AP Calculus BC Summer Math Packet

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Dustin Morgan
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1 AP Calculus BC Summer Math Packet This is the summer review a preparatio packet for stuets eterig AP Calculus BC. Dear Bear Creek Calculus Stuet, The first page is the aswer sheet for the attache problems. They are ue the first ay of school. Be sure to keep track of how log it takes you to complete the problem set a which problems you coul t o. Fill i this iformatio o the frot page where it is requeste. MOOC stas for miutes outsie of class which i this case is all the time you spe completig the problem set. If you ee some etra help i August, scheule a time to see me at school the week before school starts. Do ot use a graphig calculator to solve ay of the problems, but o check your solutios with a graphig calculator where appropriate. You will also fi a two-page back-to-back summary of math facts that you will ee to kow by memory. Please review this sheet a make sure that you o spe the time eee to recommit these facts to memory. You will have a quiz o the first ay of Calculus over the material i this review packet. Now, to make sure you have rea all of these irectios, you are to se me a cofirmatio message whe you have starte o the questios i August, bmackeey@tbcs.org. This has to be oe from a accout that I ca use to cotact you. Be sure to sig your with the ame you woul like me to call you. There is etra creit aware if you iclue a iterestig picture (.jpeg) of you that I ca use as a picture of the ay for your class o the overhea. Please make it a picture that you woul like to tell about i the class. I look forwar to teachig you Calculus urig the et school year. Have a woerful summer. Mr. MacKeey Remember: This packet is for your beefit a is itee to help you succee as you move through more avace math classes. Name:

2 The Bear Creek School Review a Preparatio Packet for AP Calculus BC Stuets Name: Aswers:. 7. (5). 5 7 y mi. 9 i 4 r uits 6. Ma:, at epoit (-,) a Mi: - at (, -) 7. = -, Mi: = 5, iflectio poit: (,-4) 8a. 0 b. 0 c. 9. check o your graphig calculator 0. proof si si C t 5. 6 t t 6 t 6. l C uits & check your graph o your graphig calculator f() c is efie, lim f( ) eists, a lim f ( ) f ( c). T. T. F 4. F 5. T 6. c c sec 5 ta cos. si 4. sec 4. 5csc5cot y y 8. 5 A t y 9. No. r is a fuctio of both r which is r t chagig a r t which is a costat. 40. Ma: 6, at epoit (4,6) a Mi: -6 at (, -6) 4. c, 4. 7 critical #' s :,,, 6 6 ecreases over:, 7,, 7 icreases over: (0, ),(, ),(, ) mi : (, 7),iflectio poits: (0,0), (,-6) 6 44a. 5 7 b. c. 45. horizotal: 4 y vertical: uits y y a y.04

3 Basic Differetiatio Rules For Elemetary Fuctios AP Calculus AB Review Checklist Name: [cu] c u. [ u v ] v u u v 5. v [ ] 8. [eu } e u u. [siu] (cosu) u 4. [cotu] (csc u) u 7. [arcsiu] u 0. u [arccotu] u u. [u v] u v. [c] 0 6. u [ u ] u, u 0 9. u [log u a u]. (l a)u [cosu] (siu) u 5. [secu] (secu tau) u 8. [arccosu] u. u [arcsecu] u 4. u u [uv] u v vu [u ] u u [lu] u u [au ] (la)a u u [tau] (sec u) u [cscu] (cscu cotu) u [arctau] u u [arccscu] u u u Basic Itegratio Rules (a > 0). kf (u)u k f (u)u. [ f (u) g(u)]u f (u)u g(u)u. u u C 4. u u u C u 5. l u C 6. u e u u e u C 7. a u u l a au C 8. siu cosu C 9. cosuusiu C 0. tauul cosu C. cotuu l siu C. secuu l secu tau C. cscuul cscu cotu C 4. sec uu tau C 5. csc uucotu C 6. secutauusecu C 7. cscucotuucscu C 8. u arcsi u C a u a u 9. u u arcta C 0. a u a a arcsec u C u u a a a Kow a be able to eplai the followig from a) Calculus b) Precalculus Defiitio of the Derivative of a Fuctio (sectio. p94) Completig the Square Theorem.: Differetiability Implies Cotiuity (p98) Biomial Theorem: epa ( y) Theorem.0: The Chai Rule Trigoometry Memory Agles Implicit a Eplicit Fuctios (sectio.5 p4) si(a+b), cos(a+b), ta(a+b) The Etreme Value Theorem (sectio. p55) Pythagorea Trigoometry Ietities Defiitio of a Critical Number (sectio. p57) Trigoometry Graphs: y = si, y = cos, y = ta Theorem. Relative Etrema Occur Oly at Critical Numbers (sectio. p57) a their iverse graphs y = arcsi, y = Rolle's Theorem (sectio. p6) arccos, y = arcta The Mea Value Theorem (sectio. p65) The First Derivative Test (sectio. p7) Defiitio of Cocavity (sectio.4 p79) Defiitio of a Limit (sectio. p5) Test for Cocavity (sectio.4 p80) The Squeeze Theorem (sectio. p6), si Poits of Iflectio (sectio.4 p8) lim, 0 Seco Derivative Test (sectio.4 p8) A Fuctio with Two Horizotal Asymptotes (sectio.5 p9) Defiitio of Cotiuity: a) at a poit, b) o a ope Newto's Metho for Approimatig the Zeros of a Fuctio (sectio.8 p5) iterval, c) everywhere (sectio.4 p67) & ) Defiitio of Differetials (sectio.9 p) o a close iterval (p70) Defiitio of Area of a Regio i the Plae (sectio 4. p58) Itermeiate Value Theorem (sectio.4 p74) Defiitio of a Defiite Itegral (sectio 4. p66) ( ) c c i Cotiuity Implies Itegrability (sectio 4. p66) i i The Fuametal Theorem of Calculus (sectio 4.4 p74) ( )( ) ( ) Mea Value Theorem for Itegrals (sectio 4.4 p77) i i 6 4 i i Defiitio of the Average Value of a Fuctio o a Iterval (sectio 4.4 p78) The Seco Fuametal Theorem of Calculus (sectio 4.4 p8) Itegratio of Eve a O Fuctios (sectio 4.5 p95) r t Balace P compou iterest times yearly The Trapezoial Rule (sectio 4.6 p00) Simpso's Rule ( is eve) (sectio 4.6 p0) lim ( ) e Defiitio of the Natural Logarithmic Fuctio (sectio 5. p) Defiitio of e (sectio 5. p4) r r The Derivative of a Iverse Fuctio (sectio 5. p) lim e Solvig a Homogeeous Differetial Equatio (sectio 5.7 p7) rt Volume by the Disk Metho a the Washer Metho (sectio 6. p47 & 49) Balace Pe cotiuous iterest Defiitio of Arc Legth (sectio 6.4 p46) Defiitio of the Area of a Surface of Revolutio (sectio 6.4 p440) Defiitio of Work Doe by a Variable Force (sectio 6.5 p446) Momets a Ceter of Mass of a Plaar Lamia (sectio 6.6 p458) Itegratio by Parts (sectio 7. p48) Ietermiate Forms a L Hopital s Rule (sectio 7.7 p5) cos lim 0 0

