Quantitative Methods Midterm Cheat Sheet

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

Multip­lic­ation
a/b * c/d = ac/bd

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

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

f(x) = ax² + bx + c

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

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

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

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

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

a^-m = 1/a^m

1/a^-m = a^m

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 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

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

a² + b² = c²

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

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

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

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)

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 are the functional inverses of the expone­ntial

y = b<sup>x is equivalent to logb(y) = x

log(mn) = log(m) + log(n)

log(m</sup>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

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