Strategies for the AP Calculus Exam

Transcription

1 Strteges for the AP Clculus Em Strteges for the AP Clculus Em Strtegy : Kow Your Stuff Ths my seem ovous ut t ees to e metoe. No mout of cochg wll help you o the em f you o t kow the mterl. Here s lst of thgs tht you shoul hve workg kowlege of efore you eter the em. Whe we sy workg kowlege we me tht you shoul kow these thgs lke you kow your ow me. Plese e vse tht ths lst s ot ehustve. However t s lst of thgs tht you cot o wthout! Pre-clculus Ut crcle vlues ( 0) ( 0) (0 ) (0 ) y

2 Trgoometrc ettes Recprocl Iettes: csc( θ ) = s θ sec( θ ) = cos ( θ) cot( θ ) = t ( θ) Doule Agle Iettes s( θ ) = s( θ) cos( θ) cos( θ ) = cos ( θ) = s ( θ) Pythgore Iettes: s θ + cos θ = ( θ ) + = ( θ) t sec ( θ ) + = ( θ) cot csc Power Reucto Iettes cos( θ) s ( θ ) = + cos θ cos ( θ ) = Lmts Cotuty Cotuty t Pot f () s cotuous t = c f the followg three cotos re stsfe.. f () s efe t = c.. lm f ( ) ests. c lm f = f c. c L Hôptl s Rule f ( ) If f () g () re fferetle lm yels y of the etermte c g( ) forms 0 or 0 the f ( ) f ( ) lm = lm. c g( ) c g ( ) Dervtves Deftos of the Dervtve Dervtve s Fucto: f ( ) ( + ) h 0 f lm h f = h f ( ) f ( c) Dervtve t pot: f ( c) = lm c c Prmetrc Form of the Dervtve (BC) y y / t g ( t) If = f ( t) y = g( t) the = =. / t f ( t ) g t vertcl tget wheever f ( t) = 0 g ( t) 0 The grph of y hs horzotl tget wheever = 0 f ( t) 0 4 AP Clculus

3 Polr Form of the Dervtve (BC) r f rcos y = rs θ the If = ( θ ) = ( θ ) ( θ) cos( θ ) + f ( θ) s( θ) y y / θ f = =. / θ f θ s θ + f θ cos θ y The grph of r = f ( θ ) hs horzotl tget wheever 0 θ = vertcl tget wheever 0 θ = y 0 θ. θ=α s tget t the pole of the grph r = f θ f f ( α ) = 0 f ( α) 0. of Dervtves of Elemetry Fuctos. ( C ) = 0 =. = cos( ). s( ) =s( ) 4. cos( ) = sec ( ) 5. t( ) ( sec ) = sec( ) t( ) 6. ( csc ) =csc( ) cot( ) 7. =csc ( ) 8. cot( ) e = e = l. l( ). log ( ) = ( ). s ( ) 4. cos ( ) = l = = Strteges for the AP Clculus Em 5

4 = ( csc ) = ( t ) = + 5. sec ( ) cot ( ) Dfferetto Theorems = + Prouct Rule: Quotet Rule Ch Rule: ( f ( ) g( ) ) = f ( ) g( ) + f ( ) g ( ) f f g f g = g g ( ( )) = f g f g g Icresg Decresg Fuctos Reltve Etrem f () s cresg f f () > 0 ecresg f f () < 0. Let f (c) e efe. = c s crtcl umer of f () f f (c) = 0 or f f (c) s uefe. Frst Dervtve Test: Seco Dervtve Test: If f () chges from ecresg to cresg t = c the f () hs reltve mmum t the pot (c f (c)). If f () chges from cresg to ecresg t = c the f () hs reltve mmum t the pot (c f (c)). If f (c) > 0 the f () hs reltve mmum t the pot (c f (c)). If f (c) < 0 the f () hs reltve mmum t the pot (c f (c)). If f (c) = 0 the the Seco Dervtve Test s coclusve. Use the Frst Dervtve Test. Cocvty Pots of Iflecto The grph of f () s cocve up f f () > 0 cocve ow f f () < 0. Let f (c) e efe. 6 AP Clculus

