Topics

  • Trigonometry and Applications
  • Vectors and Matrices
  • Polynomial and Rational Functions
  • Conic Sections (Parabolas, Ellipses, and Hyperbolas)
  • Discrete Mathematics (Combinatorics, Probability)
  • Exponential, Logarithm, and Logistic Functions
  • Introduction to Calculus: Derivatives and Applications

Semester 1 Topics

OSP: OpenStax Precalculus
OSC1: OpenStax Calculus Volume 1
SZ: Stitz Zeager Precalculus
DWFK: Precalculus by Demana, Waits, Foley, Kennedy, 8th ed
FDWK: Calculus by Finney, Demana, Waits, Kennedy, 5th Edition

Unit Topic Old Chapters Open Chapters
1 Trigonometry DWFK Ch 4 OSP Ch 5-6, SZ Ch 10
2 More Trigonometry DWFK Ch 5 OSP Ch 7, SZ Ch 10-11
3 Trig Applications DWFK Ch 6-7 OSP Ch 8, SZ Ch 11, 8
4 Polynomials DWFK Ch 1-2 OSP Ch 3, SZ Ch 1-5
5 Conic Sections DWFK Ch 8 OSP Ch 10, SZ Ch 7

Semester 2 Topics

Unit Topic Old Chapters Open Chapters
6 Discrete Mathematics DWFK Ch 9 OSP Ch 11, SZ Ch 9
7 Exponentials / Logarithms DWFK Ch 5 OSP Ch 4, SZ Ch 6
8 Calculus: Derivatives FDWK Ch 2-3 OSC1 Ch 2-3
9 Calculus: More Derivatives FDWK Ch 4 OSC1 Ch 3
10 Calculus: Applications FDWK Ch 5 OSC1 Ch 4

Unit 1: Trigonometry

  • 1.1 Unit Circle
    • DWFK 4.2-4.3, SZ 10.2, OSP 5.2-5.3
    • sin/cos from special triangles
    • unit circle with sin/cos values at special angles
  • 1.2 Trig graphs
    • DWFK 4.4-4.5, SZ 10.5, OSP 6.1-6.2
    • graph sin/cos
    • graph tan/cot/sec/csc
    • graph transformations of the trig functions (scale x/y, shift x/y)
  • 1.3 Inverse trig functions
    • DWFK 4.7, SZ 10.6, OSP 6.3
    • calculating inverse trig functions for values corresponding to special angles
  • 1.4 Trig word problems
    • DWFK 4.8, SZ 11.1, OSP 5.4, 7.6
    • calculating distances with trigonometry
    • angle of elevation / depression
    • modeling with a sinusoid

Unit 2: More Trigonometry

  • 2.1 Trig identities
    • DWFK 5.1-5.2, SZ 10.3, OSP 7.1
    • basic, Pythagorean, cofunction, and odd/even identities
    • prove trigonometric identities from left to right
  • 2.2 Sum/Difference formulas
    • DWFK 5.3, SZ 10.4, OSP 7.2
    • sum/difference formulas
    • derivation of sum/difference formula for tan from the formulas for sin/cos
  • 2.3 Multiple angle formulas
    • DWFK 5.4, SZ 10.4, OSP 7.3
    • multiple angle formulas (derivation from the sum/difference formulas)
    • derive the power-reducing and half-angle formulas from the multiple angle formulas
    • use any of the formulas to calculate trig function values
  • 2.4 Law of Sines/Cosines
    • DWFK 5.5-5.6, SZ 11.2, OSP 8.1-8.2
    • Laws of Sines/Cosines to solve triangles, including any ambiguous cases

