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Harold’s AP Calculus NotesCheat Sheet14 April 2024LimitsDefinition of LimitLet f be a function defined on an open interval containing c and let L be a real number. The statement:limx→cfx=Lmeans that for each ?>0 there exists a δ>0 such that if x-c<δ,then fx-L<?Tip:Direct substitution: Plugin fc and see if it provides a legal answer. If so then L=fc.The Existence of a LimitThe limit of fx as x approaches c is L if and only if:limx→c-fx=L and limx→c+fx=LDefinition of ContinuityA function f is continuous at c if for every ε>0 there exists a δ>0 such that x-c<δ and f(x)-f(c)<ε.Tip:Rearrange fx-fc to have x-c as a factor. Since x-c<δ we can find an equation that relates both δ and ε together.Prove that fx=x2-1 is a continuous function. fx-fc=x2-1-c2-1=x2-1-c2+1=x2-c2=x+c x-c=x+c x-c =x+c x-c Since x+c≤ 2cfx-fc≤2c x-c<εSo, given ε>0, we can choose δ=12cε>0 in the Definition of Continuity. So, substituting the chosen δ for x-c we get:fx-fc≤2c12cε=εSince both conditions are met, fx is continuous.Two Special Trig Limitslimx→0sinxx=1 and limx→01-cosxx=0Derivatives(See Larson’s 1-pager of common derivatives)Definitions of a Derivative of a Function(Slope Function)f'x=limh→0fx+h-fxhf'c=limx→cfx-f(c)x-cDerivatives Notationdydx, y', f'x, f(n)x, ddxfx, Dx[y]1. Chain Ruleddxfgx=f'gx?g'xdydx=dydt·dtdx2. Constant Ruleddxc=03. Constant Multiple Ruleddxcf(x)=cf'(x)4. Sum and Difference Ruleddxfx±g(x)=f'(x)±g'(x)5. Product Ruleddxfg=fg'+g f'6. Quotient Ruleddxfg=gf'-fg'g27. Power Ruleddxxn=nxn-18. General Power Ruleddxun=nun-1 u' where u=u(x)9. Power Rule for xddxx=1 (think x=x1 and x0=1)10. Absolute Valueddxx=xx11. Natural Exponential Ruleddxex=ex12. General Natural Exponential Ruleddxeg(x)=eg(x)g'(x)13. Exponential Ruleddxax=(lna)ax14. General Exponential Ruleddxag(x)=lna ag(x) g'(x)15. Natural Logorithm Ruleddxlnx=1x16. General Natural Logorithm Ruleddxlnf(x)=1f(x)?f'(x)17. Logorithm Ruleddxlogax=1(lna) x18. General Logorithm Ruleddxlogaf(x)=1lnx? f'(x)f(x)19. Sineddxsinx=cosx20. Cosineddxcos(x)=-sin(x)21. Tangentddxtanx=sec2x22. Cotangentddxcot(x)=-csc2(x)23. Secantddxsecx=secxtan(x)24. Cosecantddxcsc(x)=-cscxcot(x)25. Arcsineddxsin-1(x)=11-x226. Arccosineddxcos-1(x)=-11-x227. Arctangentddxtan-1(x)=11+x228. Arccotangentddxcot-1(x)=-11+x229. Arcsecantddxsec-1(x)=1x x2-130. Arccosecantddxcsc-1(x)=-1x x2-131. Hyperbolic Sine ex-e-x2ddxsinhx=coshx32. Hyperbolic Cosine ex+e-x2ddxcosh(x)=sinh(x)33. Hyperbolic Tangentddxtanhx=sech2x34. Hyperbolic Cotangentddxcoth(x)=-csch2(x)35. Hyperbolic Secantddxsechx=-sechxtanh(x)36. Hyperbolic Cosecantddxcsch(x)=-cschxcoth(x)37. Hyperbolic Arcsineddxsinh-1(x)=1x2+138. Hyperbolic Arccosineddxcosh-1(x)=1x2-139. Hyperbolic Arctangentddxtanh-1(x)=11-x240. Hyperbolic Arccotangentddxcoth-1(x)=11-x241. Hyperbolic Arcsecantddxsech-1(x)=-1x 1-x242. Hyperbolic Arccosecantddxcsch-1(x)=-1x 1+x2PhysicsTranslational MotionPosition Functionst=12gt2+v0t+s0xt=12axt2+v0,xt+x0yt=12ayt2+v0,yt+y0Velocity Functionvt=s't=gt+v0Acceleration Functionat=v't=s''tJerk Functionjt=a't=v''t=s(3)tGravitational Constant (g)g≈-9.81ms2g≈-32.2 fts2 Analyzing the Graph of a Function(See Harold’s Illegals and Graphing Rationals Cheat Sheet)x-Intercepts (Zeros or Roots)fx=0y-Interceptf0=yDomainValid x valuesRangeValid y valuesContinuityNo division by 0, no negative square roots or logsVertical Asymptotes (VA)x = division by 0 or undefinedHorizontal Asymptotes (HA)limx→∞-f(x)→y and limx→∞+f(x)→yInfinite Limits at Infinitylimx→∞-f(x)→∞ and limx→∞+f(x)→∞DifferentiabilityLimit from both directions arrives at the same slopeRelative ExtremaCreate a table with domains: fx, f'x, f''(x)ConcavityIf f''(x)→+, then cup up ?If f''x→-, then cup down ?Points of Inflectionf''x=0, then concavity changesGraphing with DerivativesTest for Increasing and Decreasing Functions1. If f'x>0, then f is increasing (slope up) ↗2. If f'x<0, then f is decreasing (slope down) ↘3. If f'x=0, then f is constant (zero slope) →First Derivative Test1. If f'x changes from – to + at c, then f has a relative minimum at (c, f(c))2. If f'x changes from + to - at c, then f has a relative maximum at (c, f(c))3. If f'x, is + c + or -c -, then fc is neitherSecond Deriviative TestLet f’(c)=0, and f”(x) exists, then1. If f''x>0, then f has a relative minimum at c,f(c)2. If f''x<0, then f has a relative maximum at c,f(c)3. If f''x=0, then the test failsTest for Concavity1. If f''x>0 for all x, then the graph is concave up ?2. If f''x<0 for all x, then the graph is concave down ?Inflection Points(Change in concavity)If c,f(c) is a point of inflection of fx, then either1. f''c=0 or2. f''x does not exist at x=cTangent LinesGeneral Formax+by+c=0Slope-Intercept Formy=mx+bPoint-Slope Formy-y0=m(x-x0)where m=f'x0 at point x0,y0Calculus Formf(x)=f'cx-c+fcSlopem=riserun=y2-y1x2-x1=?y?x→dydx=f'x Differentiation & DifferentialsRolle’s TheoremAssume f is continuous on the closed interval [a,b], and f is differentiable on the open interval (a,b).If fa=fb, then there exists at least one number c in (a,b) such that f'c=0.Mean Value TheoremIf f meets the conditions of Rolle’s Theorem, then you can find ‘c’.f'c=fb-f(a)b-a=?y?xf(b)=fa+b-af'(c)Intermediate Value TheoremAssume f is a continuous function with the interval [a, b] as its domain.If f takes values fa and fb at each end of the interval, then it also takes any value between fa and fb at some point within the interval.Calculating Differentials(Tanget line approximation)fx+?x≈fx+?ysince dy=f'x dxthen ?y≈f'x ?xsofx+?x≈fx+f'x ?xRelative Error=?ff in %Example: 482fx=4xfx+?x=f81+1Newton’s MethodFinds zeros of f, or finds c if f(c)=0.xn+1=xn-fxnf'xnExample: 482fx=x4-82=0start with x0=3Related RatesSteps to solve:Identify the known variables and rates of change.x=15 my=20 mx'=2msy'=______msConstruct an equation relating these quantities.x2+y2=r2(Often uses the Pythagorean Theorem.)Differentiate both sides of the equation.2xx'+2yy'=0Solve for the desired rate of change.y'=-xy x'Substitute the known rates of change and quantities into the equation.y'=-1520? 2=1.5 msL’H?pital’s RuleIf limx→cfx=limx→cPxQx=00,∞∞,0?∞,1∞,00,∞0,∞-∞, but not 0∞, thenlimx→cPxQx=limx→cP'xQ'x=limx→cP''xQ''x=…Numerical MethodsRiemann SumP0x=abfx dx=limP→0i=1nfxi* ?xiwhere a=x0<x1<x2<…<xn=band ?xi=xi-xi-1 and P=max?xi Types: Left Sum (LHS)Middle Sum (MHS)Right Sum (RHS)Midpoint Rule(Middle Sum)P0x=abfx dx ≈i=1nfxi ?x=?x fx1+fx2+fx3+…+fxnwhere ?x=b-anand xi=12xi-1+xi=midpoint of xi-1,xiError Bounds: EM≤ K(b-a)324n2Trapezoidal RuleP1x=abfx dx ≈?x2fx0+2fx1+2fx3+…+2fxn-1+fxnwhere ?x=b-anand xi=a+i?xError Bounds: ET≤ K(b-a)312n2Simpson’s RuleP2x=abfx dx≈?x3fx0+4fx1+2fx2+4fx3+…+2fxn-2+4fxn-1+fxnWhere n is evenand ?x=b-anand xi=a+i?xError Bounds: ES≤ K(b-a)5180n4TI-84 