BC Calculus Path to a Five Problems

Transcription

1 BC Clculus Pth to Five Problems # Topic Completed U -Substitutio Rule Itegrtio by Prts 3 Prtil Frctios 4 Improper Itegrls 5 Arc Legth 6 Euler s Method 7 Logistic Growth 8 Vectors & Prmetrics 9 Polr Grphig Bsics 0 Slopes of Polr Curves Are of Polr Regios Polr Grphs d Motio 3 Power Series 4 Tylor Polyomils 5 Tylor Series 6 Algebric Mipultio of Series 7 Clculus Mipultio of Series 8 Geometric Series 9 th Term Test 0 Altertig Series Test Itegrl Test p-series Test 3 Rtio Test 4 Direct Compriso Test 5 Limit Compriso Test 6 Rdius d Itervl of Covergece 7 Altertig Series Remider 8 Lgrge Error Boud

2 PTFs #BC 0 U-Substitutio Rule. Let u ier fuctio.. Fid du, the solve for. 3. Substitute u & du ito the itegrd (it should ow fit oe of the itegrtio rules). 4. Itegrte. 5. Substitute the ier fuctio bck for u. x x x 8. Itegrte Itegrte e t x cos x. Itegrte si 3xcos3 x 5. Itegrte x e e x 3. Itegrte e 3x dy 6. If cos( x), the y?

3 PTFs #BC 0 Itegrtio by Prts u dv uv v du. Choose u by usig LIPET s guide.. Differetite u to fid du. 3. Itegrte dv to fid v. 4. My eed to repet the process if the ew itegrl is still ot possible. 5. If u is d degree polyomil or higher, use the tbulr method to solve.. x xe 4. x si x. xsec x 5. t ( ) x 3. l x

4 PTFs #BC 03 Prtil Frctios Use prtil frctios whe the umertor of the rtiol fuctio is NOT multiple of the derivtive of the deomitor. 6. Fctor the deomitor. 7. Decompose the frctio. 8. Itegrte. *The frctio hs to be bottom hevy, if ot, use log divisio to fid the remider. Itegrte the quotiet s usul d the use prtil frctios o the remider.. 5x 3. x x3 3 3 x x

5 PTFs #BC 04 Improper Itegrls # b b # f ( x) lim f ( x) # # # # f ( x) lim f ( x) f ( x) lim f ( x) lim f ( x) If f( x ) hs discotiuity t, the If f( x ) hs discotiuity t, the # # b b b # b # f ( x) lim f ( x) f ( x) lim f ( x) If f( x ) hs discotiuity t, which is betwee the itervl, the brek up the itervl ito two itervls d evlute usig limits. Evlute ech itegrl.. x x x e. 5 x x

6 PTFs #BC 05 Arc Legth Let the fuctio give by f( x ) represet smooth curve o the itervl b,. The rc legth of f betwee d b is: b s f '( x). Set up itegrl to represet the 3 legth of y x o [0, 5]. Use your clcultor to fid the legth of the curve. 3. The legth of the curve y l sec x from x 0 to x b my be represeted by which of the followig itegrls? () (b) b 0 b 0 sec x sec x b (c) sec x t 0 x. Set up itegrl to represet the x x y e e o [0, 3]. Fid the legth without clcultor. legth of the grph of 4. Usig your clcultor, fid the legth 3 of the curve y x from (,) to (4,8).

7 PTFs #BC 06 Euler s Method. Strt t the give poit o the grph.. Clculte the slope t this poit by usig the differetil equtio. 3. For give vlue of x, clculte the vlue of 4. Add x to x d y to y to get ew poit. 5. Repet the process util you rech your desired x vlue. y, usig the equtio dy y x.. For dy x, use Euler s Method y strtig t f (0) 3 with three steps to pproximte f (.5).. Let dy cos x. If (, ) is o the si y grph of y, wht is the vlue of y whe x.0 usig Euler s method? Let x b c d e..005

8 dp If y is differetible fuctio of t such tht kp L P dt, the L P Lkt Ce lim P( t) L (L is the crryig cpcity.) t PTFs #BC 07 Logistic Growth dp is t its mximum (the rte of growth is icresig the fstest) whe the dt fuctio reches hlf its crryig cpcity, L. (this is lso the poit of iflectio o Pt ()). The popultio, P of species stisfies the logistic differetil dp P equtio P dt 5000, where P(0) 3000 d t is the time i yers. Wht is lim Pt ( )? t. A rumor spreds t the rte dp P P where P is the portio dt of the popultio tht hs herd the rumor t time t. Wht portio of the popultio hs herd the rumor whe it is spredig the fstest?

