Integration By Parts
∫udv = uv -∫ [vdu]
Geometric Series
infinite series of ar^n
a= first term, r =common ratio
from n=0 to infinity of ∑ ar^n
Taylor Series Formula
Maclaurin Series Formula
Maclaurin series is a Taylor series but not vice versa
sin(x) Maclaurin Series
cos(x) Maclaurin Series
1(1+x) Maclaurin Series
arctan(x) Maclaurin Series
1/(1-x) Maclaurin Series
ln(1+x) Maclaurin Series
e^x Maclaurin Series
Lagrange Error Bound
Alternating Series Error Bound
Divergence/nth term test
Ratio Test
nth Root Test
Integral Test
Alternating Series Test
P-series Test
Direct Comparison Test
Limit Comparison Test
Absolute Convergence Test
Power Series General Form
In Polar Cords what is x and y equal to?
tanθ = ?
x = rcos(θ) y= rsin(θ)
tan(θ) = y/x
Also
r² = x²+y²
θ = tan-1(y/x)
Area for Polar Curve
A = ½ ∫r²dθ
slope for a polar curve
dy/dx = (dy/dθ)/(dx/dθ)
Slope for a parametric curve
dy/dx = (dy/dt)/(dx/dt)
Second derivative of a parametric curve
Position, Velocity and Acceleration Vectors.
P = <x(t), y(t)>
V =<x’(t), y’(t)>
A = <x’’(t), y’’(t)>
Arc Length Formula
Remember Arc length parametric terms is simply distance formula
Speed Formula
Distance Formula
When is the Particle moving to the left and moving to the right?
Velocity = (-) , left
Velocity =(+), right
When is the particle speeding up/moving away from origin?
When Velocity and Acceleration are the same sign
Logistic Differential Equation General Form
dp/dt = kP(1 - P/C)
P = population at time t
C = carrying capacity
k = constant
Logistic Solution Form
P = c/(1+Ae^-kt)
Limacons Variations and General Equation
r = a±bcosθ or r = a±bsinθ
Inner Loop
Cardioid
Dimpled
Convex
Rose Curves and General Equation
acos(nθ) or asin(nθ)
If n is odd, there are n petals
If n is even, there are 2n petals
First petal positions:
Cos curve - On the x-axis
Sin curve - First Quadrant
Limacon Inner Loop Points
Leminiscate General Equation
r = a²sinθ or a²cosθ
Circle General Equation
r = acosθ or asinθ
Spiral General Equation
r = θ
Infinite Sum
Finite sum
Telescoping Series
First and last term are the only remaining terms within series.
Euler’s Method
Inflection Point/POI
f’’(x) =0 or und and changes sign
Critical Point
f’(x) = 0 or und
Relative Min/Max
Min: f’ changes from (+) to (-) and f’ = 0 or und
f’ changes from (-) to (+) and f’ = 0 or und
What does concave up/down mean?
f’’ > 0 for concave up, f’’<0 for concave down
Tangent Line Eq/ Normal Line eq
Tangent Line: y-y1=m(x-x1)
Norm Line: y-y1=1/m(x-x1)
Riemann Sums
Right hand - overestimate for increasing functions and under for decreasing functions
Vice versa for Left hand
The Trapezoidal sum is the same as the average of the left and right Riemann sums (Goldilocks for estimating)
MVT
continuous on [a,b], and differentiable on open,
then exists x=c where f’(c ) =f(b)-f(a)/(b-a)
EVT
If F(x) is continuous on [a,b], then f has both an abs minimum and an abs max
IVT
If f(x) is continuous on [a,b], and k is between f(a) and f(b), and a<C<b then F(C) = k
Cross Section Volume Identities
Avg Value