Area, Volume, Rotations, Newton s Method

Transcription

1 Are, Volume, Rottio, Newto Method Are etwee curve d the i A ( ) d Are etwee curve d the y i A ( y) yd yc Are etwee curve A ( ) g( ) d where ( ) i the "top" d g( ) i the "ottom" yd Are etwee curve A ( y) g( y) where ( y) i the "right" d g( y) i the "le" yc R ) Cylidricl Dic Method, Rottio Volume V ( For ll rottio, " R" i lwy the uctio rthet rom the i o rottio, " r" i the cloet. d yd Rottio etwee curve V R( ) r( ) d or V R( y) r( y) R r( ) Rottio roud horizotl lie V ( ) L L yd yc y L L Rottio roud verticl lie V R( ) r( y) yc Cro Sectiol Volume Newto' Method or pproimtig A V Deiite itegrtio produce " iged " re re. Whe itegrtig i poitive directio ( or ), regio ove the i will hve poitive iged re, regio elow the i will hv e egtive iged re. Whe itegrtig i egtive directio or, ll ig re the oppoite. Alo, or A y i the ormul or the re o the cro ectio d 0. yd A( ) d or V A( y) yc d zero o uctio ( itercept) ' For the grph to the le, the ollowig re true. d 3 d 6 d 3 d 3 d 3 d 4 The more itertio (cycle), the more ccurte the pproimtio.

2 Whe ' 0, i icreig. Whe '( ) 0, i decreig. Whe ' i icreig, i cocve up Whe '( ) i decreig, i CD Whe "( ) 0, i cocve up. Whe "( ) 0, i cocve dow. Grph o, Etrem, Sig Chrt, Cocvity ' ' Whe "( ) 0, i icreig. Whe "( ) 0, i decreig. The t Derivtive tet or etrem. I '( ) chge rom poitive to egtive, the h m I I '( ) chge rom egtive to poitive, the h mi '( ) doe ot chge ig, ut c io i trictly mootoic c c c The d Derivtive tet or etrem or t c. I ' c 0 d " c 0 the h miimum t c. I ' c 0 d " c 0 the h mimum t c. I ' I ' 0 d " 0, the we kow othig. 0 d ' c eit the NOT etrem t c. etrem etreme poit o grph, ll mimum or miimum glol (olute) mimum the highet poit (check edpoit) glol (olute) miimum the lowet poit (check edpoit) locl (reltive) mimum poit higher th eighor locl (reltive) miimum poit lower th eighor O grph o, i : 0 the i ove i 0 the i elow i ' 0 the i icreig ' 0 the i decreig " 0 the i cocve up " 0 the i cocve dow d FTC I g( ) ( t) the g '( ) ( u) u ' u C 5 g( ) ( t) 8 g '( ) (5 ) 0

3 IVT, MVT, Limit, Cotiuity, Dieretiility Itermedite Vlue Theorem (IVT) I i cotiuou o [, ] d ( ) ( ) the there eit poit c i (, ) uch tht ( ) () c ( ) Me Vlue Theorem (MVT) I i dieretile d cotiou o [, ] the there eit poit c i (, ) uch tht '( c) ( ) ( ). lim vlue o you pproch rom the right lim the vlue o you pproch rom the le I lim lim the lim eit I m lim li the i cotiuou t I lim lim the lim DNE Rolle' Theorem (Icluded i MVT) I i dieretile d cotiou o [, ] d ( ) ( ) the there eit poit c i (, ) uch th t '( c) 0. The Limit Deiitio o the Derivtive ' h 0 lim h h The t derivtive diguied limit h lim me ' h0 h Dieretile le to tke the derivtive, o hrp tur, verticl ymptote, or hole Cotiuou o rek i the grph, you c drw the grph without liig your pecil L ' Hopitl ' Rule Give o (,], where c Iterior Cotiuity: i cotiuou t c i d oly i lim c c 0 ' I lim or The lim lim g 0 g g ' Edpoit Cotiuity i deied y tkig oe ided limit. would e cotiou t ut ot o (, ] I verticl ymptote t c the lim,, or DNE ote : or the AP tet, put DNE Type o Dicotiuitie: oremovle Verticl Aymptote e. 3, VA t 3 oremovle piecewie dicoect, 4 e., dicoect t 4 5, 4 removle ctor d ccel e., removle t c

4 Diclimer: the world i ot meured ecluively i eet d ecod, ue commo ee. Uit d Clculu t poitio vt velocity t ccelertio d me multiply y me multiply y Limit ( d ) re time i the uctio i i term o time t t t t dv rte o chge o Volume with repect to time da rte o chge o Are with repect to time dr rte o chge o rdiu More with uit with repect to time '( t) v( t) d rte o chge o ge l rdi "( t) v '( t) ( t) with repect to time More with uit More with uit d t t vt C vt t C d v t t t v v vt Speed i Velocity without directio i the vg. ditce, i eet, rom your iitil poitio. t i the vg. poitio, i, etwee time t d t. i the chge i poitio, i eet, rom time t to t. v t v t i the poitio, i eet, t time t. i the totl ditce trveled, i eet, rom time t to t. vt i the vg. velocity, i, etwee time t d t. i the chge i velocity, i, rom time t to t. t i the velocity, i, t time t. t i the vg. peed, i, etwee time t d t. i the vg. ccelertio, i Poitio, Velocity, Accelertio, d Uit, rom time t to t. 3 t peed i icreig i: t d t t hre the me ig t e. 0 d t 0 e. v 0 d peed i decreig i: d ped e e. 0 d t e. peed i cott i 0 hve oppoite ig: 0 d t peed i 0 i vt 0 t v v

5 Nottio d Termiology Commo Domi Retrictio A "geerl olutio" h " C " i the olutio. you c't divide y zero For "prticulr olutio" you mut olve or C. 5 e. y, " y " me tte your wer eplicitly i term o. you c't tke the eve root o egtive i term o y e. y 3, 3 or 3, i term o z y z you c oly tke the log o poitive z "Eplicit" me your wer h the orm y e. y l, or, "Implicit" me tht your wer doe ot eed the orm y d me y or y ' or ' Eplicit : y 5 3 Implicit : 5 y 3 0 d d d y d A "True Sttemet" implie tht poit ll me or y" or " d d d o the curve, e. 4 4, 0 0, 3 3 y m lope itercept A "Fle Sttemet" implie tht poit doe y y m( ) poit lope ot ll o the curve, e. 5 4, 0 3, '( ) y ' m lope o the tget lie To write the equtio o tget lie to uctio, d you eed poit (, ( ) ) d the lope '( ) lope o the tget lie = itteou rte o chge the plug ito poit lope: lope o the ect lie = vg. rte o chge y y m or y ( ) '( ) Sect Lie lie tht iterect poit o curve Sect lie pproimtio id the equtio o the ect To write the equtio o ect lie to uctio, lie d plug i the give vlue you eed poit (, ( )) d (, ( )). Firt id the lope o the lie etwee the poit d Locl Lier Approimtio = Tget Lie Approimtio Fid the equtio o tget lie d plug i the - vlue give. the plug ito poit lope: m y ( ) m I " 0 the loclly: A tget lie "overpproimte" A ect lie "uderpproimte" Averge Vlue o o [, ] ( ) d Averge Rte o chge o o [, ] lope o the ect lie ( ) ( ) y y A horizotl lie h lope = 0 A verticl lie h lope = udeied, Prllel lie hve the me lope. C 0 Perpediculr lie hve oppoite reciprocl lope. yitercept hppe whe 0 itercept hppe whe 0 I " 0 the loclly: A tget lie "uderpproimte" A ect lie "overpproimte"