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 * a^m = a^n+m

b. aⁿ/ a^m = a^n-m

c. (aⁿ )^m = a^n*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. a_nxⁿ + a_n-1x^n-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. 2^x

b. 10^x

c. (1/2)^x

3 Graphs

a. 2^x

i. a>1, a^x increasing

b. (1/3)^x

i. a<1, a^x decreasing

c. 3*2^x-2

i. Shifts, scaling etc.

d. e^x

4 The special case of e^x

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

5. Applications
1. Wave motion
2. Heat equation

i. Heat

ii. Other diffusions

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

i. 3 forms

3. 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

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

i. cis

ii. Exponential

1. Euler’s formula

4. Roots
1. Powers

i. DeMoivre’s theorem

2. Roots
3. 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

2. 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

3. 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

3. Gaussian elimination
1. Triangular form
2. Row echelon form
3. Reduced row echelon form
4. 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

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

i. Associative

ii. Distributive

iii. NOT commutative!

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

i. Augmented form

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

XVIII Summary