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Copyright © 2015 by Harold Toomey, WyzAnt Tutor1Harold’sCalculus NotesCheat Sheet15 December 2015AP CalculusLimitsDefinition of LimitLetfbe a function defined on an openinterval containing c and letLbe a realnumber.The statement:lim𝑥→??(?) = 𝐿means that for each𝜖 > 0there exists a? > 0such thatif0 < |? − ?| < ?, then |?(?) − 𝐿| < 𝜖Tip :Direct substitution: Plug in?(?)and see ifit provides a legal answer.If so thenL =?(?).The Existence of a LimitThe limit of?(?)as?approachesaisLifand only if:lim𝑥→??(?) = 𝐿lim𝑥→?+?(?) = 𝐿Definition of ContinuityA functionf is continuousat c if forevery? > 0there exists a? > 0such that|? − ?| < ?and|?(?) − ?(?)| < ?.Tip: Rearrange|?(?) − ?(?)|to have|(? − ?)|as a factor.Since|? − ?| < ?wecan find an equation that relates both?and?together.Prove that?(𝒙) = 𝒙?− ?is a continuous function.|?(?) − ?(?)|= |(?2− 1) − (?2− 1)|= |?2− 1− ?2+ 1|= |?2− ?2|= |(? + ?)(? − ?)|= |(? + ?)| |(? − ?)|Since|(? + ?)| ≤ |2?||?(?) − ?(?)| ≤ |2?||(? − ?)| < ?So given? > 0, we canchoose? = |???| ? > ?in theDefinition of Continuity.So substituting the chosen?for|(? − ?)|we get:|?(?) − ?(?)| ≤ |2?| (|12?| ?) = ?Since both conditions are met,?(?)is continuous.Two Special Trig Limits???𝑥→0??? ??= 1???𝑥→01 − ??? ??= 0
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Copyright © 2015 by Harold Toomey, WyzAnt Tutor2Derivatives(See Larson’s 1-pager of common derivatives)Definition of a Derivative of a FunctionSlope Function?(?) = limℎ→0?(? + ℎ) − ?(?)?(?) = lim𝑥→??(?) − ?(?)? − ?Notation for Derivatives?(?), ?(𝑛)(?),????, ?,???[?(?)], ?𝑥[?]The Constant Rule???[?] = 0The Power Rule???[?𝑛] = ??𝑛−1???[?] = 1 (?ℎ??? ? = ?1??? ?0= 1)The General Power Rule???[?𝑛] = ??𝑛−1??ℎ??? ? = ?(?)The Constant Multiple Rule???[??(?)] = ??(?)The Sum and Difference Rule???[?(?) ± ?(?)] = ?(?) ± ?(?)Position Function?(?) =12??2+ ?0? + ?0Velocity Function?(?) = ?(?) = ?? + ?0Acceleration Function?(?) = ?(?) = ?′′(?)Jerk Function?(?) = ?(?) = ?′′(?) = ?(3)(?)The Product Rule???[??] = ??+ ? ?The Quotient Rule???[??] =??− ???2The Chain Rule???[?(?(?))] = ?(?(?))?(?)????=????·????Exponentials(?𝒙, 𝒂𝒙)???[?𝑥] = ?𝑥,???[?𝑥] = (ln ?) ?𝑥Logorithms(𝐥? 𝒙 , 𝐥?𝐠𝒂𝒙)???[ln ?] =1?,???[log??] =1(ln ?) ?Sine???[???(?)] = cos(?)Cosine???[???(?)] = −???(?)Tangent???[???(?)] = ???2(?)Secent???[???(?)] = ???(?) ???(?)Cosecent???[???(?)] = − ???(?) ???(?)Cotangent???[???(?)] = −???2(?)
Copyright © 2015 by Harold Toomey, WyzAnt Tutor3Applications of DifferentiationRolle’s Theoremfis continuous on the closed interval [a,b],andfis differentiable on the open interval (a,b).Iff(a) = f(b), then there exists at least one numbercin(a,b)such thatf’(c) = 0.Mean Value TheoremIffmeets the conditions of Rolle’sTheorem, then?(?) =?(?) − ?(?)? − ??(?) = ?(?) + (? − ?)?′(?)Find ‘c’.L’Hôpital’s Rule𝐼? lim𝑥→??(?) = lim𝑥→??(?)?(?)={00,, 0 • ∞, 1, 00, ∞0, ∞ − ∞} , ??? ??? {0},?ℎ?? lim𝑥→??(?)?(?)= lim𝑥→??(?)?(?)= lim𝑥→??′′(?)?′′(?)= ⋯Graphing with DerivativesTest for Increasing and DecreasingFunctions1.Iff’(x) > 0, thenfis increasing (slope up)2.Iff’(x)< 0, thenfis decreasing (slope down)3.Iff’(x)= 0, thenfis constant (zero slope)The First Derivative Test1.Iff’(x)changes fromto + atc, thenfhas arelativeminimumat(c, f(c))2.Iff’(x)changes from + to - atc, thenfhas arelativemaximumat(c, f(c))3.Iff’(x), is +c+ or -c-, thenf(c)is neitherThe Second Deriviative TestLetf’(c)=0, andf”(x)exists, then1.

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