Calculus 2 Final Exam Review – Flashcards
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McLaurin Series for 1/(1-x)
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?x?=1+x+x²+x³+x?+... R=1
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McLaurin Series for e^x
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?x?/n!=1+x/1!+x²/2!+x³/3!+... R=infinity
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McLaurin Series for sinx
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?(-1)?(x²??¹)/(2n+1)!=x-x³/3!+x?/5!-x?/7!+... R=infinity
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McLaurin Series for cosx
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?(-1)?(x²?)/(2n)!=1-x²/2!+x?/4!-x?/6!+... R=infinity
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Conditionally Convergent
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A series is ? a_n is called conditionally convergent if it is convergent but not absolutely convergent.
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Absolutely Convergent
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A series ? a_n is called Absolutely convergent if the series of absolute values ? |a_n| is convergent also if the series is absolutely convergent it is also convergent.
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Root Test
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L=1 (nothing) L1 (diverges)
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Ratio Test
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L=1 (nothing) L1 (diverges)
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SA Rotation about x-axis
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f(x)
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SA Rotating about y-axis
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x
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Integral Test Remainder Estimate
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For integral remainder 1/sqru(n^4+1) < 1/Sq(n^4) = 1/n^2 take integral of last one and b=inf and a=10
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The Comparison Test
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(i) If ?bn is convergent and an?bn for all n, then ? an is also convergent. (ii) If ?bn is divergent and an?bn for all n, then ? an is also divergent .
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Washer Method
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pi?(outer radius)²-(inner radius)² (x-axis)
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Shell Method
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2pi?rh (y-axis)
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Integration by Parts
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uv-?vdu
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Arc Length Formula
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??(1+(dy/dx)^2)
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?lnu
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ulnu-u
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sin2x
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2sinxcosx
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sin²x
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(1-cos2x)/2
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cos²x
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(1+cos2x)/2
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?secxdx
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ln|secx+tanx|
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Derivative of secx
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secxtanx
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Derivative of tanx
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sec²x
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Derivative of cscx
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-cscxcotx
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Derivative of cotx
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-csc²x
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SA Rotating about y-axis
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x
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Length of Polar Curve
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??(r²+(dr/d?)²)d?
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derivative of sinx
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cosx
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derivative of cosx
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-sinx
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integral of sinx
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-cosx
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integral of cosx
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sinx
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Sin(A) Sin(B)
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1/2[Cos(A-B)-Cos(A+B)]
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Cos(A) Cos(B)
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1/2[Cos(A-B)+Cos(A+B)]
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Sin(A) Cos(B)
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1/2[Sin(A-B)+Sin(A+B)]
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f average
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(1/B-A)?f(x)dx
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Area
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b ?[f(x)-g(x)]dx a
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?(a²-x²)
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x=aSin?
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?(x²-a²)
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x=aSec?
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?(a²+x²)
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x=aTan?
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Surface area
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b S=? 2? f(x) ?(1+[f'(x)]²) dx a
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Absolutely Convergent
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A series ? a_n is called Absolutely convergent if the series of absolute values ? |a_n| is convergent also if the series is absolutely convergent it is also convergent.
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Conditionally Convergent
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A series is ? a_n is called conditionally convergent if it is convergent but not absolutely convergent.
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Limit of arctan(n) n-->?
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?/2
