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Quantitative Methods Midterm Cheat Sheet by

Fraction Rules

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

Multip­lic­ation
a/b * c/d = ac/bd

Reciprocal
1/(a/b) = b/a

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 = 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

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

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

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
 

Negative Exponents

a-m = 1/am

1/a-m = am

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)

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

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

Pythag­orean Theorem

a2 + b2 = c2

Unit Circle

Secant, Cosecant, Cotangent

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

Trigon­ometric Identities

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

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

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)

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

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)

e2.3 = 10, e12 = e2.3*5.2 = (e2.3­)5.2 = 105.2 = 2E5

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

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