AP CALCULUS AB & BC Course Description Clculus is n dvnced mthemtics course tht uses meningful problems nd pproprite technology to develop concepts nd pplictions relted to continuity nd discontinuity of functions nd differentition, nd integrtion. The dvnced plcement courses will use curriculum bsed upon the curriculum of the stte of Tennessee nd the guidelines of the College Bord. Items for the BC course will be denoted with n sterisk. Section references pper in brckets fter ech student performnce indictor. They re for our dopted textbook: Clculus: Grphicl, Numericl, Algebric by Finney, et. l.
Stndrd 1.0: Functions, Grphs, nd limits Lerning Expecttions: Students will: 1.1 Anlyze the grphs of functions nd reltions; 1. Evlute the limits of functions (including one-sided limits); 1.3 Anlyze symptotic nd unbounded behvior; 1.4 Understnd continuity s property of functions; 1.5 * Anlyze prmetric, polr, nd vector functions. Student Performnce Indictors: 1.1. Grph functions on grphing clcultor using the pproprite windows. [1.1 1.6] 1.1.b Recognize functions by type: liner, qudrtic, polynomil, rtionl, exponentil, logrithmic, power, roots, bsolute vlue, [1.1 1.6] 1.1.c * Recognize functions by type: prmetric, polr, nd vector. [10.1, 10.3, 10.5] 1.1.d Predict nd explin the observed locl nd globl behvior of function. [4.1 4.3] 1.. Demonstrte n intuitive understnding of the limiting process. [.1] 1..b Clculte limits using lgebr. [.1] 1..c Estimte limits from grphs or tbles of dt. [.1] 1.3. Demonstrte n understnding of symptotes in terms of grphicl behvior. [.] 1.3.b Describe symptotic behvior in terms of limits involving infinity. [.] 1.3.c Compre reltive mgnitudes of functions nd their rtes of chnge. (For exmple, contrsting exponentil growth, polynomil growth, nd logrithmic growth.) [8.] 1.4. Develop n intuitive understnding of continuity. (Close vlues of the domin led to close vlues of the rnge.) [3.3] 1.4.b Develop nd understnding of continuity in terms of limits. [.4] 1.4.c Develop geometric understnding of grphs of continuous functions. (Intermedite Vlue Theorem nd Extreme Vlue Theorem.) [.3 nd 4.1] 1.5. * Anlyze nd grph curves in prmetric form. [10.1] 1.5.b * Anlyze nd grph curves in polr form. [10.5] 1.5.c * Anlyze nd grph curves in vector form. [10.3]
Stndrd.0: Derivtives Lerning Expecttions: Students will:.1 Develop the concepts of the derivtive;. Hve n understnding of the derivtive t point;.3 Investigte the derivtive s function;.4 Explore second derivtives;.5 Apply derivtives;.6 Compute derivtives. Student Performnce Indictors:.1. Compute derivtives grphiclly, numericlly, nd nlyticlly. [3.1 3.3].1.b Interpret derivtive s n instntneous rte of chnge. [.4].1.c Define the derivtive s the limit of the difference quotient. [3.1].1.d Understnd the reltionship between differentibility nd continuity. [3.].. Determine the slope of curve t point. Exmples to emphsize include points t which there re verticl tngents nd points t which there re no tngents. [3.1 3.]..b Determine tngent lines to curve t point nd locl liner pproximtions. [3.1, 4.5]..c Compute the instntneous rte of chnge s the limit of the verge rte of chnge. [.8, 3.1, 3.4]..d Approximte the rte of chnge from grphs nd tbles of vlues. [3.1].3. Determine the corresponding chrcteristics of the grphs of f nd f. [3.1, 3.4].3.b Determine the reltionship between the incresing nd decresing behvior of f nd the sign of f. [4.].3.c Investigte the Men Vlue Theorem nd its geometric consequences. [4.].3.d Explore equtions involving derivtives. Verbl descriptions re to be trnslted into equtions involving derivtives nd vice vers. [3.4, 4.4, 4.6].4. Explore the corresponding chrcteristics of the grphs of f, f, nd f. [4.3].4.b Determine the reltionship between the concvity of f nd the sign of f. [4.3].4.c Find the points of inflection s plces where the concvity chnges. [4.3].5. Anlyze curves, including the notions of monotoncity nd concvity. [4.3].5.b * Anlysis of plnr curves given in prmetric form, polr form, nd vector form, including velocity nd ccelertion vectors. [10.3, 10.6].5.c Perform optimiztion for both bsolute (globl) nd reltive (locl) extrem. [4.4].5.d Model rtes of chnge, including relted rtes problems. [4.6].5.e Use implicit differentition to find the derivtive of n inverse function. [3.8]
