ST 550


Course syllabus


I         Basic Concepts

1                 The Real numbers

a.      Counting numbers

b.     Whole numbers

c.     Rational numbers

d.     Irrational numbers & the Reals

2                 Exponents

a.      an * am = an+m

b.     an/ am = an-m

c.     (an )m = an*m

d.     (a*b)n = an * bn

e.      (a/b)n = an/ bn

3                 Scientific notation

a.      M % 10n, 1[M<10, n an integer

b.     170000 = 1.7 % 105

c.     .000017 = 1.7 % 10-5

d.     On calculators, often given as M E n, i.e., 1.7 E 5

4                 Functions

a.      A rule – must be consistent and single-valued

b.     Usually a rule for a number, given a number

c.     Graphs in the x-y plane

                                                               i.      Vertical lines hit at most once

                                                              ii.      Points are of the form (x, f(x))

d.     Shifts

                                                               i.      Vertical shifts: f(x) + b

                                                              ii.      Horizontal shifts: f(x-b)

e.      Scaling

                                                               i.      Stretching – multiply by a>1

                                                              ii.      Compressing – multiply by 0<a<1

                                                            iii.      Flip – multiply by -1

II        Functions

1                 Shifts & scaling

a.      Vertical

                                                               i.      Shifts

                                                              ii.      Scaling

1.     Stretching

2.     Compressing

3.     Flips

b.     Horizontal

                                                               i.      Shifts

                                                              ii.      Scaling

1.     Stretching

2.     Compressing

3.     Flips

2                 Lines

a.      Slope

b.     Point-slope form

c.     Slope-intercept form

d.     Parallel and perpendicular lines

3                 Polynomials

a.      General form

b.     Order

c.     Special names

d.     Special properties

III       More Functions

1                 Curve fitting

a.      Linear

                                                               i.      Positive correlation

                                                              ii.      Negative correlation

b.     Nonlinear

2                 Distance

a.      Pythagoras

b.     Distance formula

3                 Symmetry

a.      About the y-axis

b.     Through the origin

c.     About the x-axis

d.     About a line

e.      Through a point

4                 Applications

a.      BMI

                                                               i.      W/H2*703

b.     Depreciation

c.     Motion under gravity

d.     Concentration of mixtures

e.      Growth rates

IV       Roots of Polynomials

1                 Lines

a.      x and y intercepts

b.     Intersection of lines

c.     Applications

                                                               i.      Celsius and Fahrenheit scales

                                                              ii.      Catching up with an object

2                 Quadratics

a.      Completing the square

                                                               i.      Vertex

b.     Factoring

c.     Quadratic formula

V        More on Quadratics

1                 Complex numbers

a.      Unsolvable equations

b.     New numbers

                                                               i.      Consistent

c.     Rules of arithmetic

                                                               i.      Conjugates

                                                              ii.      Division

d.     The complex plane

2                 Arithmetic in the complex plane

a.      Addition

b.     Multiplication

c.     Roots

3                 Applications

a.      Projectile motion

                                                               i.      Depth of a well

b.     Fenced areas

VI       General Polynomials

1                 Definition

a.      anxn + an-1xn-1 + … +a0

b.     Degree or order

c.     Special properties

2                 Factoring

a.      Zero-product theorem

b.     Rational roots

c.     Factoring over the reals

d.     Factoring over the complex numbers

3                 Division

a.      Long division

b.     Remainders

VII      Rational Functions

1                 Definition

a.      Polynomial/polynomial

b.     Domain

c.     Range

2                 Graphing

a.      Zeroes

b.     Poles

c.     Asymptotes

                                                               i.      Vertical

                                                              ii.      Horizontal

                                                            iii.      Slant or oblique

3                 Applications

a.      Michaelis-Menton Kinetics

                                                               i.      Growth in a chemostat

b.     Waves

                                                               i.      Flood waves

                                                              ii.      s*(h2-h)*(h-h1)*(h-H)/(h3-B)

                                                            iii.      H<h1, h2.  Interested in h between h1 and h2

c.     Unit cost

                                                               i.      Fixed and variable expenses

                                                              ii.      Profit


VIII    Applications of Rational Functions

1                 More graphing

a.      Zeroes

b.     Poles

c.     Asymptotes

d.     Long division

                                                               i.      Remainder term

                                                              ii.      Small order/larger order

2                 Minimum surface area

a.      Open top box

                                                               i.      Fixed volume

                                                              ii.      Square base

b.     Beverage can

                                                               i.      Fixed volume

                                                              ii.      What if the top costs more?

3                 Michaelis-Menton

a.      f(0) = 0

b.     Approaches a limit

c.     Guess linear/linear

4                 Average cost

a.      Fixed costs and unit costs

b.     Approaches a limit – what does this mean?

