2. Precalculus

数学英语



周维祺

Precalculus

High school materials that prepare students for calculus courses

Powers and Exponents

  • : to the power of
    : to (the) -th power
    : -th power of

  • : square

  • : cube

Polynomials

  • are coefficients
  • is the constant term (pronounciation: a naught)
  • : monomial
  • is the degree/order of the polynomial
    : quadratic polynomial
    : cubic polynomial

Roots and Logarithms

  • : -th root of
    : square root of
    : cubic root of

  • : log (the logarithm of) with base

  • : (natural) log

Trigonometry and Polar Coordinates

  • Trigonometric functions: , , , , ,

  • Cartesian coordinates:

  • Polar coordinates:

  • Spherical coordinates, cylindrical coordinates, etc.

Cartesian Coordinates

  • Descartes -> des Cartes -> Cartes -> Cartesian

  • In all coordinate systems there are origin and axis (plural: axes)

  • In Cartesian coordinate systems there are
    quadrants ()
    octants()
    hyperoctants()

Complex Numbers: Rectangular Form

  • plus

  • is the real part

  • is the imaginary part

  • is the complex conjugate of

Complex Numbers: Polar Form

  • is the radius
  • is the phase (factor)

Complex Numbers: Special Sets

  • (-th) roots of unity:

  • (open) unit disk / disc:

  • unit circle:

Game

  • Shuffle the 52-cards deck and place 10 cards in a row on the table

  • Two players take turns to pick cards. In each turn the player chooses one end of the row and pick a card from that end

  • The player with more points after 10 turns wins. Play and find a good strategy for the first player

Round 1

Round 2

Round 3

Round 4

Functions and Maps


  • : Domain
    : Range/Image

  • : a univariate function (a funtion with only variable)
    : a multivariate function (with multiple variables)

Roots and Support Sets

  • If , then is a root/zero of
    Alternatively: vanishes at
    In certain context: annihilates

  • The closure of the set is called the support of

Injectivity and Surjectivity

  • Injectivity: If implies then is injective.

  • Surjectivity: If every has a pre-image so that , then is surjective.

  • Bijectivtity: If is both injective and surjective, then is bijective.

Onto and Into

  • maps (from ) onto : is surjective

  • maps (from ) into : the range of is contained in

Equations and Expressions

  • Equation: a+b=c
    Inequality: a+b>c
    Expression: a+b

  • , are operands, while is the operator

Laws of Operations

  • A binary operation is Commutative if
    ab=ba

  • A binary operation is Associative if
    a+(b+c)=(a+b)+c

  • Given binary operations and , is Distributive over if

Parentheses, Brackets and Braces

  • : (curly) braces

  • : (angle) brackets
    : (square) brackets

  • : parentheses or round brakcets

Sufficiency and Necessity

  • :
    implies , can be inferred from
    is sufficient for , is necessary for
    is a sufficient condition of
    is a necessary condition of

  • :
    is sufficient and necessary for (and so is )
    they are equivalent

Definition(Dyck Paths). A Dyck path of length is a path which begins at the origin , ends at , and consists of steps (which are vectors) (North-East), called rises, and (1,-1) (South-East), called falls.

Example and Nonexample

Definition(Catalan Numbers). The Catalan number of non-negative integers is



Theorem. The total number of Dyck paths of length is the Catalan number

Conventions

  • A definition should be both precise and minimal

  • If a defintion is not straightforward, or counterintuitive or concerns subtle delicate cases, then it would be a good practice to include examples and/or nonexamples

  • It is often good to explain the motivation

Types of Example

  • Example

  • Nonexample

  • Counterexample

Tasks

Give a definition for continuous functions on
provide some examples and nonexamples

Summary

  • Elementary functions

  • Complex numbers

  • Coordinates and expressions

  • Definitions, examples, nonexamples