.5.f Interpret the derivtive s rte of chnge in vried pplied contexts, including velocity, speed, nd ccelertion. [3.4].5.g Investigte the geometric interprettion of differentil equtions vi slope fields nd the reltionship between slope fields nd solution curves for differentil equtions. [6.1].5.h * Compute the numericl solution of differentil equtions using Euler s method. [6.6].5.i * Apply L H ô pitl s Rule, including its use in determining limits nd convergence of improper integrls nd series. [8.1].6. Attin knowledge of derivtives of bsic functions, including power, exponentil, logrithmic, trigonometric, nd inverse trigonometric functions. [3.3, 3.5, 3.8, 3.9].6.b Apply the bsic rules for the derivtive of sums, products, nd quotients of functions. [3.3].6.c Apply the chin rule nd implicit differentition. [3.6, 3.7].6.d * Evlute the derivtives of prmetric, polr, nd vector functions. [10.1, 10.3, 10.6]
Stndrd 3.0: Integrls Lerning Expecttions: Students will: 3.1 Discover the interprettions nd the properties of the definite integrl; 3. Apply integrls; 3.3 Discover the Fundmentl Theorem of Clculus; 3.4 Explore techniques of ntidifferentition; 3.5 Apply ntiderivtives; 3.6 Compute numericl pproximtions to definite integrls. Student Performnce Indictors: 3.1. Compute Riemnn sums using left, right, nd midpoint evlution points. [5.1] 3.1.b Determine the definite integrl s limit of Riemnn sums over equl subdivisions. [5.] 3.1.c Determine the definite integrl of the rte of chnge of quntity over n intervl interpreted s the chnge of the quntity over the intervl: f '( x) = f ( b) f ( ) [7.1] 3.1.d Explore the bsic properties of definite integrls. (Exmples include dditivity nd linerity.) [6.1] b 3.. * Use pproprite integrls in vriety of pplictions to model physicl, biologicl, or economic situtions. [7.5] 3..b Use knowledge nd techniques for solving pplictions nd dpt this knowledge to solve similr ppliction problems. [7.1 7.5] 3..c Use the integrl of rte of chnge to give ccumulted chnge. [7.1 7.5] 3..d Represent Riemnn sum s definite integrl. [5.4] 3..e Find the re of region. [5., 5.3, 7.] 3..f * Find the re of region bounded by polr curves. [10.6] 3..g Find the volume of solid with known cross sections. [7.3] 3..h Find the verge vlue of function. [5.3] 3..i Find the distnce trveled by prticle long line. [7.1] 3..j * Find the length of curve. [7.4] 3..k * Find the length of curve in prmetric form nd polr form. [10.1, 10.6] 3.3. Use the Fundmentl Theorem to evlute definite integrls. [5.3, 5.4] 3.3.b Use the Fundmentl Theorem to represent prticulr ntiderivtives, nd the nlytic nd grphicl nlysis of functions so defined. [5.3, 5.4] 3.4. Explore how ntiderivtives follow directly from derivtives of bsic functions. [6.1]
3.4.b * Determine ntiderivtives by substitution of vribles (including chnge of limits for definite integrls), prts, nd simple prtil frctions (nonrepeting liner fctors only). [6., 6.3, 8.4] 3.4.c * Determine improper integrls (s limits of definite integrls). [8.3] 3.5. Find specific ntiderivtives using initil conditions, including pplictions to motion long line. [6.1, 7.1] 3.5.b Solve seprble differentil equtions nd using them in modeling. In prticulr, studying the eqution y = ky nd exponentil growth. [6.1, 6.4] 3.5.c * Solve logistic differentil equtions nd using them in modeling. [6.5] 3.6. Use Riemnn nd trpezoidl sums to pproximte definite integrls of functions represented lgebriclly, grphiclly, nd by tbles of vlues. [5.1, 5., 5.5]
