A.P. Calculus Formulas Hanford High School, Richland, Washington revised 8/25/08

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1 A.P. Clls Formls Hfor High Shool, Rihl, Wshigto revise 8/5/08. floor ftio (ef) Gretest iteger tht is less th or eql to.. (grph) 3. eilig ftio (ef) Lest iteger tht is greter th or eql to. 4. (grph) b g h b + b b + b 6. b b + b + b 3 3 b g h of Hfor High Shool Clls Rihl, Wshigto

2 7. f ( ) (grph) p4 Chge of bse rle for logs: log l l 9. p579 Cirle forml: ( ) ( ) h + y k r 0. p580 Prbol forml: ( h) 4 p ( y k ). p583 Ellipse forml:. p585 Hyperbol forml: y + b b y + b b 3. p59 eetriity: e 4. si + os 5. + t se 6. + ot s 7. si ± v 8. os ± v 9. t ± v b g si os ± os si b g os os si si b g v v v v t ± tv t tv 0. si( ) si os of Hfor High Shool Clls Rihl, Wshigto

3 . os( ) os si. t( ) 3. si 4. os 5. t 6. si si v 7. os osv 8. si osv 9. os si v 30. lw of sies: 3. lw of osies: t t os + os os + os b g b g os v os + v b g b g os v + os + v b g b g si + v + si v b g b g si + v si v b si A si B si C + b bosc 3. re of trigle sig trig. Are si B 33. p3 prmeteritio of ellipse: 34. p60 lim si p7 lim si y + beomes os t, y bsi t b 0 3 of Hfor High Shool Clls Rihl, Wshigto

4 36. p83 Itermeite Vle Theorem If ftio is otios betwee b, the it tkes o every vle betwee f ( ) f ( b). 37. p99 efiitio of erivtive f + h f f ( ) lim ( ) ( ) h 0 h 38. p6 39. ( ) 0 b g 40. p7 b g p7 43. p9 44. p0 45. p4 46. p4 47. p p p p45 b g ± v ± v ( v ) v + v F HG I vk J v v v si os os si t se ot s se se t s s ot 4 of Hfor High Shool Clls Rihl, Wshigto

5 5. p5 slope of prmetrie rve: y y t t 5. p65 erivtive forml for iverses f f ( ) f 53. p p67 si os t ot p68 se - > s > 59. p69 ot ( ) 60. p69 se ( ) 6. p69 s ( ) π t ( ) F os H G I K J F si H G I K J 6. p7 63. p74 e l e 5 of Hfor High Shool Clls Rihl, Wshigto

6 64. p73 l 65. p88 Etreme Vle Theorem If f is otios over lose itervl, the f hs mimm miimm vle over tht itervl. 66. p96 Me Vle Theorem If f ( ) is ifferetible ftio over, b, (for erivtives) the t some poit betwee b : f ( b) f ( ) f ( ) b 67. p33 lieritio forml L( ) f ( ) + f ( ) ( ) 69. p85 k f ( ) ( ) k f 70. p85 f ( ) ± g( ) f ( ) g( ) 7. p88 Me Vle Theorem If f is otios o [, b ], the t some (for efiite itegrls) poit i [, b ], ( ) ( ) ± b f f b 7. p94 First fmetl theorem: f ( t) t f ( ) h 73. p307 Trpeoil Rle: T y 0 + y + y y + y b g p p33 si os p33 os si p33 se t + 6 of Hfor High Shool Clls Rihl, Wshigto

7 80. p33 s ot + 8. p33 se t se + 8. p33 s ot s p33 l p33 e e p33 l 86. t l os ot l si se l se + t s l s + ot rsi + + rt + rse p34 Itegrtio by prts (forml): v v v 94. p34 orer for hoosig i LIPET logs, iverse trig., polyomil, itegrtio by prts: epoetil, trig. y y e 95. p35 epoetil hge: 0 kt 96. p353 hlf-life 7 of l k Hfor High Shool Clls Rihl, Wshigto

8 97. otios ompo iterest: A( t) A e P 98. p365 logistis ifferetil eqtio: kp ( M P ) t o rt 99. p367 logistis growth moel M P + Ae ( Mk ) t b y 00. p409 srfe re bot is (Crtesi): S π y + 0. p43 legth of rve (Crtesi): b y L p450 lim l lim 05. lim 06. lim 07. lim 0 < h F I HG K J + e 08. lim! 09. k k 0 ( +) 0. k k ( + )( + ) 6 8 of Hfor High Shool Clls Rihl, Wshigto

9 . k k 3 F HG ( + ) I K J. prtil sm of geometri series: S r r h 3. p475 Wht series? r geometri, overges to r if r < f ( ) f ( ) 4. p487 Mlri Series: P( ) f ( ) + f ( ) ! 3! f ( ) 5. p489 Tylor Series: P ( ) f ( ) + f ( )( ) + +! f ( ) 3 b g ! b g 6. p49 Mlri Series for p49 Mlri Series for p49 Mlri Series for e 3 4 e ! 3! 4! p49 Mlri Series for si si ! 5! 7! p49 Mlri Series for os os ! 4! 6! 3 4. p49 Mlri Series for l( + ) l( + ) p49 Mlri Series for t ( ) : t ( ) p496 Lgrge form of remier ( ) ( + f ) ( ) ( ) ( + )! R + 9 of Hfor High Shool Clls Rihl, Wshigto

10 4. Tylor s Ieqlity ( ) M R! ( + ) + 5. p498 Wht series? reiprol of ftorils, overges to e 0! + 6. p50 Wht series? b b b g telesopig series, overges to b lim b + 7. p54 Wht series? p series, overges if p > p 8. p54 Wht series? hrmoi, iverges 9. p57 Wht series? b g + ltertig hrmoi, overges 30. p53 eriv. of prmetrie rve: y y t t 3. p533 legth of rve (prmetri): L b F H G I K J F + H G I K J b F y 3. srfe re (prmetri): S y t H G I t K J F + H G I π t K J 33. positio vetor (str form): r ( t) f ( t) + g ( t) + h( t) 34. p54 spee from veloity vetor: spee vbtg 35. p54 iretio from veloity vetor: iretio t y t t i j k veloity vetor spee 36. p55 polr to Crtesi: r osθ, y r siθ b g v t v t r siθ r osθ 38. p55 slope of polr grph: slope t ( r, θ) + r osθ r siθ 0 of Hfor High Shool Clls Rihl, Wshigto

11 39. slope of polr grph t origi: slope tθ 40. p553 re isie polr rve: β A r α α 4. legth of rve (polr): L r θ r θ F β + H G I K J β 4. srfe re (polr): S πr si θ r α θ r θ + F H G I K J θ of Hfor High Shool Clls Rihl, Wshigto

A.P. Calculus Formulas. 1. floor function (def) Greatest integer that is less than or equal to x.

A.P. Calculus Formulas. 1. floor function (def) Greatest integer that is less than or equal to x. A.P. Clls Formls. floor ftio (ef) Gretest iteger tht is less th or eql to.. (grph). eilig ftio (ef) Lest iteger tht is greter th or eql to. 4. (grph) 5. 6. g h g h Pge 7. f ( ) (grph) - - - - - - 8. Chge