ALGEBRA Popeties of Asoute Vue Fo e umes : 0, 0 + + Tige Iequity Popeties of Itege Epoets Ris Assume tt m e positive iteges, tt e oegtive, tt eomitos e ozeo. See Appeies B D fo gps fute isussio. + ( ) m m m m m m m m m m m Spei Pout Fomus ( A B) ( A+ B) A B + + A+ B A AB B + A B A AB B ( A+ B) A + AB+ AB + B A B A A B AB B + Ftoig Spei Biomis A B A B A B ( + ) + + + A B A B A AB B A + B A+ B A AB B Quti Fomu Te soutios of te equtio + + 0 e: ± 4 Diste Fomu Te iste etwee two poits, y Mipoit Fomu + y + y, Sope of Lie y y m ( ) (, y) is: ( ) + ( ) y y Hoizot ies y ve sope 0. Veti ies ve uefie sope. Pe Pepeiu Lies Give ie wit sope m: sope of pe ie m sope of pepeiu ie m Foms of Equtios of Lie St Fom: + y Sope-Iteept Fom: y m +, wee m is te sope is te y-iteept ( ) is poit o te ie Poit-Sope Fom: y y m sope, y, wee m is te Popeties of Logitms Let,,, y e positive e umes wit, et e y e ume. See Appei B fo gps fute isussio. og y og 0 og og og ( ) og ( y) og + og y og og og y og ( ) og og og og y e equivet y Cge of se fomu
GEOMETRY A e, C iumfeee, SA sufe e o te e, V voume Retge Cie Tige A w A p C p A w Peogm Tpezoi A A + Heo, s Fomu: ( )( ) A s s s s wee s + + Retgu Pism Spee Retgu Pymi V w SA + w + w V 4 π SA 4p V w w w V Rigt Cyie Rigt Ciu Cyie Coe Ae of Bse V p SA p + p V π SA π + π + Tigoometi Hypeoi Futios: Defiitios, Gps, Ietities See Appei C.
Eipse CONIC SECTIONS Po Hypeo Defiitio of Limit Let f e futio efie o ope itev otiig, eept possiy t itsef. We sy tt te imit of f s ppoes is L, wite im f L, if fo evey ume e > 0 tee is ume > 0 su tt f L < ε weeve stisfies 0 < < δ. Bsi Limit Lws Sum Lw: + + im f g im f im g Diffeee Lw: im f g im f im g Costt Mutipe Lw: im kf kim f Pout Lw: ( k, ) im f g im f im g Quotiet Lw: f im f im, im g g povie 0 im g Let, > 0 wit. Cete: ( k, ) Mjo is egt: Mio is egt: St fom of equtio:. ( ) y k + mjo is is oizot ( ) y k. + mjo is is veti Foi: o mjo is, uits wy fom te ete, wee - Dieti ( k, ) Let p 0. Vete: ( k, ) St fom of equtio: ( ). 4 p y k vetiy oiete : k, + p Dieti: y k p ( ). y k 4 p oizoty oiete : + p, k Dieti: p LIMITS Te Squeeze Teoem If g f otiig, eept possiy t itsef, if img im L, te im f L s we. fo i some ope itev Cotiuity t Poit Give futio f efie o ope itev otiig, we sy f is otiuous t if im f f. L Hôpit s Rue Suppose f g e iffeetie t poits of ope itev I otiig, tt g 0 fo I eept possiy t. Suppose fute tt eite im f 0 im g 0 o Te Vete ± ± im f im g. ( k, ) f f im im, g g Vete Let, > 0. Cete: ( k, ) St fom of equtio: ( ) ( y k). foi e ige oizoty Asymptotes: y k ± ( ) ( y k) ( ). foi e ige vetiy Asymptotes: y k ± ( ) Foi: uits wy fom te ete, wee + Veties: uits wy fom te ete ssumig te imit o te igt is e ume o o -.
DERIVATIVES Te Deivtive of Futio Te eivtive of f, eote f, is te futio wose vue t te poit is Eemety Diffeetitio Rues Costt Rue: k 0 Costt Mutipe Rue: kf k f ( ) Sum Rue: f + g f g + Diffeee Rue: f + f f im, 0 povie te imit eists. f g f g Deivtives of Epoeti Logitmi Futios e e Pout Rue: f g f g f + g ( ) Quotiet Rue: f g Powe Rue: Ci Rue: ( ) ( og ) g f g f g f ( g ) f g g Deivtives of Tigoometi Futios ( si) os ( s)s ot ( os)si ( se) se t ( t) se ( ot)s Deivtives of Ivese Tigoometi Futios si os ( t ) + s se ( ot ) + Deivtives of Hypeoi Futios ( si ) os ( s )s ot ( os ) si ( se )se t ( t ) se ( ot )s
Deivtives of Ivese Hypeoi Futios si + ( os ), > ( t ), < s + ( se ), 0< < ( ot ), > Te Deivtive Rue fo Ivese Futios If futio f is iffeetie o itev (, ), if f 0 fo (, ), te f ot eists is iffeetie o te imge of te itev (, ) ue f, eote s f ((, ) ) i te fomu eow. Fute, if,, te f f, f if f (, ), te f f f. ( ) Te Me Vue Teoem If f is otiuous o te ose itev, iffeetie o,,, fo wi [ ] te tee is t est oe poit f f f ( ). INTEGRATION Popeties of te Defiite Iteg Give te itege futios f g o te itev, [ ] y ostt k, te foowig popeties o.. f 0. f f. k k 4. kf ( ) k f ± ± 5. f g f g Te Fumet Teoem of Cuus Pt I Give otiuous futio f o itev I fie poit I, efie te futio F o I y F f () t t. Te F f ( ) fo I. Te Sustitutio Rue If u g te itev I, if f is otiuous o I, te is iffeetie futio wose ge is ( ) f g g f u u. Hee, if F is tieivtive of f o I, f ( g ) g ( ) F( g )+ C. +, ssumig 6. f f f e iteg eists o [, ], te 7. If f g f g. 8. If m mi f M m f, te ( ) m f M. Pt II If f is otiuous futio o te itev, if F is y tieivtive of f o,, [ ] [ ] te f F F. Itegtio y Pts Give iffeetie futios f g, f g f g g f. If we et u f v g, te u f v g esiy ememee iffeeti fom te equtio tkes o te moe uv uv vu.
SEQUENCES AND SERIES Summtio Fts Fomus Costt Rue fo Fiite Sums:,fo y ostt Costt Mutipe Rue fo Fiite Sums:,fo y ostt i i Sum/Diffeee Rue fo Fiite Sums: ( ± ) ± i i i i Sum of te Fist Positive Iteges: + i Sum of te Fist Sques: + i 6 ( + ) Sum of te Fist Cues: ( + ) i 4 Geometi Seies Fo geometi sequee Pti Sum: s Ifiite Sum:, if 0,, if < Biomi Seies Fo y e ume m - < <, { } wit ommo tio : m m ( + ) 0 mm ( ) mm ( ) ( m ) + m + +!! m( m) ( m + ) + +! wee m m m mm m 0,,,! m mm m +! fo + Tyo Seies Mui Seies Give futio f wit eivtives of oes tougout ope itev otiig, te powe seies f f f ( ) f + f ( )+ ( ) + (!!! 0 ) ( ) + is e te Tyo seies geete y f out. Te Tyo seies geete y f out 0 is so kow s te Mui seies geete y f.