Stndrd 4.0: * Polynomil Approximtions nd Series Lerning Expecttions: Students will: 4.1 Develop the concepts of series; 4. Explore series of constnts. Student Performnce Indictors: 4.1.. Define series s sequence of prtil sums. 4.1.b. Define convergence in terms of the limit of the sequence of prtil sums. 4.1.c. Use technology to explore convergence or divergence. 4... Use motivting exmples such s deciml expnsion. [9.1] 4..b. Investigte geometric series with pplictions. [9.1] 4..c. Define the hrmonic series. [9.5] 4..d. Find the error bound of n lternting series. [9.5] 4..e. Relte improper integrls to series. [9.5] 4..f. Use the integrl test to determine the convergence of p-series. [9.5] 4..g Use the rtio to determine convergence or divergence. [9.4] 4..h Compre series to test for convergence or divergence. [9.4, 9.5]
Section Summry And Pcing Guide For Clculus AB To cover the course outline for Clculus AB, techer must cover: Chpter : Sections 1,, 3, nd 4 Chpter 3: Sections 1,, 3, 4, 5, 6, 7, 8, nd 9 Chpter 4: Sections 1,, 3, 4, nd 6 Chpter 5: Sections 1,, 3, 4, nd 5 Chpter 6: Sections 1,, nd 4 Chpter 7: Sections 1,, nd 3 Wht follows is suggested pcing guide tht ttempts to complete the course content for AP Clculus AB in one semester - specificlly, the spring semester when there re fewer dys due to the plcement of the exm dy. Dy Section Topic 1 Quiz Quiz: Sections 1.1 1.6. Skip section 1.4.1 Rtes of Chnge nd Limits 3. Limits Involving Infinity 4 Quiz Quiz: Limits 5.3 Continuity 6.4 Rtes of Chnge nd Tngent Lines 7 Quiz Quiz: Chpter 8 3.1 Derivtive of Function 9 Derivtive of Function (continued) 10 3. Differentibility 11 Review 1 Test Test: Chpter nd Sections 3.1 3. 13 3.3 Rules of Differentition 14 3.4 Velocity nd Other Rtes of Chnge 15 Velocity nd Other Rtes of Chnge (continued) 16 3.5 Derivtives of Trigonometric Functions 17 Derivtives of Trigonometric Functions (continued) 18 3.6 Chin Rule 19 Chin Rule (continued) 0 Review 1 Quiz Quiz: 3.4 3.6 3.7 Implicit Differentition 3 Implicit Differentition (continued) 4 3.8 Derivtives of Inverse Trigonometric Functions 5 3.9 Derivtives of Exponentil nd Logrithmic Functions 6 Derivtives of Exponentil nd Logrithmic Functions (continued) 7 Concept Connections 8 Review 9 Test Test: Chpter 3 30 4.1 Extreme Vlues of Functions, Optionl - Section 4.5 31 4. Men Vlue Theorem
3 4.3 Connecting Derivtives to the Grph of Function 33 Connecting Derivtives to the Grph of Function (continued) 34 Concept Connections 35 Quiz Quiz: 4.1 4.3 36 4.4 Modeling nd Optimiztion 37 Modeling nd Optimiztion (continued) 38 4.6 Relted Rtes 39 Relted Rtes (continued) 40 Relted Rtes (continued) 41 Review 4 Test Test: Chpter 4 43 5.1 Estimting with Finite Sums 44 5. Definite Integrls 45 5.3 Definite Integrls nd Antiderivtives 46 Concept Connections 47 5.4 Fundmentl Theorem of Clculus 48 Concept Connections 49 5.5 Trpezoidl Rule (Skip Simpson s Rule) 50 Review 51 Test Test: Chpter 5 5 6.1 Antiderivtives nd Slope Fields 53 6. Integrtion by Substitution 54 Integrtion by Substitution (continued) 55 6.4 Exponentil Growth nd Decy 56 Exponentil Growth nd Decy (continued) 57 Quiz Quiz: Chpter 6 58 7.1 Integrl s Net Chnge 59 7. Ares in Plne 60 7.3 Volumes 61 Volumes (continued) 6 Volumes (continued) 63 Quiz Quiz 7.1-7.3 64 Review 65 Review 66 Review 67 Review 68 Review 69 Review 70 Review 71 Review 7 Review 73 Review 74 AP Exm/Semester Exm
