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Copyright Β© 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:limπ‘₯β†’??(?) = 𝐿means that for eachπœ– > 0there exists a? > 0such thatif0 < |? βˆ’ ?| < ?, then |?(?) βˆ’ 𝐿| < πœ–Tip :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:limπ‘₯β†’?βˆ’?(?) = 𝐿limπ‘₯β†’?+?(?) = 𝐿Definition of ContinuityA functionf is continuousat c if forevery? > 0there exists a? > 0such that|? βˆ’ ?| < ?and|?(?) βˆ’ ?(?)| < ?.Tip: Rearrange|?(?) βˆ’ ?(?)|to have|(? βˆ’ ?)|as a factor.Since|? βˆ’ ?| < ?wecan find an equation that relates both?and?together.Prove that?(𝒙) = 𝒙?βˆ’ ?is a continuous function.|?(?) βˆ’ ?(?)|= |(?2βˆ’ 1) βˆ’ (?2βˆ’ 1)|= |?2βˆ’ 1βˆ’ ?2+ 1|= |?2βˆ’ ?2|= |(? + ?)(? βˆ’ ?)|= |(? + ?)| |(? βˆ’ ?)|Since|(? + ?)| ≀ |2?||?(?) βˆ’ ?(?)| ≀ |2?||(? βˆ’ ?)| < ?So given? > 0, we canchoose? = |???| ? > ?in theDefinition of Continuity.So substituting the chosen?for|(? βˆ’ ?)|we get:|?(?) βˆ’ ?(?)| ≀ |2?| (|12?| ?) = ?Since both conditions are met,?(?)is continuous.Two Special Trig Limits???π‘₯β†’0??? ??= 1???π‘₯β†’01 βˆ’ ??? ??= 0
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Copyright Β© 2015 by Harold Toomey, WyzAnt Tutor2Derivatives(See Larson’s 1-pager of common derivatives)Definition of a Derivative of a FunctionSlope Function?β€²(?) = limβ„Žβ†’0?(? + β„Ž) βˆ’ ?(?)β„Ž?β€²(?) = limπ‘₯β†’??(?) βˆ’ ?(?)? βˆ’ ?Notation for Derivatives?β€²(?), ?(𝑛)(?),????, ?β€²,???[?(?)], ?π‘₯[?]The Constant Rule???[?] = 0The Power Rule???[?𝑛] = ??π‘›βˆ’1???[?] = 1 (?β„Ž??? ? = ?1??? ?0= 1)The General Power Rule???[?𝑛] = ??π‘›βˆ’1?β€²?β„Ž??? ? = ?(?)The Constant Multiple Rule???[??(?)] = ??β€²(?)The Sum and Difference Rule???[?(?) Β± ?(?)] = ?β€²(?) Β± ?β€²(?)Position Function?(?) =12??2+ ?0? + ?0Velocity Function?(?) = ?β€²(?) = ?? + ?0Acceleration Function?(?) = ?β€²(?) = ?β€²β€²(?)Jerk Function?(?) = ?β€²(?) = ?β€²β€²(?) = ?(3)(?)The Product Rule???[??] = ??β€²+ ? ?β€²The Quotient Rule???[??] =??β€²βˆ’ ??β€²?2The Chain Rule???[?(?(?))] = ?β€²(?(?))?β€²(?)????=????Β·????Exponentials(?𝒙, 𝒂𝒙)???[?π‘₯] = ?π‘₯,???[?π‘₯] = (ln ?) ?π‘₯Logorithms(π₯? 𝒙 , π₯?𝐠𝒂𝒙)???[ln ?] =1?,???[log??] =1(ln ?) ?Sine???[???(?)] = cos(?)Cosine???[???(?)] = βˆ’???(?)Tangent???[???(?)] = ???2(?)Secent???[???(?)] = ???(?) ???(?)Cosecent???[???(?)] = βˆ’ ???(?) ???(?)Cotangent???[???(?)] = βˆ’???2(?)
Copyright Β© 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?β€²(?) =?(?) βˆ’ ?(?)? βˆ’ ??(?) = ?(?) + (? βˆ’ ?)?β€²(?)Find β€˜c’.L’HΓ΄pital’s Rule𝐼? limπ‘₯β†’??(?) = limπ‘₯β†’??(?)?(?)={00,∞∞, 0 β€’ ∞, 1∞, 00, ∞0, ∞ βˆ’ ∞} , ??? ??? {0∞},?β„Ž?? limπ‘₯β†’??(?)?(?)= limπ‘₯β†’??β€²(?)?β€²(?)= limπ‘₯β†’??β€²β€²(?)?β€²β€²(?)= β‹―Graphing with DerivativesTest for Increasing and DecreasingFunctions1.Iff’(x) > 0, thenfis increasing (slope up)β†—2.Iff’(x)< 0, thenfis decreasing (slope down)β†˜3.Iff’(x)= 0, thenfis constant (zero slope)β†’The First Derivative Test1.Iff’(x)changes from–to + 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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