4 Summer Review Practice for stuets eterig AP Calculus BC August Practice Name: Do ot use a graphig calculator to compute ay of your aswers. Check your aswers with a graphig calculator where appropriate. Show all your work o each problem clearly a clealy. Use a pecil a a eraser. Keep track of how log you spe o these problems a which oes you ca't o correctly. Tur i this complete August Practice o the first ay of class a be prepare for a quiz over this review.. Use the efiitio of the erivative to fi f () for f ( ) 7... Use the efiitio of the erivative to fi f () for 5 f ( ).. Fi the equatio of the taget lie (i the form y = m + b) to the graph of f ( ) 5 for =. 4. The raius r of a sphere is icreasig at a rate of iches per miute. Fi the rate of chage of the volume of the sphere whe r = 4 iches.

5 5. Fi the volume of the largest right circular cylier that ca be iscribe i a sphere of raius r Determie the absolute etremes of the fuctio f ( ) 6 8 a the -values i the iterval [ -, ] where they occur. Sketch f() over [-,]. 7. Fi all relative etrema a poit of iflectio for f ( ) 6 5. Lim si b) Lim cos 8. a) 0 c) Lim si 0 9. Graph f ( ) Illustrate a erive Newto's Metho formula for approimatig the zeros of a ifferetiable fuctio.

6 . Fi F () : F( ) t t. Evaluate the iefiite itegral: si 4 cos. Fi ( f ) () for the fuctio f ( ) Fi the erivative of f ( t) arccos( t ) 5 Fi the erivative of f ( t) arcta( t ) l() 6. Evaluate the iefiite itegral. 7. Sketch the regio boue by the graphs of the algebraic fuctios a fi the area of the regio f ( ) 6, g ( ) 7.

7 8-9.. Fi the oe-sie limits. 8. lim 9. lim 0. List the three coitios for a fuctio f to be cotiuous at a poit c. Number them i the proper orer. True or False (T or F)? If the statemet is false, eplai why or give a eample that shows that it is false.. If y is a ifferetiable fuctio of u, u is a ifferetiable fuctio of v, a v is a ifferetiable fuctio of, the. If a fuctio is ifferetiable at a poit, the it is cotiuous at that poit. y y u v. u v. If a fuctio is cotiuous at a poit, the it is ifferetiable at that poit. 4. If a fuctio has a erivative from both the right a the left at a poit, the it is ifferetiable at that poit. 5. If the velocity of a object is costat, the its acceleratio is zero Fi the erivative, i simplifie factore form where appropriate. 6. f() = f ( ) (5 ) 8. f ( ) sec (5 ). 9. y 4 6

8 0. y Fi the erivative.. y = si. y = cos4. y = ta 4. y = csc5 5. y = Fi the limit a show how you arrive at your coclusio. For full creit, give a accurate o-calculator eplaatio. ta( ) ta( ) Lim y 7. Fi by implicit ifferetiatio. + y + y = 0

9 8. Use your aswer for #7 a fi the equatio for the lie taget to the curve at (0,) i y = m + b form. 9. Let A be the area of a circle of raius r that is chagig with respect to time. If r t is costat, is A t costat? Eplai. 40. Determie the absolute etrema of f ( ) a the -values i [0,4] where they occur. 4. Fi the poit(s) guaratee by the Mea Value Theorem for f ( ) ( ) over [-,]. 4. Fi the critical umbers for f ( ) si si over (0,) a the ope itervals o which the fuctio is icreasig or ecreasig.

10 4. Fi all relative etrema a poits of iflectio for f ( ) ( 4). 44. Fi the limit. a. lim b. lim 4 9 c. lim cos 45. Fi the vertical a horizotal asymptotes of the graph of f ( ). 46. Fi the area of the largest rectagle with lower base o the -ais a upper vertices o the curve y = Use Newto's Metho to approimate a zero of f ( ) cos over [-,] to the earest (i.e. where = cos )

11 48. Fi the ifferetial for y Fi y a y for y = for = a = = 0.0

CHAPTER 4 Integration

CHAPTER 4 Integration CHAPTER Itegratio Sectio. Atierivatives a Iefiite Itegratio......... 77 Sectio. Area............................. 8 Sectio. Riema Sums a Defiite Itegrals........... 88 Sectio. The Fuametal Theorem of Calculus..........