Section Summry And Pcing Guide For Clculus BC To cover the course outline for Clculus BC, techer must cover: Chpter 1: Section 4 Chpter : Sections 1,, 3, nd 4 Chpter 3: Sections 1,, 3, 4, 5, 6, 7, 8, nd 9 Chpter 4: Sections 1,, 3, 4, nd 6 Chpter 5: Sections 1,, 3, 4, nd 5 Chpter 6: Sections 1,, 3, 4, 5, nd 6 Chpter 7: Sections 1,, 3, 4, nd 5 Chpter 8: Sections 1,, 3, nd 4 Chpter 9: Sections 1,, 3, 4, nd 5 Chpter 10: Sections 1,, 3, 5, nd 6 Wht follows is suggested pcing guide tht ttempts to complete the course content for AP Clculus BC in two semesters. Dy Section Topic 1 1.4 Prmetric Equtions Quiz Quiz: Sections 1.1 1.6. 3.1 Rtes of Chnge nd Limits 4. Limits Involving Infinity 5 Quiz Quiz: Limits 6.3 Continuity 7.4 Rtes of Chnge nd Tngent Lines 8 Quiz Quiz: Chpter 9 3.1 Derivtive of Function 10 Derivtive of Function (continued) 11 3. Differentibility 1 Review 13 Test Test: Chpter nd Sections 3.1 3. 14 3.3 Rules of Differentition 15 3.4 Velocity nd Other Rtes of Chnge 16 Velocity nd Other Rtes of Chnge (continued) 17 3.5 Derivtives of Trigonometric Functions 18 Derivtives of Trigonometric Functions (continued) 19 3.6 Chin Rule 0 Chin Rule (continued) 1 Review Quiz Quiz: 3.4 3.6 3 3.7 Implicit Differentition 4 Implicit Differentition (continued) 5 3.8 Derivtives of Inverse Trigonometric Functions 6 3.9 Derivtives of Exponentil nd Logrithmic Functions 7 Derivtives of Exponentil nd Logrithmic Functions (continued)
8 Concept Connections 9 Review 30 Test Test: Chpter 3 31 4.1 Extreme Vlues of Functions 3 4. Men Vlue Theorem 33 4.3 Connecting Derivtives to the Grph of Function 34 Connecting Derivtives to the Grph of Function (continued) 35 Concept Connections 36 Quiz Quiz: 4.1 4.3 37 4.4 Modeling nd Optimiztion 38 Modeling nd Optimiztion (continued) 39 4.5 Lineriztion nd Newton s Method (optionl) 40 4.6 Relted Rtes 41 Relted Rtes (continued) 4 Relted Rtes (continued) 43 Review 44 Test Test: Chpter 4 45 5.1 Estimting with Finite Sums 46 5. Definite Integrls 47 5.3 Definite Integrls nd Antiderivtives 48 Concept Connections 49 5.4 Fundmentl Theorem of Clculus 50 Concept Connections 51 5.5 Trpezoidl Rule (Skip Simpson s Rule) 5 Review 53 Test Test: Chpter 5 54 6.1 Antiderivtives nd Slope Fields 55 6. Integrtion by Substitution 56 Integrtion by Substitution (continued) 57 6.3 Integrtion by Prts 58 Integrtion by Prts (continued) 59 Quiz Quiz: 6.1 6.3 60 6.4 Exponentil Growth nd Decy 61 Exponentil Growth nd Decy (continued) 6 8.4 Prtil Frctions (Introduce non-repeted liner fctors) 63 6.5 Popultion Growth 64 Popultion Growth (continued) 65 6.6 Numericl Methods (Euler s Method only) 66 Review 67 Test Test: Chpter 6 68 7.1 Integrl s Net Chnge 69 7. Ares in Plne 70 7.3 Volumes 71 Volumes (continued) 7 Volumes (continued)
73 Quiz Quiz 7.1-7.3 74 7.4 Lengths of Curves 75 7.5 Applictions from Science nd Sttistics (work) 76 Review 77 Test Test: Chpter 7 78 Review 79 Review 80 Review 81 Review 8 Review 83 Review 84 Review 85 Test Semester Exm 86 8.1 L H ô pitl s Rule 87 L H ô pitl s Rule (continued) 88 8. Reltive Rtes of Growth (Skip Oh-nottion) 89 Reltive Rtes of Growth (continued) 90 Quiz Quiz: 8.1 8. 91 8.3 Improper Integrls 9 Improper Integrls (continued) 93 8.4 Prtil Frctions (Only liner non-repeting is required) 94 Prtil Frctions (Non liner nd repeting optionl) 95 Prtil Frctions (Trigonometric substitution optionl) 96 Review 97 Test Test: Chpter 8 98 9.1 Power Series 99 Power Series (continued) 100 Power Series (continued) 101 9. Tylor Series 10 Tylor Series (continued) 103 Tylor Series (continued) 104 9.3 Tylor s Theorem 105 Tylor s Theorem (continued) 106 Tylor s Theorem (continued) 107 9.4 Rdius of Convergence 108 Rdius of Convergence (continued) 109 Rdius of Convergence (continued) 110 9.5 Testing Convergence t Endpoints 111 Testing Convergence t Endpoints (continued) 11 Testing Convergence t Endpoints (continued) 113 Review 114 Test Test: Chpter 9 115 1.4 Prmetric Equtions 116 Prmetric Equtions (continued) 117 10.1 Prmetric Functions
118 Prmetric Functions (continued) 119 10. Vectors in the Plne 10 Vectors in the Plne (continued) 11 10.3 Vector-vlued Functions 1 Vector-vlued Functions (continued) 13 Vector-vlued Functions (continued) 14 10.4 Modeling Projectile Motion (optionl) 15 10.5 Polr Coordintes nd Polr Grphs 16 Polr Coordintes nd Polr Grphs (continued) 17 Polr Coordintes nd Polr Grphs (continued) 18 10.6 Clculus of Polr Curves 19 Clculus of Polr Curves (continued) 130 Clculus of Polr Curves (continued) 131 Review 13 Test Test: Chpter 10 133 Review Once chpter 10 is completed, the rest of the semester is devoted to review for the dvnced plcement exm. The dte for the exm is usully during the first week of My. It should be noted tht some techers mix their exm review dys throughout the term while others sve it for the end of the term s noted in the outline bove.
