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Harold’s Calculus NotesCheat Sheet4 April 2020AP CalculusLimitsDefinition of LimitLet f be a function defined on an open interval containing c and let L be a real number. The statement:limx→afx=Lmeans that for each ?>0 there exists a δ>0 such that if 0<x-a<δ,then fx-L<?Tip?:Direct substitution: Plug in fa and see if it provides a legal answer. If so then L = fa.The Existence of a LimitThe limit of fx as x approaches a is L if and only if:limx→a-fx=Llimx→a+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+cx-c=x+c x-c Since x+c≤ 2cfx-fc≤2cx-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=1limx→01-cosxx=0Derivatives(See Larson’s 1-pager of common derivatives)Definition of a Derivative of a Function Slope Functionf'x=limh→0fx+h-fxhf'c=limx→cfx-f(c)x-cNotation for Derivativesf'x, f(n)x, dydx, y', ddxfx, Dx[y]0. The Chain Ruleddxfgx=f'gxg'xdydx=dydt·dtdx1. The Constant Multiple Ruleddxcf(x)=cf'(x)2. The Sum and Difference Ruleddxfx±g(x)=f'(x)±g'(x)3. The Product Ruleddxfg=fg'+g f'4. The Quotient Ruleddxfg=gf'-fg'g25. The Constant Ruleddxc=06a. The Power Ruleddxxn=nxn-16b. The General Power Ruleddxun=nun-1 u' where u=u(x)7. The Power Rule for xddxx=1 (think x=x1 and x0=1)8. Absolute Valueddxx=xx9. Natural Logorithmddxlnx=1x10. Natural Exponentialddxex=ex11. Logorithmddxlogax=1(lna) x12. Exponentialddxax=(lna)ax13. Sineddxsinx=cosx14. Cosineddxcos(x)=-sin(x)15. Tangentddxtanx=sec2x16. Cotangentddxcot(x)=-csc2(x)17. Secantddxsecx=secxtan(x)Derivatives(See Larson’s 1-pager of common derivatives)18. Cosecantddxcsc(x)=-cscxcot(x)19. Arcsineddxsin-1(x)=11-x220. Arccosineddxcos-1(x)=-11-x221. Arctangentddxtan-1(x)=11+x222. Arccotangentddxcot-1(x)=-11+x223. Arcsecantddxsec-1(x)=1x x2-124. Arccosecantddxcsc-1(x)=-1x x2-125. Hyperbolic Sine ex-e-x2ddxsinhx=coshx26. Hyperbolic Cosine ex+e-x2ddxcosh(x)=sinh(x)27. Hyperbolic Tangentddxtanhx=sech2x28. Hyperbolic Cotangentddxcoth(x)=-csch2(x)29. Hyperbolic Secantddxsechx=-sechxtanh(x)30. Hyperbolic Cosecantddxcsch(x)=-cschxcoth(x)31. Hyperbolic Arcsineddxsinh-1(x)=1x2+132. Hyperbolic Arccosineddxcosh-1(x)=1x2-133. Hyperbolic Arctangentddxtanh-1(x)=11-x234. Hyperbolic Arccotangentddxcoth-1(x)=11-x235. Hyperbolic Arcsecantddxsech-1(x)=-1x 1-x236. Hyperbolic Arccosecantddxcsch-1(x)=-1x 1+x2Position Functionst=12gt2+v0t+s0Velocity Functionvt=s't=gt+v0Acceleration Functionat=v't=s''tJerk Functionjt=a't=v''t=s(3)tAnalyzing 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 (concavity changes)Graphing 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) →The 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 neitherThe Second Deriviative Test Let 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 fails (See 1st derivative test)Test for Concavity1. If f''x>0 for all x, then the graph is concave up ?2. If f''x<0for all x, then the graph is concave down ?Points of Inflection Change in concavityIf c,f(c) is a point of inflection of fx, then either1. f''c=0 or2. f''x does not exist at x = cTangent LinesGenreal Formax+by+c=0Slope-Intercept Formy=mx+bPoint-Slope Formy-y0=m(x-x0)Calculus Formy=f'cx-c+fcSlopem=riserun=?y?x=y2-y1x2-x1=dydx=f'xDifferentiation & DifferentialsRolle’s Theoremf 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 Theoremf is a continuous function with an 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+?y=fx+f'x ?xdy=f'x dx so ?y=f'x ?xRelative Error=?ff in %Example: 482→fx=4x, fx+?x=f81+1Newton’s Method Finds zeros of f, or finds c if f(c) = 0.xn+1=xn-fxnf'xnExample: 482→fx=x4-82=0, xn=3Related RatesSteps to solve:Identify the known variables and rates of change.x=15 m; y=20 m; x'=2ms; y'= ?Construct an equation relating these quantities.x2+y2=r2Differentiate 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=32 msL’H?pital’s RuleIf limx→cfx=limx→cPxQx and00,∞∞,0?∞,1∞,00,∞0,∞-∞, but not 0∞,then limx→cPxQx=limx→cP'xQ'x=limx→cP''xQ''x=…Summation FormulasSum of Powersi=1nc=cni=1ni=n(n+1)2=n22+n2i=1ni2=n(n+1)(2n+1)6=n33+n22+n6i=1ni3=i=1ni2=n2(n+1)24=n44+n32+n24i=1ni4=n(n+1)(2n+1)(3n2+3n-1)30=n55+n42+n33-n30i=1ni5=n2(n+1)2(2n2+2n-1)12=n66+n52+5n412-n212i=1ni6=n(n+1)(2n+1)(3n4+6n3-3n+1)42i=1ni7=n2n+123n4+6n3-n2-4n+224Skn=i=1nik=n+1k+1k+1-1k+1r=0k-1k+1rSr(n)Interesting Summation Formulasi=1ni(i+1)=i=1ni2+i=1ni=n(n+1)(n+2)3i=1n1i(i+1)=nn+1i=1n1i(i+1)(i+2)=n(n+3)4(n+1)(n+2)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=?xfx1+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=abfxdx ≈?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 RulesIntegration 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=kxnxndx=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≤xi?=?x=b-anDefinition of a Definite IntegralArea under curvelim?→0i=1nf(ci)?xi=abfx dxSwap Boundsabfx dx=-bafx dxAdditive Interval Propertyabfx dx=acfx dx+ cbfx dxThe Fundamental Theorem of Calculusabfx dx=Fb-F(a)The 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’.The Average Value for a Function1b-aabfx dxIntegration Methods1. MemorizedSee 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 LIATED Rule:L?–?Logarithmic : lnx, logbxI?–?Inverse Trig.: tan-1x, sec-1x, etc.A?–?Algebraic:? x2, 3x60, etc.T?–?Trigonometric: sinx, tanx, etc.E?–?Exponential:? ex, 19xD?– Derivative of:? dydx4. Partial FractionsP(x)Q(x) dxwhere Px and Qx are polynomialsCase 1: If degree of Px≥Qxthen do long division firstCase 2: If degree of Px<Qxthen do partial fraction expansion5a. 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. Table of IntegralsCRC Standard Mathematical Tables book7. Computer Algebra Systems (CAS)TI-Nspire CX CAS Graphing CalculatorTI –Nspire CAS iPad app8. Numerical MethodsRiemann Sum, Midpoint Rule, Trapezoidal Rule, Simpson’s Rule, TI-84, etc.9. WolframAlphaGoogle of mathematics. Shows steps. Free.Partial Fractions(See Harold’s Partial Fractions Cheat Sheet)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 (worst case scenario x*)and limx→+∞Rnx=0 ................
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