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AP CALC BC
lim𝑥→𝑎
𝑓 (𝑥 )𝑔(𝑥 )
=lim𝑥→𝑎
𝑓 ′ (𝑥)𝑔 ′ (𝑥 )
Length of a CurveIf the function is continuous and differentiable on , then the length of the curve from to is…
𝐿=∫𝑎
𝑏
√1+(𝑑𝑦𝑑𝑥 )2
𝑑𝑥
Length of a CurveThe length of the
curve from to is…
𝐿=∫𝑐
𝑑
√1+( 𝑑𝑥𝑑𝑦 )2
𝑑𝑦
The formula for the length of a curve defined parametrically from to is…
𝐿=∫𝑎
𝑏
√(𝑑𝑥𝑑𝑡 )2
+( 𝑑𝑦𝑑𝑡 )2
𝑑𝑡
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 made…
𝑑𝑦= 𝑓 ′ (𝑥 ) 𝑑𝑥
𝑑𝑑𝑥ln𝑢=¿ ¿
1𝑢𝑑𝑢𝑑𝑥
∫ 1𝑢𝑑𝑢=¿¿
𝑙𝑛|𝑢|+𝐶
∫ 𝑙𝑛𝑥𝑑𝑥=¿¿
𝑥𝑙𝑛𝑥−𝑥+𝐶
When converting Cartesian coordinates to Polar coordinates, what does equal?
𝑥=𝑟𝑐𝑜𝑠𝜃
When converting Cartesian coordinates to Polar coordinates, what does equal?
𝑦=𝑟𝑠𝑖𝑛 𝜃
When converting Polar coordinates to Cartesian coordinates, what does equal?
𝑟=±√𝑥2+𝑦2
When converting Polar coordinates to Cartesian coordinates, what does equal?
𝜃=tan− 1𝑦𝑥
GRAPHING POLAR CURVES
The equation of a circle centered at the origin is…
𝑟=𝐴
GRAPHING POLAR CURVES
The equation of a spiral is…
Or
GRAPHING POLAR CURVES
The equation of a rose curve is…
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 is…
Or
GRAPHING POLAR CURVES
If a limacon curve is heart shaped (cardioid) then…
𝐴=𝐵
GRAPHING POLAR CURVES
If a limacon curve has an inner loop then…
𝐴<𝐵
GRAPHING POLAR CURVES
If a limacon curve is dimpled then…
𝐵<𝐴<2𝐵
The formula for finding the slope of a polar curve is…
𝑑𝑦𝑑𝑥
=𝑓 ′ (𝜃 )𝑠𝑖𝑛𝜃+ 𝑓 (𝜃 )𝑐𝑜𝑠𝜃𝑓 ′ (𝜃 )𝑐𝑜𝑠 𝜃− 𝑓 (𝜃 )𝑠𝑖𝑛𝜃
A series of the form
is called a…
𝑝−𝑠𝑒𝑟𝑖𝑒𝑠
Harmonic Series
Does the harmonic series converge or diverge?
The harmonic series diverges!
The series
is known as….
The Alternating Harmonic Series.
Does the alternating harmonic series converge or diverge?
The alternating harmonic series
converges!
A p-series
converges if…
𝑝>1
A p-series
diverges if…
𝑝 ≤1
A series of the form
is called a…
Geometric series
THE GEOMETRIC SERIES TEST
A geometric series converges if and only if…
|𝑟|<1
Sum of an Infinite Geometric Series
𝑆=𝑎11−𝑟
THE nth TERM TEST
h𝑡 𝑒𝑛∑𝑎𝑛𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑠
The Ratio Test for Convergence of a
Series…
(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.Given…
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:
∑𝑘=0
∞ 𝑓 𝑘 (𝑎 )𝑘!
(𝑥−𝑎)𝑘= 𝑓 (𝑎)+ 𝑓 ′ (𝑎 ) (𝑥−𝑎 )+ 𝑓′ ′ (𝑎)2 !
¿
The Maclaurin Series is…
A special case of the Taylor Series for
∑𝑘=0
∞ 𝑓 𝑘 (0 )𝑘 !
(𝑥)𝑘= 𝑓 (0 )+ 𝑓 ′ (0 ) (𝑥 )+ 𝑓 ′ ′ (0 )2 !
¿
The Taylor polynomial expansion for is…
1+𝑥+ 𝑥2
2 !+ 𝑥
3
3 !+…
The Taylor polynomial expansion for is…
x −𝑥3
3 !+𝑥
5
5 !+…
The Taylor polynomial expansion for is…
1−𝑥2
2!+ 𝑥
4
4 !+…
The Taylor polynomial expansion for is…
x −𝑥2
2!+ 𝑥
3
3 !−𝑥4
4 !+…
Harmonic Series
∑𝑛=1
∞ (−1)𝑛+1
𝑛
Alternating Harmonic
Series
∑𝑛=0
∞
𝑎𝑟 𝑛
Geometric Series
Converge or Diverge?