9 PTFs #BC 08 Vectors & Prmetrics Positio Vector: r( t) x( t), y( t) Velocity Vector: v( t) x'( t), y'( t) Accelertio Vector: ( t) x''( t), y''( t) Speed (mgitude of velocity vector): speed x'( t) y '( t) Totl distce trveled (rc legth): distce '( ) '( ) Slope of the curve: Secod derivtive: dy y '( t) x '( t) d dy dy '( ) '( ) d y dt x t x t ' t t x t y t dt. The positio of prticle is give by x si t y cos t o the itervl d 0,.. Fid the velocity vector. b. For wht vlues of t is the prticle t rest?. The velocity of roller coster t time t secods c be modeled prmetriclly by x'( t) 0 4cos t d y '( t) 0 t si t cos t for 0 t 0 secods. At time t 0 the rollercoster is meter high.. Fid the time t t which the cr is t its mx height. c. Write the rectgulr equtio of the pth of the prticle i x d y oly. b. Fid the speed, i m/sec, of the cr t this time.

10 PTFs #BC 09 Polr Grphig Bsics Polr-Rectgulr Coversio Formuls: x rcos r x y y rsi t y x. Grph the poit P, o coordite ple. Covert P to polr coordite d grph the polr coordite o differet set of xes. 3. Show tht the polr equtio r c be writte i cos si rectgulr form s the equtio x y.. Covert r 9si to rectgulr equtio. 4. A curve drw i the xy ple is described by the equtio i polr r si for coordites 0. Fid the gle tht correspods to the poit o the curve with x coordite -.

11 PTFs #BC 0 Slopes of Polr Curves The slope of polr curve is still the slope of the tget lie: dy d r si dy d d d r cos d d dy Horizotl tgets whe 0 d. Verticl tgets whe 0 d. dy No coclusio if both 0 d d d 0.. Fid the poits where the horizotl d verticl tget lies occur for r si for 0.. Cosider the polr curve r si 3 for 0. Fid the slope of the curve t the poit where. 4

12 PTFs #BC Are of Polr Regios The re of polr regio is: A r d. Fid the re iside the curve of r 3cos 3 for Fid the re of the regio tht is iside r 4 4cos d outside of r 6.. The re of the regio bouded by the polr grph of r 3 cos is give by the itegrl: cos d b. 3 cos d c. 3 cos d 0 d. 3 cos d e. 3 cos d 0

13 PTFs #BC Polr Grphs d Motio Vrible d its directio of movemet: R = rdius dr dt mesures movemet goig towrd/wy from the origi X = horizotl distce dt mesures movemet goig towrd/wy from y-xis Y = verticl distce dy dt mesures movemet goig towrd/wy from x-xis ***You must check both the loctio d the directio of movemet!*** Problems work just like vectors, just with trig fuctios. All vector formuls pply. A prticle moves log the polr curve r 4 si so tht t time t secods, t. (Clc.) c) Fid the positio vector i terms of t. Fid the velocity vector t time t.5. ) Fid the time t i the itervl t for which the x-coordite of the prticle s positio is -. b) Is the prticle movig towrds or wy from the y-xis t the time foud i prt ()? Show the lysis tht leds to your swer.

14 PTFs #BC 3 Power Series A Power Series c be used to represet fuctio, but oly o specified domi or itervl. It ivolves vrible, ormlly x, isted of just costts. A Power series cetered t x 0 is of the form x 0 x x.... A Power series cetered t x c is of the form x c 0 x c x c.... Fid the first four o-zero terms d the geerl terms of the series.. cetered t x 0 x x cetered t x. x cetered t x 0

15 PTFs #BC 4 Tylor Polyomils The Tylor Polyomil of order for f cetered t x c is give by: f ''( c) f ( c) P ( x) f ( c) f '( c) x c x c... x c!! The Mcluri Polyomil of order for f cetered t x 0 is give by: f ''(0) f (0) P ( x) f (0) f '(0) x x... x!! To costruct Tylor polyomil or series:. Evlute f( x ) d it s first derivtives t x c.. Plug ito oe of the formuls bove to write the series/polyomil. 3. If it is series, mke sure to iclude the... d to write the geerl term.. Fid the first four terms of the Tylor polyomil for f( x) x bout x. 3. Fid the fourth-degree Tylor polyomil for f ( x) l( x) bout x d use it to pproximte the vlue of l(.).. Let f be fuctio d f () 3, f '() 5, f ''() 3 d f '''() 8. Write third degree Tylor polyomil for f bout x d use it to pproximte f.5.

