Transcript
AP CALC BC
limπ₯βπ
π (π₯ )π(π₯ )
=limπ₯βπ
π β² (π₯)π β² (π₯ )
Length of a CurveIf the function is continuous and differentiable on , then the length of the curve from to isβ¦
πΏ=β«π
π
β1+(ππ¦ππ₯ )2
ππ₯
Length of a CurveThe length of the
curve from to isβ¦
πΏ=β«π
π
β1+( ππ₯ππ¦ )2
ππ¦
The formula for the length of a curve defined parametrically from to isβ¦
πΏ=β«π
π
β(ππ₯ππ‘ )2
+( ππ¦ππ‘ )2
ππ‘
Integration by Parts Formula
Differentials Formula
Differentials Formula to estimate the error in a measurement, or to find the effect on a formula when a small change in measurement is madeβ¦
ππ¦= π β² (π₯ ) ππ₯
πππ₯lnπ’=ΒΏ ΒΏ
1π’ππ’ππ₯
β« 1π’ππ’=ΒΏΒΏ
ππ|π’|+πΆ
β« πππ₯ππ₯=ΒΏΒΏ
π₯πππ₯βπ₯+πΆ
When converting Cartesian coordinates to Polar coordinates, what does equal?
π₯=ππππ π
When converting Cartesian coordinates to Polar coordinates, what does equal?
π¦=ππ ππ π
When converting Polar coordinates to Cartesian coordinates, what does equal?
π=Β±βπ₯2+π¦2
When converting Polar coordinates to Cartesian coordinates, what does equal?
π=tanβ 1π¦π₯
GRAPHING POLAR CURVES
The equation of a circle centered at the origin isβ¦
π=π΄
GRAPHING POLAR CURVES
The equation of a spiral isβ¦
Or
GRAPHING POLAR CURVES
The equation of a rose curve isβ¦
Or
*If B is an odd it is equal to the number of petals*If B is even, double it to get the number of petals
*A is the length of the pedals
GRAPHING POLAR CURVES
The equation of a limacon curve isβ¦
Or
GRAPHING POLAR CURVES
If a limacon curve is heart shaped (cardioid) thenβ¦
π΄=π΅
GRAPHING POLAR CURVES
If a limacon curve has an inner loop thenβ¦
π΄<π΅
GRAPHING POLAR CURVES
If a limacon curve is dimpled thenβ¦
π΅<π΄<2π΅
The formula for finding the slope of a polar curve isβ¦
ππ¦ππ₯
=π β² (π )π πππ+ π (π )πππ ππ β² (π )πππ πβ π (π )π πππ
A series of the form
is called aβ¦
πβπ πππππ
Harmonic Series
Does the harmonic series converge or diverge?
The harmonic series diverges!
The series
is known asβ¦.
The Alternating Harmonic Series.
Does the alternating harmonic series converge or diverge?
The alternating harmonic series
converges!
A p-series
converges ifβ¦
π>1
A p-series
diverges ifβ¦
π β€1
A series of the form
is called aβ¦
Geometric series
THE GEOMETRIC SERIES TEST
A geometric series converges if and only ifβ¦
|π|<1
Sum of an Infinite Geometric Series
π=π11βπ
THE nth TERM TEST
hπ‘ ππβπππππ£πππππ
The Ratio Test for Convergence of a
Seriesβ¦
(a) If the series converges.(b) If the series diverges.(c) If the test fails.
*Often helpful with series that involving exponentials or factorials.
The Alternating Series Test for Convergence of a Series.Givenβ¦
If
then the alternating series converges.
ROOT TEST
*Helpful with series raise to the nth power
If a function has derivatives of all orders at , then the Taylor series for about is:
βπ=0
β π π (π )π!
(π₯βπ)π= π (π)+ π β² (π ) (π₯βπ )+ πβ² β² (π)2 !
ΒΏ
The Maclaurin Series isβ¦
A special case of the Taylor Series for
βπ=0
β π π (0 )π !
(π₯)π= π (0 )+ π β² (0 ) (π₯ )+ π β² β² (0 )2 !
ΒΏ
The Taylor polynomial expansion for isβ¦
1+π₯+ π₯2
2 !+ π₯
3
3 !+β¦
The Taylor polynomial expansion for isβ¦
x βπ₯3
3 !+π₯
5
5 !+β¦
The Taylor polynomial expansion for isβ¦
1βπ₯2
2!+ π₯
4
4 !+β¦
The Taylor polynomial expansion for isβ¦
x βπ₯2
2!+ π₯
3
3 !βπ₯4
4 !+β¦
Harmonic Series
βπ=1
β (β1)π+1
π
Alternating Harmonic
Series
βπ=0
β
ππ π
Geometric Series
Converge or Diverge?
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