AP Clculus Resources Among the mterils to help students prepre for the A.P. Exm, it is recommended tht the techer hve ccess to previous exms. The free-response portions re vilble fter the exm ech yer. Solutions to the previous few yers cn be viewed t the dvnced plcement website (http://pcentrl.collegebord.com). The multiplechoice portion of the exm cn be reviewed through relesed exms tht cn be purchsed from the College Bord. They relese certin exms t irregulr intervls. In ddition to these ctul exms, there re mny A.P. review texts vilble for purchse. As is typicl of stndrdized test reviews, mny of these texts tend to be more difficult thn the ctul exm. We encourge techers new to A.P. Clculus to spend time nd gther resources from those techers who hve tught the course previously. Wht follows is n overview of wht students should know before tking the exm. Techers might wnt to provide their students with copy.
STUFF YOU SHOULD KNOW BEFORE TAKING THE CALCULUS ADVANCED PLACEMENT EXAM 1. ALL OF THE BASIC DIFFERENTIATION AND INTEGRATION RULES. DIFFERENTIATION 1. d [cu] = c u' * you cn fctor constnt out. d [u + v] = u' + v' 3. d [uv] = uv' + vu' *product rule 4. d [ u vu' -uv' ] = * quotient rule: ho d[hi] - hi d[ho] over ho ho v v 5. d [c] = 0 6. d [un ] = n u (n-1) u' *power rule 7. d [x] = 1 8. d [ u ] = u u' *rrely used u 9. d [ln u] = u ' u 10. d [eu ] = e u u' 11. d [log u] = u' (ln u ) *works like ln, but you divide by (ln ) 1. d [u ] = u (ln ) u' *works like e, but you mult by (ln ) 13. d [sin u] = (cos u) u' 14. d [cos u] = -(sin u) u' 15. d [tn u] = (sec u) u' 16. d [cot u] = -(csc u) u'
17. d [sec u] = (sec u tn u) u' 18. d [csc u] = -(csc u cot u) u' *every other trig is negtive, but similr 19. d [rcsin u] = u' 1 u 0. d [rccos u] = u' 1 u 1. d [rctn u] = u' u + 1. d [rccot u] = u' u + 1 3. d [rcsec u] = u' u u 1 4. d [rccsc u] = u' u u 1 *every other inverse trig is the opposite 1. k*f(u) du = k* f(u) du. [f(u) + g(u)] du = f(u) du + 3. du = u + c n+ 1 INTEGRATION 4. un du = u n + 1 + c du 5. u = ln u + c 6. eu du = e u + c 1 7. u du = ln u + c 8. sin u du = - cos u + c 9. cos u du = sin u + c 10. tn u du = - ln cos u + c 11. cot u du = ln sin u + c 1. sec u du = ln sec u + tn u + c 13. csc u du = - ln csc u + cot u + c g(u) du
14. sec u du = tn u + c 15. csc u du = - cot u + c 16. sec u * tn u du = sec u + c 17. csc u * cot u du = - csc u + c du 18. = rcsin u u + c du 19. = 1 + u rctn u + c du 0. = 1 u u rcsec u + c. LIMITS AND CONTINUITY: A) Two specil limits: lim sin x = 1, B) lim 1 cos x = 0 x 0 x x 0 x B) In generl, if you cn plug limit it in, then plug it in C) If you cnnot just plug limit in, then you cn investigte the limit in vriety of wys I) Numericlly - mke tble of vlues II) Anlyticlly - remove ny discontinuity nd then plug in III) Grphiclly - where is the vlue of y heding for your prticulr x? D) Sometimes limits do not exist: I) diverge to infinity II) oscillting functions III) when the limit on the left is not equl to the limit on the right E) Limits t infinity (lso horizontl symptotes). Check the degree on the top (T) nd the bottom (B): I) if T>B, then it diverges to infinity II) if T=B, then it is the rtio of the leding coefficients III) if T<B, then it is zero Note tht it is possible tht the left nd right end behvior my differ. As n exmple: lim 3x = - 3 while lim 3x = 3 x 4x + 5 x 4x + 5 F) A point is continuous if the limit on the left is the limit on the right nd these equl the vlue of the function. An intervl is continuous if every point on the intervl is continuous. G) A nmed theorem: The Intermedite Vlue Theorem. If f(x) is continuous function on [,b], then for ny y 0 between f() nd f(b) there exists c within [,b] such tht f(c) = y 0.
