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Quantitative Methods Midterm Cheat Sheet by

Fraction Rules

Common Denomi­nators
a/b + c/d = ad + bc/bd

a/b * c/d = ac/bd

1/(a/b) = b/a

Theory of Geometric Series

S = p + pr + pr2 + pr3 + ... + prn
Sr = p + pr + pr2 + pr3 + ... + prn + prn+1
S - Sr = p - pr^n+1
S = p-prn+1/1-r


Given point (c,d) and slope m, the unique line that satisfies this is the set of all points (x,y) such that
m = y-d/x-c

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


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) = ax2 + bx + c

Zeros given by the quadratic formula:
-b +/- sqrt(b2 - 4ac)/2a

Coordinate point (-b/2a,( -b2 +4ac)/4a)

Geometric Equations

Circles: Circum­ference 2(pi)r Area (pi)r2

Cylinders: Surface area 2(pi)r x h + 2(pi)r2 Volume: (pi)r2h

Sphere: Surface area 4(pi)r2 Volume 4/3(pi)r3

Negative Exponents

a-m = 1/am

1/a-m = am

Trigon­ometric Identities

cos2x + sin2x = 1
tan-1x = arctan x
cot(x) = 1/tan x
e10 = cos x + isin x

Secant, Cosecant, Cotangent

Secant: 1/cos(x)
Cosecant = 1/sin(x)
Cotangent = 1/tan(x) = cos(x)­/sin(x)

Unit Circle

Pythag­orean Theorem

a2 + b2 = c2

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.

|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+­7-10)


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)

e2.3 = 10, e12 = e2.3*5.2 = (e2.3)5.2 = 105.2 = 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 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 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 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

|A-B| = sqrt((­Xa-Xb)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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