5 The grph of f () my hve pot of flecto (c f (c)) f f (c) = 0 or f f (c) s uefe. The grph of f () oes hve pot of flecto t (c f (c)) f t chges cocvty t = c. Itermete Vlue Theorem If f () s cotuous o [] k s etwee f () f () the there ests umer c [] such tht f (c) = k. Rolle s Theorem If f s cotuous o [] fferetle o () f f () = f () the there ests umer c () such tht f (c) = 0. Me Vlue Theorem for Dervtves If f s cotuous o [] fferetle o () the there ests umer c f ( ) f () such tht f ( c) =. Itegrls Atervtves of Elemetry Fuctos. kf ( ) = k f ( ) f ± g = f ±g.. 0 = C 4. k = k + C 5. = + C ( ) 6. + = l + C 7. = + e e C 8. = + C > l 9. s cos = + C 0. cos s = + C. sec t = + C ( 0 ) Strteges for the AP Clculus Em 7

6 . csc cot = + C. sec t sec = + C 4. csc cot csc = + C = rct C 0 + > + 5. = rcs + C ( > 0) = rcsec C + 8. t l cos = + C 9. cot l s = + C 0. sec l sec t = + + C. csc l csc cot = + + C. f g g =f u u Itegrto y Prts (BC) Defto of Defte Itegrl uv = uv vu * * lm lm m 0 = = = = f f f 8 AP Clculus

7 Appromte Defte Itegrto Left Rem Sum: f lm f = Rght Rem Sum: Mpot Rem Sum: Trpezol Sum: f lm f = f f lm = + f ( ) f f f f ( 0)+ + + ( )+ The Fumetl Theorems of Clculus If f s cotuous o [] F s tervtve of f o [] the =. f F F If f s cotuous o ope tervl cotg the for ll the tervl =. f t t f Furthermore f () () re fferetle ( ) β f ( t ) t = f β ( ) β f α α α ( ) ( ). Me Vlue Theorem for Itegrls If f s cotuous o [] the there ests umer c [] such tht = ( ) f f c. f = f. The me vlue of f o [] s Strteges for the AP Clculus Em 9

8 Applctos of Itegrto Arc Legth: Dsk Metho: Wsher Metho: ( ) s = + f ( ) V = R (( ) ( ) ) V = R r Shell Metho: V = r h Prmetrc Arc Legth (BC): ( ) ( ) s = f t + g t t Polr Are (BC): β A = ( f ( θ) ) θ α The Shell Metho o loger eplctly ppers o the AP Clculus Em ut t coul t hurt to kow t. Sometmes t s eser to use th ether the Dsk or Wsher Methos. Moto D: vt = ( t) = = t v t t Avg Velocty = t D: r( t) = t yt v t t y t = t t y t = Avg Velocty r = t Dstce = v t t Dsplcemet = v t t ( ) Dstce = t + y t t Dsplcemet = t y t t 0 AP Clculus

9 Euler s Metho y f ( y) = y ( ) = + h = y. 0 0 y = y + f ( y ) h Seres Tests of Ifte Seres Test Seres Covergece Coto Dvergece Coto th-term = Noe lm 0 Geometrc Seres = 0 r r < S = r r Telescopg Seres ( + ) = lm = L S = L Noe p-seres = p p > p Altertg = Seres ( ) 0 lm = 0 + Noe Itegrl Root = f = f () cotuous postve ecresg = f ( ) coverges f verges lm < lm > Rto lm lm > + + < = Strteges for the AP Clculus Em

10 Test Drect Comprso Lmt Comprso = Seres Covergece Coto 0 < coverges = = lm = L > 0 = coverges Dvergece Coto 0 < verges = lm = L > 0 = verges Altertg Seres Error Bou If the ltertg seres ( ) ( 0 = s the N th prtl sum of the seres the S SN N +. Mclur Seres for Elemetry Fuctos ) coverges to sum S SN = ( ). e = = = 0!!! ( ) s( ) = = + + = 0 ( + )!! 5! 7!. 4 6 ( ) cos( ) = = + + = 0 ( )!! 4! 6! 4. = = = l( + ) = = + + = 4 Tylor Seres ( f ) ( c) If f s smooth t = c the f = c s the Tylor seres for f cetere = 0! N ( f ) ( c) t c PN = c s the N th egree polyoml for f cetere t c. = 0! Lgrge Error Bou If P N () s the N th + egree polyoml for f cetere t c f t M for ll t M + etwee c the f ( ) PN ( ) c. ( +! ) Strtegy : Tme Mgemet I ths prt of the ook we go over some test-tkg tps to help you mmze your score. The em s ve to two sectos. Secto I cots Multple-Choce Questos (MCQs) Secto II cots Free-Respose Questos (FRQs). We wll N = AP Clculus