This document contains a summary of key concepts in algebra, geometry, and trigonometry:
1) Algebra topics include arithmetic operations, factoring, exponents, binomials, and the quadratic formula.
2) Geometry topics cover lines, triangles, circles, spheres, cones, cylinders, sectors, and trapezoids including formulas for area, perimeter, volume, and surface area.
3) Trigonometry definitions and formulas are provided for sine, cosine, tangent, cotangent, addition, subtraction, and half-angle identities.
A crash coarse in stochastic Lyapunov theory for Markov processes (emphasis is on continuous time)
See also the survey for models in discrete time,
https://netfiles.uiuc.edu/meyn/www/spm_files/MarkovTutorial/MarkovTutorialUCSB2010.html
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
A crash coarse in stochastic Lyapunov theory for Markov processes (emphasis is on continuous time)
See also the survey for models in discrete time,
https://netfiles.uiuc.edu/meyn/www/spm_files/MarkovTutorial/MarkovTutorialUCSB2010.html
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
Scientific Computing with Python Webinar 9/18/2009:Curve FittingEnthought, Inc.
This webinar will provide an overview of the tools that SciPy and NumPy provide for regression analysis including linear and non-linear least-squares and a brief look at handling other error metrics. We will also demonstrate simple GUI tools that can make some problems easier and provide a quick overview of the new Scikits package statsmodels whose API is maturing in a separate package but should be incorporated into SciPy in the future.
1. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH
GTBL001-front˙end GTBL001-Smith-v16.cls October 17, 2005 20:2
Algebra Geometry
Arithmetic Triangle Circle
a+b a b a c ad + bc Area = 1 bh Area = πr 2
= + + = 2
c c c b d bd c2 = a 2 + b2 − 2ab cos θ C = 2πr
a
b a d ad
= = a
c b c bc c r
h
d
b
Factoring
x 2 − y 2 = (x − y)(x + y) x 3 − y 3 = (x − y)(x 2 + x y + y 2 ) Sector of a Circle Trapezoid
x 3 + y 3 = (x + y)(x 2 − x y + y 2 ) x 4 − y 4 = (x − y)(x + y)(x 2 + y 2 )
Area = 1 r 2 θ
2 Area = 1 (a + b)h
2
s = rθ
Binomial (for θ in radians only) a
(x + y)2 = x 2 + 2x y + y 2 (x + y)3 = x 3 + 3x 2 y + 3x y 2 + y 3
h
s
Exponents b
xn
xn xm = x n+m = x n−m (x n )m = x nm
xm r
n
1 x xn
x −n = n (x y)n = x n yn = n
x y y Sphere Cone
√
√ √ √ √ x x n
x n/m = m
xn n xy = n
xny n = √ Volume = 4 πr 3 Volume = 1 πr 2 h
y n y
3 3 √
Surface Area = 4πr 2 Surface Area = πr r 2 + h 2
Lines
Slope m of line through (x0 , y0 ) and (x1 , y1 )
r
h
y1 − y0
m=
x1 − x0
r
Through (x0 , y0 ), slope m
y − y0 = m(x − x0 )
Slope m, y-intercept b
Cylinder
y = mx + b Volume = πr 2 h
Surface Area = 2πr h
Quadratic Formula
If ax 2 + bx + c = 0 then
√
−b ± b2 − 4ac h
x=
2a r
Distance
Distance d between (x1 , y1 ) and (x2 , y2 )
d= (x2 − x1 )2 + (y2 − y1 )2
1
2. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH
GTBL001-front˙end GTBL001-Smith-v16.cls October 17, 2005 20:2
Trigonometry
(x, y) sin θ =
y Half-Angle
r
1 − cos 2θ 1 + cos 2θ
r sin2 θ = cos2 θ =
x 2 2
cos θ =
r
y
Addition
tan θ =
x sin(a + b) = sin a cos b + cos a sin b cos(a + b) = cos a cos b − sin a sin b
Subtraction
sin θ =
opp sin(a − b) = sin a cos b − cos a sin b cos(a − b) = cos a cos b + sin a sin b
hyp
hyp adj Sum
opp cos θ =
hyp u+v u−v
sin u + sin v = 2 sin cos
2 2
opp u+v u−v
tan θ = cos u + cos v = 2 cos cos
adj 2 2
adj
Product
sin u sin v = 1 [cos(u − v) − cos(u + v)]
2
Reciprocals cos u cos v = 1 [cos(u − v) + cos(u + v)]
2
1 1 1 sin u cos v = 1 [sin(u + v) + sin(u − v)]
2
cot θ = sec θ = csc θ =
tan θ cos θ sin θ cos u sin v = 1 [sin(u + v) − sin(u − v)]
2
Definitions π/2
2π/3 π/3
cos θ 1 1 π/4
cot θ = sec θ = csc θ = 3π/4
sin θ cos θ sin θ
5π/6 π/6
Pythagorean π 0
sin2 θ + cos2 θ = 1 tan2 θ + 1 = sec2 θ 1 + cot2 θ = csc2 θ Radians
sin(0) = 0 cos(0) = 1
√
π π
Cofunction sin 6 = 1
2 cos 6 = 2
3
√ √
π π
π π π sin = 2
cos = 2
sin 2 − θ = cos θ cos 2 − θ = sin θ tan 2 − θ = cot θ 4 2 4 2
√
π π
sin 3 = 2
3
cos 3 = 1
2
π π
sin =1 cos =0
Even/Odd 2
√
2
sin 2π
3 = 2
3
cos 2π
3 = −1
2
sin(−θ ) = −sin θ cos(−θ) = cos θ tan(−θ ) = −tan θ √ √
sin 3π
4 = 2
2
cos 3π
4 =− 2
2
√
sin 5π
6 = 1
2 cos 5π
6 = − 23
Double-Angle sin(π) = 0 cos(π ) = −1
sin 2θ = 2 sin θ cos θ cos 2θ = cos2 θ − sin2 θ cos 2θ = 1 − 2 sin2 θ sin(2π) = 0 cos(2π) = 1
2