Cheatography
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Fraction Rules
Common Denominators
a/b + c/d = ad + bc/bd
Multiplication
a/b * c/d = ac/bd
Reciprocal
1/(a/b) = b/a 
Theory of Geometric Series
S = p + pr + pr^{2} + pr^{3} + ... + pr^{n}
Sr = p + pr + pr^{2} + pr^{3} + ... + pr^{n} + pr^{n+1}
S  Sr = p  pr^n+1
S = ppr^{n+1}/1r 
Lines
Given point (c,d) and slope m, the unique line that satisfies this is the set of all points (x,y) such that
m = yd/xc
slope is change in y/change in x
Parallel, Perpendicular
y=2x+4 and y=2x+3 are parallel
y=2x+4 and 2y=4x+8 are same line
y=2x+4 and y = 1/2x+6 meet perpendicularly
Parallel lines never meet, same slope
Perpendicular lines meet once at right angles and slopes are negative reciprocals 
Geometric Formulas
SOHCAHTOA
SOH = sin(x) = a/c
CAH = cos(x) = b/c
TOA = tan(x) = a/b
180 degrees = (pi)rad
Divide arclength by radius to get radian measure
Special Angles
Degree Radian Cosine Sine Tangent
0 0 1 0 0
30 (pi)/6 sqrt3/2 1/2 1/sqrt3
45 (pi)/4 1/sqrt(2) 1/sqrt(2) 1
60 (pi)/3 1/2 sqrt(3)/2 sqrt(3)
90 (pi) 0 1 undefined 
Quadratic Functions: Parabolas
f(x) = ax^{2} + bx + c
Zeros given by the quadratic formula:
b +/ sqrt(b^{2}  4ac)/2a
Coordinate point (b/2a,( b^{2} +4ac)/4a) 
Geometric Equations
Circles: Circumference 2(pi)r Area (pi)r^{2}
Cylinders: Surface area 2(pi)r x h + 2(pi)r^{2} Volume: (pi)r^{2}h
Sphere: Surface area 4(pi)r^{2} Volume 4/3(pi)r^{3} 


Negative Exponents
a^{m} = 1/a^{m}
1/a^{m} = a^{m} 
Trigonometric Identities
cos^{2}x + sin^{2}x = 1
tan^{1}x = arctan x
cot(x) = 1/tan x
e^{10} = cos x + isin x 
Secant, Cosecant, Cotangent
Secant: 1/cos(x)
Cosecant = 1/sin(x)
Cotangent = 1/tan(x) = cos(x)/sin(x) 
Exponential Rules
exp(a)exp(b) = exp(a+b)
[exp(a)]^b = exp(ab)
exp(a) = 1/exp(a)
Domain: all real numbers
Range: all positive numbers 
Absolute Values
Absolute Values often generate "and" and "or" situations.
Examples:
x<1: 1<x and x<1 1<x<1
x>1: x>1 or x<1
2x+3> 1: 2x+3>1 or (2x+3)>1
x>1 or x<2 
Multiplication and Scientific Notation
Multiplication and Division:
 Convert into scientific notation
 Add/subtract exponents
 Multiply/divide coefficients
 Convert to scientific notation
 2.3E4 x 9.5E7/1.6E10 = 2.3x9.5/1.6E(4+710) 


Logarithms
Logarithms are the functional inverses of the exponential
y = b^x is equivalent to logb(y) = x
log(mn) = log(m) + log(n)
log(m^n) = nlog(m)
log(1/m) = log(m)
loga(m) = logb(m)/logb(a)
exp(a)exp(b) = exp(ab)
exp(a) = 1/exp(a)
e^{2.3 = 10, e}12 = e^{2.3*5.2 = (e}2.3)^{5.2 = 10}5.2 = 2E5 
Function Variables
Domain = valid inputs to function
Range = what can the function produce
Zeros or Roots = where is f(x)=0
Intersections = Where is f(x) = g(x)
Local maximum is largest value around itself
Local minimum is smallest value around itself
Global is largest overall 
Sine
Sine is the y component as theta spins
Domain is all real numbers
Range 1</ y </ 1
Maxima at pi/2 + 2kpi, Minima at 3pi/2 + 2kpi
Zeros at kpi
Period 2pi 
Cosine
Cosine is the x component as theta spins
Domain is all real numbers
Range is 1</ y </ 1
Max is 2kpi, Minima (2k + 1)pi
Zeros at pi/2 + kpi
Period is 2pi 
Tangent
Tangent is the slope of the line with angle theta
Domain is all real numbers except pi/2 + kpi
Range is all real numbers
No max or min. asymptotes at undefined points
Zeros at kpi
Period is pi 
Distance Between Points
The distance between two points on the plane is based on the Pythagorean Theorem
AB = sqrt((XaXb)^{2} +(Ya  Yb)^{2})
A=(Xa,Ya) B=(Xb,Yb) 
Basic Facts
Total human population: 7 billion
USA population: 300 million
Distance from NY to LA: 2500 miles
Distance to the moon: 2.4E5 miles
Distance to the Sun: 1E8 miles
Distance around the equator: 2.5E4 miles
Area of the US: 4E6 square miles
Surface area of the Earth: 2E8 square miles 

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