IX       Compositions and Inverses

1                 Definition & notation

a.      Why?

b.     Order matters

c.     On the calculator

d.     f)g(x) or f(g(x))

2                 Examples

a.      Example functions

b.     Iteration

c.     Related rates

d.     Profit

3                 Inverses

a.      Definition

b.     Solving

c.     Symmetry

d.     Restricting the domain

e.      Examples

                                                               i.      Simple

                                                              ii.      Radius from volume

X        Exponentials

1                 Definition

a.      Exponents with different bases

b.     Keeping the base, and changing the power

                                                               i.      Integer

                                                              ii.      Rational

                                                            iii.      Real

2                 Examples

a.      2x

b.     10x

c.     (1/2)x

3                 Graphs

a.      2x

                                                               i.      a>1, ax increasing

b.     (1/3)x

                                                               i.      a<1, ax decreasing

c.     3*2x-2

                                                               i.      Shifts, scaling etc.

d.     ex

4                 The special case of ex

a.      e = 2.718281828459….

b.     limit (1 + 1/n)n

c.     Why?

                                                               i.      Applications

                                                              ii.      Calculus

5                 Applications

a.      Exponential growth

b.     Half-life

c.     Compound interest



XI       Logarithms

1.     Definition

a.      Inverse of exponential

b.     logarithm = exponent

2.     Rules of Logarithms

a.      log(a*b)

b.     log(a/b)

c.     log(an)

d.     Each corresponds to a rule for exponents!

3.     Different Bases

a.      Two main bases used

                                                               i.      Base 10

                                                              ii.      Natural logs, base e.

b.     Change of base formula

                                                               i.      Derived from definition

4.     Applications

a.      Doubling time & half-life

b.     Curve fitting

c.     pH

d.     Richter scale


XII     Trigonometry

  1. Triangles
    1. Right triangles
    2. 6 possible ratios of sides
  2. Solving triangles
    1. Law of sines
    2. Law of cosines
  3. Extended definition
    1. Unit circle
    2. Radian measure
  4. Periodic functions
    1. Periods of basic trig functions
    2. Scaling


XIII    Trigonometry 2

  1. Special triangles
    1. 45-45-90
    2. 30-60-90
  1. Applications
    1. Wave motion
    2. Heat equation

                                                     i.     Heat

                                                    ii.     Other diffusions

  1. Circle Identity
    1. Sine and cosine as points on the unit circle
    2. Basic identity
  2. Sum & difference
    1. sin(A+B)
    2. cos(A+B)
    3. sin(A-B)
    4. cos(A-B)
    5. Check some special cases
  3. Double & half angles
    1. sin(2A) = sin(A+A)
    2. cos(2A)

                                                     i.     3 forms

    1. Half angles

                                                     i.     From cos(2A)


XIV    Polar Coordinates

  1. Definition
    1. Distance and direction
    2. Connection to rectangular coordinates
  2. Arithmetic
    1. Adding
    2. Subtracting
    3. Multiplying

                                                     i.     In the complex plane

  1. Polar coordinates in the Complex plane
    1. Arithmetic
    2. Notation

                                                     i.     cis

                                                    ii.     Exponential

1.     Euler’s formula

  1. Roots
    1. Powers

                                                     i.     DeMoivre’s theorem

    1. Roots
    2. Roots of unity


XV      Systems of Linear Equations

  1. 2 equations, 2 unknowns
    1. Intersections of lines

                                                     i.     3 possibilities

1.     Independent

2.     Dependent, inconsistent

3.     Dependent, consistent

    1. Substitution method
  1. 3 equations, 3 unknowns
    1. Planes in three space
    2. Same three possibilities, more possible pictures
    3. Gaussian elimination

                                                     i.     Switch two equations

                                                    ii.     Multiply an equation by a non-zero constant

                                                  iii.     Add a multiple of one equation to another

  1. Applications
    1. Combined prices
    2. Fitting quadratics


XVI    Systems of Linear Equations, 2

  1. Applications
    1. Mixture problems
    2. Speed and motion
    3. Combined prices
    4. Nutrition
    5. Fitting quadratics
  2. Matrix Notation
    1. Definition
    2. Relation to linear systems
    3. Row operations

                                                     i.     Augmented form

                                                    ii.     Interchange rows

                                                  iii.     Multiply a row

                                                  iv.     Add a multiple of a row to another

  1. Gaussian elimination
    1. Triangular form
    2. Row echelon form
    3. Reduced row echelon form
  2. Matrix Operations
    1. Addition & subtraction
    2. Multiplication by a scalar

                                                     i.     Commutative

                                                    ii.     Associative

                                                  iii.     Distributive

                                                  iv.     Identity and inverses


XVII   Matrix Multiplication

  1. Definition
    1. Matrix times a column vector
    2. Matrix times a matrix

                                                     i.     Sizes must match

  1. Change of variables
    1. x & y linear combinations of u & v
    2. Substitute.
    3. Associativity
  2. Properties
    1. Matrix multiplication

                                                     i.     Associative

                                                    ii.     Distributive

                                                  iii.     NOT commutative!

  1. Inverses
    1. The identity matrix
    2. Definition of an inverse
    3. Finding an inverse

                                                     i.     Augmented form

    1. Using an inverse
  1. Applications
    1. Solving systems
    2. Eigenvectors


XVIII Summary