3. THE DEFINITION OF THE DERIVATIVE A) f '(x) = lim h 0 difference quotient. f ( x+ h) f ( x) h if the limit exists. This is the limit of the B) You should lso be fmilir with the lternte definition of the derivtive t point : f '() = lim f ( x) f ( ) x x 4. DERIVATIVE THOUGHTS A) If function is differentible on n intervl, then it is continuous s well. B) The derivtive evluted t point is the slope of the line tngent to the function t tht point. C) The derivtive of the position function is the velocity function nd the derivtive of the velocity function is the ccelertion function. D) The grph f '(x) will hve zeros t the reltive extrem of differentible function f(x). f '(x) will be bove the x-xis where f(x) is rising nd below the x-xis where f(x) is flling. E) For piece-wise defined function, if the derivtive is the sme on both sides then it is considered smooth nd the derivtive exists. F) The chin rule for the derivtive of composite function y = f(g(x)) is d y = f ' (g(x))*g ' (x). In nother nottion, if u = g(x), then y = f(u), so the derivtive w.r.t. x is: dy dy du = du 5. CRITICAL NUMBERS AND THE FIRST AND SECOND DERIVATIVE Criticl numbers exist where the first derivtive = 0 or does not exist for points tht re within the domin (i.e. points tht exist). The first derivtive is used to find intervls where the function is incresing or decresing [include the endpoints if they exist]. The second derivtive indictes intervls where the function is concve up or down (lwys use open intervls). Points of inflection exist where the intervls of concvity chnge on either side of the point. 6. RELATIVE AND ABSOLUTE MAX AND MIN A) Reltive extrem re on n open intervl. A reltive mximum exists where the intervl on the left of criticl point is incresing while the intervl on the right is decresing (This is the first derivtive test for reltive extrem). A reltive mximum
lso occurs where criticl point exists in n intervl tht is concve down (This is the second derivtive test for reltive extrem). *Mins re just the opposite*. B) Absolute mximums nd minimums lso occur either t the reltive extrem or t closed endpoints. This is nother nmed theorem: The Extreme Vlue Theorem. If f is continuous on closed intervl [, b], then f hs both mximum vlue nd minimum vlue on the intervl. 7. DERIVATIVE APPLICATIONS A) Another nmed theorem: The Men Vlue Theorem. There is point, c, in the closed intervl [,b] on differentible function f(x) such tht f '(c) = f ( b ) f ( ). b B) The M.V.T enbles us to find the point where the instntneous rte of chnge is equl to the verge rte of chnge. It is where the slope of the tngent line is equl to the slope of the secnt line. C) A technique for optimiztion (mx/min) problems: Write the eqution of wht is to be mximized or minimized. Substitute until the right side is in terms of one vrible, (the left side will be the vrible you will wnt to optimize). Tke the derivtive, set it equl to zero, nd solve. D) A technique for relted rtes problems Find n eqution relting the vribles you re interested in Get the eqution in terms of two vribles (you my hve to use secondry eqution) Tke the derivtive (implicit) of both sides with respect to t One of the derivtives will be given, plug it in Solve for the other derivtive Answer the question - you my hve to work to get the vlue of the vrible you re going to plug in. 8. CURVE SKETCHING CONSIDERATIONS A) Discontinuities - removble (holes), non-removble (verticl symptotes) B) Limits t infinity - possible horizontl symptotes C) x nd y intercepts - set the other = 0 nd solve D) Reltive mx nd mins - occur t criticl points. Use 1st or nd derivtive test. E) Concvity - up when f ' ' (x)>0, down when f ' ' (x)<0 F) One sided limits t the verticl symptotes - will be + or -, grph or plug in G) Trnsltions nd diltions nd reflections - see #9 below. 9. Mnipultion of functions (from Pre-Clculus): Consider y = f(x) A) y = f(x - h) shifts h units horizontlly B) y = f(x) + k shifts k units verticlly C) y = A*f(x) stretches verticlly by fctor of A
