ( 0 ÅÅÅÅ 0 or ) for lim xø ÅÅÅ s well. ghxl Slope; Criticl Points: Any c in the omin of f sch tht either f ' (c) = 0 or f ' (c) is nefine is clle criticl point or criticl vle of f. Tngents n Normls The eqtion of the tngent line to the crve y = f(x) t x = is y - f() = f ' () (x - ) The tngent line to grph cn e se to pproximte fnction vle t points very ner the point of tngency. This is known s locl liner pproximtions. Mke sre yo se º inste of = when yo pproximte fnction. The eqtion of the line norml(perpeniclr) to the crve y = f(x) t x = is y - f() = - Å Hx - L f ' HL Incresing n Decresing Fnctions A fnction y = f(x) is si to e incresing/ecresing on n intervl if its erivtive is positive/negtive on the intervl. Mximm, Minimm, n Inflection Points The crve y = f(x) hs locl (reltive) minimm t point where x = c if the first erivtive chnges signs from negtive to positive t c. The crve y = f(x) hs locl mximm t point where x = c if the first eivtive chnges signs from positive to negtive. The crve y = f(x) is si to e concve pwr on n intervl if the secon erivtive is positive on tht intervl. Note tht this wol men tht the first erivtive is incresing on tht intervl. The crve y = f(x) is si to e concve ownwr on n intervl if the secon erivtive is negtive on tht intervl. Note tht this wol men tht the first erivtive is ecresing on tht intervl. The point where the concvity of y = f(x) chnges is clle point of inflection. The crve y = f(x) hs glol (solte) minimm vle t x = c on [, ] if f(c) is less thn ll y vles on the intervl.

3 ABCrmSheet.n 3 Similrly, y = f(x) hs glol mximm vle t x = c on [, ] if f (c) is greter thn ll y vles on the intervl. The glol mximm or minimm vle will occr t criticl point or one of the enpoints. Relte Rtes: If severl vriles tht re fnctions of time t re relte y n eqtion (sch s the Pythgoren Theorem or other forml), we cn otin reltion involving their (time)rtes of chnge y ifferentiting with respect to t. Approximting Ares: It is lwys possile to pproximte the vle of efinite integrl, even when n integrn cnnot e expresse in terms of elementry fnctions. If f is nonnegtive on [, ], we interpret s the re one ove y y = f(x), elow y the x-xis, n verticlly y the lines x = n x =. The vle of the efinite integrl is then pproximte y iviing the re into n strips, pproximting the re of ech strip y rectngle or other geometric figre, then smming these pproximtions. For or iscssion we will ivie the intervl from to into n strips of eql with, Dx. The for methos we lerne this yer re liste elow. Left sm: n- i=0 f Ht i L Dt = f Ht 0 L Dt+ f Ht i L Dt+ f Ht 2 L Dt+ + f Ht n- L Dt, sing the vle of f t the left enpoint of ech sintervl. n Right sm: i= sintervl. n Mipoint sm: i=0 ech sintervl. f Ht i L Dt = f Ht L Dt+ f Ht 2 L Dt+ f Ht 3 L Dt+ + f Ht n L Dtsing the vle of f t the right enpoint of ech f H ÅÅ t i+t i+ L Dt = f H Å t 0+t L Dt + f H Å t +t 2 L Dt +ÿ ÿ ÿ+f H ÅÅÅÅ t n-+t n L Dtsing the vle of f t the mipoint of Trpezoil Rle: ÅÅÅÅ 2 H f Ht 0L + f Ht LL Dt + ÅÅÅÅ 2 H f Ht L + f Ht 2 LL Dt +ÿ ÿ ÿ+ ÅÅÅÅ 2 H f Ht n-l + f Ht n LL Dt. Note tht the trpezoil pproximtion is the verge of the left n right sm pproximtions. Antierivtives: The ntierivtive or inefinite integrl of fnction f(x) is fnction F(x) whose erivtive is f(x). Since the erivtive of constnt is zero, the ntierivtive of f(x) is not niqe; tht is, if F(x) is n integrl of f(x), then so is F(x) + C, where C is ny constnt. Rememer when yo re integrting fnction, f(x), yo re fining fmily of fnctions F(x) +C whose erivtives re f(x). Integrtion Formls: Ÿ kfhxl x = k Ÿ ghxld x = Ÿ Ÿ ghxl x Ÿ n = ÅÅÅ n+ n+ + C Ÿ ÅÅÅÅ = ln»» +C Ÿ cos = sin + C Ÿ sin =-cos + C

4 ABCrmSheet.n 4 Ÿ tn = ln» sec» +C Ÿ sec 2 = tn + C Ÿ e = e + C = ÅÅÅ ln + C Ÿ ÅÅÅ è!!!!!!!!!!!! = sin - ÅÅÅÅ C Ÿ ÅÅ = ÅÅÅÅ tn- ÅÅÅÅ + C The Fnmentl Theorems The First Fnmentl Theroem of Clcls sttes If f is continos on the close intervl [, ] n F ' = f, then, = FHL - FHL The Secon Fnmentl Theorem of Clcls Sttes If f is continos on [, ], then the fnction F(x) = x f HtL t hs erivtive t every point in [, ]. n F ' (x) = x x f HtL t = Definite Integrl Properties (in ition to the inefinite integrl properties). = 0 2. =- Ÿ 3. = c + Ÿ c Ares: If f(x) is positive for some vles of x on [, ] n negtive for others, then represents the cmltive sm of the signe res etween the grph of y = f(x) n the x-xis (where the res ove the x- xis re conte positively n the res elow the x-xis re conte negtively)

5 ABCrmSheet.n 5 Ths,»» x represents the ctl re etween the crve n the x-xis. The re etween the grphs of f(x) n g(x) where f(x) g(x) on [, ] is given - ghxld x Volmes: The volme of soli of revoltion (consisting of isks) is given y rightenpoint Volme = pÿ left enpoint HrisL 2 xhor yl The volme of soli of revoltion (consisting of wshers) is given y rightenpoint Volme = pÿ left risl 2 - Hinsie ris L 2 D xhor yl The volme of soli of known cross-sectionl res is given y rightenpoint Volme = Ÿ left enpoint Hcross sectionl rel xhor yl Arc Length: If the erivtive of fnction y = f(x) is continos on the intervl [, ], then the length s of the rc of the crve of y = f(x) from the point where x = to the point where x = is given y s = $%%%%%%%%%%%%%%%%% + I y "###################### x M2 x or + H L 2 x