Mathematical Methods

Algebra, the study of relationships

Probability, the study of uncertainty/chance

Geometry, the study of shapes

Calculus, the study of change

Measurement

Trigonometry

Ratios that are relationships between sides(lengths) and (angles)

Length, area, volume, units etc;
Studying geometric shapes like square, triangles, rectangles and functions

Sine, cosine and tangent
SOHCAHTOA

Pronumerals and numerals

Relates one variable to another
Eg: y = x

The x,y coordinate plane
Has an equation that plots all the possible values of that make the equation hold true, this forms a shape or rather a family of shapes

Functions

The Unit Circle

  • Sine, cosine and tangent are periodic functions
  • Has an angle which is measured from the positive x-axis

Families:

  • Polynomials
  • Trigonometric
  • Exponential/logarithmic

Arithmetic

  • Basic operations(addition, subtraction etc;)
  • N: Counting numbers
  • Z: Integers
  • Q; Rational numbers
  • Q': Irrational numbers
  • R: Real numbers
  • I: Imaginary numbers
  • C: Complex numbers

Rate of change between two or more variables

Average rate of change:

  • Two points, that occur some distance aparts
  • Ratio of rise over run

Instantaneous rate of change

  • Rate of change at a single point
  • Limit of average rate of change approaches 0

Derivatives

Integrals

Find the rate of change of a function

  • Notation: d/dx, f'(x)

Shows how one variable varies with another

Find the net change over a period given the rate of change

  • Area under the graph
  • Sum of infitesimally small geometric shapes

How often is something like to happen?

Basic probability:

  • All probabilities add up to 1
  • U - union of two events, either one or the other or both
  • Upside down "U" - intersection of two events, only both(simultaneuously)
  • Can be shown using venn diagrams, tree diagrams, two way tables

Probability Distribution types:

  • Discrete, can only take up specific values
  • Continuous, can take up any possible value within given domain

Transformations:

  • Can dilate, translate or reflect the function
  • Using matrices, function notation

Discrete:

  • Mean(Average), E(X) = sum(x*Pr(X=x))
  • Median: Middle value(50th percentile)
  • Mode: Most common value
  • Variance: Average area of spread
  • Standard deviation: Average distance from where the values would fall from the mean

Continuous:

  • Mean(Average), integral(x*Pr(X=x))
  • Median: 50th percentile
  • Mode: Most common value
  • Variance: Average area of spread
  • Standard deviation: Average distance from where the values would fall from the mean

Domain: The extent of all possible x-values
Range: The extent of all possible y-values

Symmetry properties(that yield the same result depending on base angle in first quadrant and sign of quadrant):

  • Q1: theta
  • Q2: pi-theta
  • Q3: pi + theta
  • Q4: 2*pi-theta

Signs:

  • Sine is positive in Q1, 2 and negative in Q3, 4
  • Cosine is positive in Q1, 4 and negative in Q2, 3
  • Tangent is positive in Q1, 3 and negative in Q2, 4

Binomial Distribution:

  • Only two outcomes
  • Has ,n, number of trials
  • Each trial is independent
  • Probability of success, p, is constant

Normal Distribution:

  • Symmetric on both sides
  • Mean= Median = Mode
  • Infinitely continuous

Find unknowns using knowns:

  • Simultaneous equations

Algebraic expansion identities:

  • (a+b)^2 = a^2 + 2ab+ b^2
  • (a+b)^3 = a^3 + 3a^2b+3ab^2+b^3
  • a^2-b^2= (a+b)(a-b)
  • a^3+b^3= (a+b)(a^2-ab+b^2)

Patterns:

  • Pascal's Triangle

Areas:

  • Rectangles: b*h
  • Triangles: (1/2)bh = (1/2)bc*sin(theta)

Chain rule:

  • For composite functions

Product rule:

  • For functions that are multiplied together

Quotient rule:

  • For functions that are divided by each other

Indefinite integrals:

  • For integrating functions without limits
  • To obtain the antiderivative

Definite Integrals:

  • To the find the change over a certain period, has limits