Calculus Bible.pdf - Calculus AB Bible(2nd most important...

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1 Calculus AB Bible(2nd most important book in the world)(Written and compiled by Doug Graham)PG.Topic24Limits5Continuity56Derivative by Definition78Derivative Formulas910Related Rates11 Properties of Derivatives12 Applications of Derivatives1314Optimization Problems1517Integrals/Substitution17 Properties of Logarithms18 Newton's method18 Separating variables18 Average Value19 Continuity/Differentiability Problem19 Rectilinear Motion20 Mean Value Theorem20 Studying the graph of fx()21 Trigonometric Identities21 Growth, Double-Life, and Half-Life formulas 2224Applications of the Integral Area, Volume, and Sums()2627Approximating Area Trapezoidal Rule, Left Endpoint, Right Endpoint, and Midpoint()28 1st Fundamental Theorem of Calculus28 2nd Fundamental Theorem of Calculus28 Integral as an accumulator29Finding Derivatives and Integrals given a graph of f(x)30Integration by Parts
2 LIMITS** When evaluating limits, we are checking around the point that we are approaching, NOT at the point.**Every time we find a limit, we need to check from the left and the right hand side (Only if there is a BREAK at that point).**Breaking Points are points on the graph that are undefined or where the graph is split into pieces.Breaking Points :1)Asymptotes (when the denominator equals 0)2)Radicals (when the radical equals 0)3)Holes (when the numerator and denominator equals 0)4)Piece-wise functions (the # where the graph is split)limxa+fx( )=right hand limit limxafx( )=left hand limit**If left and right hand limits DISAGREE, then the limit Does Not Exist (DNE) at that point. **If left and right hand limits AGREE, then the limit exists at that point as that value.**Even if you can plug in the value, the limit might not exist at that point. It might not exist from the left or right side or the two sides will not agree.For example: fx( )=3 for x11 for x<1limx1fx( )=DNEbecause limx1+fx( )=3 and limx1fx( )=1Note : In general when doing limits,#x0=#x0=−∞#x→ ∞=0LIMITSLIMITS AT NON -BREAKING POINTSVery easy. Just plug in the#()EX#1:limx1x3+x5=3 EX#2 :limx2x+7=9= 3 EX#3 :limx12x1x+1=12HOLES IN THE GRAPH00()Factor and cancel or multiply by the conjugate and cancel, then plug in #()EX#1:limx2x2+3x10x2=limx2x+5()x2()x2=limx2x+5()=7 EX#2 :limx→−2x+113x+2=limx→−2x+113()x+11+3()x+2()x+11+3()=limx→−2x+21x+2()1x+11+3()=16
3 RADICALSYou must first check that the limit exists on the side(s) you are checking()If a # makes a radical negative, the limit will not exist at that #.When we check at the breaking point (the # that makes the radical zero) there are two possible answers:1)0 if the limit works from the side that you are checking.

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