D) y = f(bx) shrinks horizontlly by fctor of B E) y = - f(x) reflects through the x-xis F) y = f(- x) reflects through the y-xis G) y = f(x) flips those portions below the x-xis up H) y = f( x ) keeps the right side nd flips it through the y-xis 10. THE DEFINITE INTEGRAL TRAP Χ = A) Estimting the re under curve by mens of LRAM, RRAM, MRAM, nd The trpezoid rule is TRAP = 1 ( y0 + y1+ y+... + yn 1+ yn)* Χ where b n b B) If f is bove the x-xis on [, b], then the re under the curve is 11. THE FUNDAMENTAL THEOREM OF CALCULUS f ( x) A) The definite integrl is the difference of the boundries evluted t the ntiderivtive: f ( x ) = f ( b) f ( ). B) Another version of the F.T.C: b I) If the upper limit of integrtion is just plin x: D [ f ( t) dt] = f ( x) II) In generl with the chin rule: D [ f ( t) dt] = f ( h( x))* h ( x) f ( g( x))* g ( x) x h( x) g( x) 1. APPLICATIONS OF INTEGRATION right A) Are between two curves ( top bottom) left B) The verge vlue of function: C) Volumes of Revolution 1 f x b ( ) I) Disks round the x-xis (or other horizontl xis, y = h) b x x
b v = π [( outside xis) ( inside xis) ] II) While this is no longer required for A.P., Shells round the y-xis (or other verticl xis, x = k) b v = π p( x)*( topcurve bottomcurve) where p(x) = D) Volumes by Cross Section x - k if the xis is on the left of the region k - x if the xis is on the right of the region I) If shpe hs known cross section whose re is function of x, A(x), you cn integrte over its length to get the volume: V = right left Ax ( ) II) If the cross-sections re between the curve nd the x-xis: right ) Squres: V = [ f ( x)] left b) Equilterl Tringles: V = c) Semi-Circles: V = 8 π right E) Rectiliner motion (motion long line): I) Totl distnce = vt () dt II) Displcement = endtime beginningtime endtime vtdt () beginningtime 3 4 [ f ( x)] left right [ f ( x)] G) An ccumultor function is function defined s the integrl from constnt to x. endtime H) RteOfSomething * dt = Net chnge in the something. beginningtime left 13. SOLVING DIFFERENTIAL EQUATIONS & GROWTH AND DECAY PROBLEMS A) In generl, seprte your vribles nd integrte both sides.
B) This is where the nturl log often creeps in. If the rte of chnge is proportionl to the mount present, (dy/dt = k*y), then you cn go right to y = ce kt. You could lso prove this by seprting vribles. For Newton s Lw of cooling, [dy/dt = k(y - q) where q is the constnt temperture], y = ce kt + q C) A slope field cn be generted for first order differentil eqution. The solution of differentil eqution is fmily of grphs tht will pss gently through the slope field. If you hve specific point you cn solve for the constnt of integrtion nd find the exct function. D) We use Euler s method to numericlly solve differentil eqution. Given n initil point (x 0,y 0 ): x n+1 = x n +, y n+1 = y n + y' n E) To sketch function f(x) when given f '(x): The zeros of f '(x) tht pss through the x-xis (s opposed to bouncing off) will be the reltive extrem of f(x), the reltive extrem of f '(x) will be points of inflection of f(x). Where f '(x) is flling f(x) is concve down. Where f '(x) is rising f(x) is concve up. 14. OTHER FUNCTIONS FOR THE BC STUDENT AND ASSOCIATED FORMULAS A) Additionl formuls for functions: right dy I) The rc length of curve: L = 1 + left II) Integrtion By Prts: udv = uv vdu III) The logistic differentil eqution dp P = kp( 1 ), hs the solution dt M M P = 1+ Ae kt B) Prmetric functions I) dy dy = dt dt II) L = b dt dy + dt dt C)Vector I) Definitions for prticle movement. ) v(t) = dr v is the prticle s velocity vector. dt b) v(t), the mgnitude of v, is the prticle s speed.