Plus[MATH] fnInt(f(x),x,a,b), [MATH] [1] [ENTER]Example: [MATH] fnInt(x^2,x,0,1)01x2 dx=13TI-Nspire CAS[MENU] [4] Calculus [3] Integral[TAB] [TAB][X] [^] [2] [TAB][TAB] [X] [ENTER]Shortcut: [ALPHA] [WINDOWS] [4]Integration(See Harold’s Fundamental Theorem of Calculus Cheat Sheet)Basic Integration Rules(Integration is the “inverse” of differentiation, and vice versa.)f'x dx=fx+Cddxfx dx=f(x)fx=00 dx=Cfx=k=kx0k dx=kx+C1. The Constant Multiple Rulek f(x) dx=kf(x) dx2. The Sum and Difference Rule[fx±gx] dx=fx dx±gx dxThe Power Rulefx=Pnx=kxn+…xndx=xn+1n+1+C, where n≠-1If n=-1, thenx-1dx=lnx+CThe General Power RuleIf u=gx, and u'=ddxg(x) thenunu'dx=un+1n+1+C, where n≠-1Reimann Sumi=1nf(ci)?xi, where xi-1≤ci≤xiDefinition of a Definite Integral(Area under curve)lim?→0i=1nfci?xi=abfx dxwhere ?=?x=b-anSwap Boundsabfx dx=-bafx dxAdditive Interval Propertyabfx dx=acfx dx+ cbfx dxFundamental Theorem of Calculusabfx dx=Fb-F(a)Second Fundamental Theorem of Calculusddx axft dt= f(x)ddx agxft dt=fgxg'xddxg(x)h(x)ft dt=fhxh'x- fgxg'(x)Mean Value Theorem for Integralsabfx dx=fcb-a Find ‘c’.Average Value of a Function1b-aabfx dxIntegration Methods1. MemorizedSee Ron Larson’s 1-pager of common integrals2. U-Substitutionfgxg'xdx=Fgx+CSet u=gx, then du=g'x dxfu du=Fu+Cu= _____ du= _____ dx3. Integration by Partsu dv=uv-v du u= _____ v= _____du= _____ dv= _____Pick ‘u’ using the LIATE Rule:L?–?Logarithmic : lnx, logbxI?–?Inverse Trig.: sin-1x, cos-1x, tan-1x, sec-1x, etc.A?–?Algebraic:? x2, 3x60, x, etc.T?–?Trigonometric: sinx, cosx, tanx, secx, cscx, cotxE?–?Exponential:? ex, 19x4. Partial FractionsP(x)Q(x) dxwhere Px and Qx are polynomials.Case 1: If degree of Px≥Qxthen do long division first.Case 2: If degree of Px<Qxthen do partial fraction expansion.5a. Trig Substitution for a2-x2a2-x2 dxSubstutution: x=asinθIdentity: 1-sin2θ=cos2θ5b. Trig Substitution for x2-a2x2-a2 dxSubstutution: x=asecθIdentity: sec2θ-1=tan2θ5c. Trig Substitution for x2+a2x2+a2 dxSubstutution: x=atanθIdentity: tan2θ+1=sec2θ6. Computer Algebra Systems (CAS)TI-Nspire CX CAS Graphing CalculatorTI –Nspire CAS iPad app7. Numerical MethodsRiemann Sum, Midpoint Rule, Trapezoidal Rule, Simpson’s Rule, various quadrature rules, TI-84 Calculator, etc.8. WolframAlphaWolframAlpha is the Google of mathematics. Shows steps. Free.9. AI ChatbotOpenAI ChatGPT, Microsoft Copilot, Google Gemini, etc.Partial Fractions(See Harold’s Partial Fraction Decomposition Cheat Sheets)Conditionfx=PxQxwhere Px and Qx are polynomialsand degree of Px<QxIf degree of Px≥Qx then do long division firstExample ExpansionPxax+bcx+d2ex2+fx+g=A(ax+b)+B(cx+d)+C(cx+d)2+Dx+E(ex2+fx+g) Typical Solutionax+b dx=a lnx+b+CSequences & Series(See Harold’s Series Cheat Sheet)Sequencelimn→∞an=L (Limit)Example: (an, an+1, an+2, …)Geometric SeriesS=limn→∞a(1-rn)1-r =a1-r only if r<1where r is the radius of convergenceand (-r, r) is the interval of convergenceConvergence Tests(See Harold’s Series Convergence Tests Cheat Sheet)Series Convergence TestsDivergence or nth TermGeometric Seriesp-SeriesAlternating SeriesIntegralRatioRootDirect ComparisonLimit ComparisonTelescoping SeriesTaylor Series(See Harold’s Taylor Series Cheat Sheet)Taylor Seriesfx=Pnx+Rnx=n=0∞fn(c)n!(x-c)n+ fn+1(x*)(n+1)!(x-c)n+1where x≤x*≤c (x* is the worst case scenario or max. value of f(x) in the range.)and limx→+∞Rnx=0 ................
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