16 The Tylor Series geerted by f t x c is defied by: ''( ) ( ) f c f c f ( c) f '( c) x c x c... x c!! The Mcluri Series geerted by f t x c is defied by: f ''(0) f (0) f (0) f '(0) x x... x!! 0 0 PTFs #BC 5 Tylor Series (*This is just Tylor series tht is cetered t x 0!) To costruct Tylor polyomil or series: 4. Evlute f( x ) d it s first derivtives t x c. 5. Plug ito oe of the formuls bove to write the series/polyomil. 6. If it is series, mke sure to iclude the... d to write the geerl term.. The Tylor series bout x 5 for certi fuctio f coverges to f( x ) for ll x i the itervl of covergece of f, d f (5) 0. The th derivtive of f t x 5 is give by f 5!. Write the first four terms d the geerl term for the Tylor series bout x 5.. The Tylor series bout x 0 for f coverges to f( x ) for ll x i the itervl of covergece. The th derivtive of f t x 0 is give by f 0 5! for. The grph of f hs horizotl tget lie t x 0, d f (0) 6. Write the first four terms of the Mcluri series for f bout x 0.

17 PTFs #BC 6 Algebric Mipultio of Series Power series tht you must kow: 3 x x x x e x...! 3!! x x x x x si x x... 3! 5!! x cos x...! 4!! x x x x x You my mipulte the series bove to fid ew series (s log s you re cetered t x 0 ) by:. Substitutig ito kow series.. Multiplyig/dividig by costt d/or vrible. 3. Addig/subtrctig two kow series.. Let f be the fuctio give by f ( x) 6 x 3 e. Fid the first four ozero terms d the geerl term for the Tylor series for f bout x Fid the first three o-zero terms d the geerl term of the Tylor polyomil for gx ( ) cetered t x 0. cos x x. Write power series d the geerl term for si( x ).

18 PTFs #BC 7 Clculus Mipultio of Series You my mipulte series to fid ew series (s log s you re cetered t x 0 ) by:. Differetitig kow series.. Itegrtig kow series.. Let 3 4 P( x) 7 3 x 4 5 x 4 x 4 6 x 4 be the fourth-degree Tylor polyomil for the fuctio f bout 4. Assume f hs derivtives for ll orders of ll rel umbers.. Write the secod degree Tylor polyomil for f '( x ) bout 4 d use it to pproximte f '(4.3).. The Mcluri series for the fuctio f is give by 3 x 4x 8x f ( x) x... o 3 0 its itervl of covergece. Fid the first four terms d the geerl term for the Mcluri series for f '( x ). b. Write fourth-degree Tylor polyomil for x g( x) f ( t) dt For f( x) x 5 5. Fid the power series for f '( x ). f ( x). b. Fid the power series for

19 PTFs #BC 8 Geometric Series Geometric Series: 3... r r r r o A geometric series with commo rtio, r will: Coverge if r (Sum: S ) r Diverge if r For #-3, determie if ech sum diverges or coverges. If it coverges, fid the sum.. 3 Fid the sum. 4. i i 3 (A) 3 3 (B) (C) 3 3 (D) 33 (E)

20 PTFs #BC 9 th Term Test If lim 0, the the series diverges. This test is test for divergece oly! It c ever be used to prove covergece. Determie if ech series diverges or if the th term test is icoclusive cos

21 PTFs #BC 0 Altertig Series Test A series of the form or is ltertig series whose terms lterte pos/eg or eg/pos d where 0. The series will coverge if both coditios re met: lim 0 (the limit of the terms is goig to zero) (the terms re decresig i mgitude) Determie if ech series coverge or diverge.. 3. Which of the followig series coverge? (You my use y of the tests to decide.) I. II. III. cos (A) Noe (B) II (C) III (D) I d II (E) I d III. 4 3

22 PTFs #BC Itegrl Test If f is cotiuous, positive d decresig for x d f ( x) the f ( x ) either both coverge or both diverge. d. Stte tht the fuctio f ( x) is cotiuous d positive d the use f '( x ) to show tht the fuctio is decresig. 3. Evlute f ( x ). If If f ( x ) diverges, the f ( x ) coverges, the diverges s well. coverges s well. 4. The sum, S, c be pproximted by fidig the th prtil sum, S d the fidig the remider, R. S S R where 0 R f ( x). Determie if the followig series coverge or diverge.. e 3. Let f be positive, cotiuous, decresig fuctio such tht f ( ). If coverges to k, which of the followig must be true? (A) lim k (B) f ( x) k. l (C) (D) f ( ) diverges f ( ) coverges (E) f ( x) k