c) (t) = dv v v dr = is the prticles ccelertion vector. dt dt d) v, unit vector, is the direction of motion v II) If the initil position is r(0) = hi + kj, then r(t) = (v 0 cos θ t + h)i + ( 1 gt + v 0 sin θ t + k) j D) Polr I) dy dy = d θ = d θ β 1 II) A = r dθ III) L = α β α r f ( θ) Sin( θ) + f ( θ) Cos( θ) f ( θ) Cos( θ) f ( θ) Sin( θ) dr + dθ dθ 15. INFINITE POSSIBILITIES FOR THE BC STUDENT A) L Hopitl s Rule; if f() = g() = 0 nd f nd g re differentible on n open intervl I contining, nd if g (x) 0 on I s long s x, then: lim f ( x) x gx ( ) = lim f ( x) x g ( x) B) L Hopitl s Rule works for = (s x goes to infinity) nd if lim f ( x) x gx ( ) = C) We tke cre of improper integrls s follows: I) If one of the limits of integrtion is infinite, we ssign it letter like nd then tke lim II) If both limits of integrtion re infinite, brek the integrl into two pieces ech hving only one infinite limit of integrtion. III) If one of the limits of integrtion is verticl symptote, we ssign it letter nd then tke the limit s the vrible pproches tht letter from the pproprite side. IV) If verticl symptote exists within the intervl, brek the integrl into two pieces ech hving only one limit of integrtion needing to be pproched by certin side.
1 D) will converge if p > 1 nd diverge otherwise. 1 x p 1 1 E) will diverge if p > 1 nd will converge if p < 1. 0 x p F) The Limit Comprison Test compres function to known function: If the positive functions f nd g re continuous on [, ) nd if lim f ( x) x gx ( ) = L, 0 < L <, then f ( x) nd g ( x) either both converge or both diverge. 16. INFINITE SERIES FOR THE BC STUDENT A) A Tylor series centered t x = is of the form: f ( ) f ( ) f ( ) f ( ) + ( x ) + ( x ) + ( x ) 3 + f ( 4 ) ( ) 4 ( x ) +... 0! 1!! 3! 4! B) A Mclurin Series is centered t x = 0. Here re few to memorize: 1 1) 1 x = 1 + x + x + x 3 +... ) e x = 1 + x + x! + x 3 3! + x 4 4! +... 3) sin x = x - x 3 3! + x5 5! -... 4) cos x = 1 - x! + x 4 4! -... C) A Tylor polynomil centered t x = is of the form n ( k f ) ( ) k P n (x) = ( x ) k = 0 k! D) Any Tylor series cn be written s: P(x) = P n (x) + R n (x) E) R n (x) cn be estimted s f ( n+ 1 ) () c n x ( n )! ( ) + 1 for some c between nd x for + 1 the series centered t x =. Its bsolute vlue is referred to s the Lgrnge error. F) At x = b, f(b) - P n (b) is the trunction error. G) To determine the rdius nd intervl of convergence of n infinite series, Set Lim n+1 < 1 nd solve. We must check the endpoints by n n plugging them in. n H) The Alternting Series Test: The series ( 1) + 1 un = u1 u + u3 u 4+... n= 1 converges if ll three of the following conditions re stisfied:
1. ech u n is positive;. u n > u n+1 for ll n > N, for some integer N; 3. Lim n u n = 0 I) The Alternting Series Estimtion Theorem: If n lternting series stisfies the conditions listed bove, then not only does it converge, but the trunction error is bounded by the next unused term. 17. CALCULATOR INFORMATION A) A grphing clcultor pproprite for use on the exmintions is expected to hve the built- in cpbility to: 1. plot the grph of function within n rbitrry viewing window,. find the zeros of functions (solve equtions numericlly), 3. numericlly clculte the derivtive of function, nd 4. numericlly clculte the vlue of definite integrl. B) You do not hve to show ny work for those spects listed bove. You do need to show work on the free response section for items not listed. As n exmple, finding reltive mximum. C) Grphing clcultors with QWERTY keybord re not llowed on the exmintion.