23 PTFs #BC p-series Test A series of the form... is p-series, where p is positive p p p p 3 costt. The series coverges if p. The series diverges if p. (Remember the grph of f( x) d how it diverged becuse it ws too fr wy x from the x- d y-xis. If the expoet is or smller, it will pull the grph further wy d the series will ot diverge. If the expoet is greter th, the terms re smller d the grph is much closer to the xes; these series coverge.) Determie if ech series coverge or diverge

24 PTFs #BC 3 Rtio Test Rtio test is useful for series tht coverge rpidly such s expoetil fuctios d fctorils. A series coverges if lim. A series diverges if lim (or ). The rtio test is icoclusive if lim. *This test is essetilly fidig the rtio of two djcet terms. It is similr to fidig the rtio of geometric series d so hs the sme covergece/divergece specifictios. Determie if the followig series coverge or diverge !

25 PTFs #BC 4 Direct Compriso Test Direct compriso test is useful to compre series tht re similr to kow coverget or diverget series. 5. Choose pproprite pret, b, series (usully geometric or p-series). 6. Determie whether the pret, b, series coverges or diverges. 7. Show tht your series fits oe of the two sttemets hold true: If 0 b d coverget series coverges.) If 0 b d diverget series diverges.) b coverges, the b diverges, the coverges. (Less th diverges. (Greter th Determie if the followig series coverge or diverge ! 4.

26 PTFs #BC 5 Limit Compriso Test This test compres messy lgebric series i questio,, with kow similr coverget or diverget series, b, by tkig the quotiet of the two series. 8. Choose pproprite pret, b, series (usully geometric or p-series). 9. Determie whether the pret, b, series coverges or diverges. 0. Set up lim d evlute to fid the limit, L. b. Show tht your limit L, fits oe of the three sttemets below: If L is fiite d positive, the diverge. If L 0 d If L d b d b coverges, the coverges. b diverges, the diverges. both coverge or both Determie if the followig series coverge or diverge si 4. 3

27 PTFs #BC 6 Rdius d Itervl of Covergece We cll R the rdius of covergece, which is how fr wy from the ceter the series will coverge. The covergece of power series hs oe of three possibilities:. The series coverges oly t x c so R 0.. The series coverges for ll vlues of x so R. 3. There exists R 0 such tht the series coverges for x c R d diverges for x c R. To fid the rdius d itervl of covergece: 7. Use the Geometric Series or the Rtio test to set up bsolute vlue iequlity. 8. Solve the bsolute vlue iequlity this will give you your rdius of covergece. 9. Test the edpoits to determie covergece d write your itervl of covergece. Fid the rdius d itervl of covergece for the followig series.. x 3. x f ( x) x x... x x!

28 PTFs #BC 7 Altertig Series Remider If ltertig series coverges, the the sum of the series, S c be pproximted by fidig the th prtil sum, S. The prtil sum is withi R, the remider, of the ctul sum. R is less th or equl to the first eglected term. R S S. For the series, fid the error i usig the first 00 terms to estimte the sum. 3. Determie the umber of terms required to pproximte the sum of the coverget series! e 0 with error of less th Estimte the mout of error ivolved i the pproximtio of f ( ) for the first three terms of x f( x). 3!

29 PTFs #BC 8 Lgrge Error Boud A fuctio f( x ) is pproximted by the sum of Tylor polyomil P( x ) d some remider R ( x ), give by f ( x) P( x) R( x). Therefore the error boud ssocited with the remider is defied s: Error = R ( x) f ( x) P ( x) with ( ) f () z R ( x) ( x c) ( )! where f () z is the mx vlue of the derivtive o the specified itervl. Remember, it is just the first term left off with the mx vlue of the derivtive used. If the mx derivtive is give, use it! If ot, fid the ext derivtive d look for mx spot to evlute.. Let f be the fuctio give by g( x) si 5x d let Px ( ) be the 4 third-degree Tylor polyomil for f bout x 0. Use the Lgrge error boud to show tht f P Let f be fuctio tht hs derivtives of ll orders for.5,3.5. Assume tht f (3), f '(3) 3, f ''(3), d f '''( x) 36 for ll x i.5,3.5.. Fid the secod-order Tylor polyomil bout x 3 for f( x ). b. Estimte f (.7) usig prt (). c. Wht is the mx possible error i estimtig prt (b)?