Unit 3: Trig Applications

  • 3.1 Vectors
    • DWFK 6.1, SZ 11.8, OSP 8.8
    • operations: addition and scalar multiplication
    • component form vs. magnitude and direction
  • 3.2 Dot Product
    • DWFK 6.2, SZ 11.9, OSP 8.8
    • component definition
    • dot product and magnitude
    • dot product with unit vectors / projection on axes
    • law of cosines => angle definition
    • orthogonality
    • finding the angle between two vectors
  • 3.3 Parametric Equations
    • DWFK 6.3, SZ 11.10, OSP 8.6-8.7
    • parametrizing a line (segment)
    • parametrizing a circle / circular motion
    • projectile motion
  • 3.4 Polar Coordinates
    • DWFK 6.4, SZ 11.4, OSP 8.3
    • rectangular/polar conversion
    • rectangular/polar equations
  • 3.5 Polar Graphs
    • DWFK 6.5, SZ 11.5, OSP 8.4
    • analysis of graphs: max r value, symmetry
  • 3.6 Linear systems, matrices
    • DWFK 7.1, 7.3, SZ 8.2, OSP 9.1-9.2, 9.6
    • solving by substitution, elimination
    • solving by diagonalization of matrices
  • 3.7 Matrix algebra
    • DWFK 7.2, SZ 8.3, 8.4, OSP 9.5
    • matrix operations: addition, scalar multiplication
    • matrix multiplication and composition
  • 3.8 Matrix inverses and determinants
    • DWFK 7.2, SZ 8.5, OSP 9.7, 9.8
    • matrix inverses
    • solving linear systems with matrix inverses
    • matrix inverse <-> nonzero determinant
    • linear transformations and the determinant

Unit 4: Polynomials

  • 4.1 Function properties
    • DWFK 1.2-1.3, SZ 1.6, OSP 1.1-1.3
    • domain/range
    • increasing/decreasing
    • continuity
    • boundedness (above/below)
    • local/global min/max
  • 4.2 Operations, transformations, composition, inverses
    • DWFK 1.4-1.6, SZ 1.5, 1.7, 5.1, 5.2, OSP 1.4-1.5
    • function operations
    • function composition and inverses
  • 4.3 Polynomials 1
    • DWFK 2.1-2.2, SZ 3.1, 3.2, OSP 3.2-3.6
    • end behavior
    • factoring and zeros
    • graphing
    • Remainder theorem
    • Factor theorem
  • 4.4 Polynomials 2, Complex numbers
    • DWFK 2.3-2.4, P.6, SZ 3.3, 3.4, OSP 3.5-3.6, 3.1
    • synthetic division
    • potential rational zeros
  • 4.5 Complex numbers, Fundamental theorem of Algebra
    • DWFK P.6, 2.5, SZ 3.4, OSP 3.6
    • complex numbers: operations, conjugation, magnitude
    • factoring over the real numbers
    • factoring over the complex numbers
  • 4.6 Rational functions
    • DWFK 2.6, SZ 4.2, OSP 3.7
    • graphing rational functions by hand
    • behavior near asymptotes
    • (asymptotic) end behavior

Unit 5: Conic Sections

  • 5.1 Parabolas
    • DWFK 8.1, SZ 7.3, OSP 10.1
    • geometric definition: focus, directrix
    • standard form <-> focus/directrix
  • 5.2 Ellipses
    • DWFK 8.1, SZ 7.4, OSP 10.2
    • geometric definition
    • algebraic definition
    • graphing
  • 5.3 Hyperbolas
    • DWFK 8.1, SZ 7.5, OSP 10.3
    • geometric definition
    • algebraic definition
    • graphing

Unit 6: Discrete mathematics

  • 6.1 Combinatorics
    • DWFK 9.1, OSP 11.5, LDM 1.1, 1.3
    • multiplicative rule for counting / tree diagrams
    • permutations
    • combinations
    • distinguishible permutations
  • 6.2 Binomial Theorem
    • DWFK 9.2, SZ 9.4, OSP 11.6
    • Pascal’s triangle / Binomial Theorem
    • full binomial expansion
    • formula for single term
  • 6.3 Probability
    • DWFK 9.3, OSP 11.7
    • basic definition (discrete)
    • “independent” events -> multiplication
    • conditional probability, independence
  • 6.4 More Probability
    • DWFK 9.3, OSP 11.7
    • binomial distribution
    • hypergeometric distribution
  • 6.5 Sequences
    • DWFK 9.4, SZ 9.1, OSP 11.1-11.3
    • sequence notation
    • recursive definition
    • explicit definition
    • limits
  • 6.6 Series
    • DWFK 9.5, SZ 9.2, OSP 11.4
    • summation notation
    • finite sums arithmetic sequences
    • finite sums geometric sequences
    • infinite geometric series

Unit 7: Exponential and logarithm

  • 7.1 Exponential / logarithm 1
    • DWFK 3.1-3.4, SZ 6.1-6.5, OSP 4.1-4.7
    • graphs of $2^x$, $10^x$, $e^x$
    • graphs of $\log_2$, $\log_{10}$, and $\log_e = \ln$
  • 7.2 Exponential / logarithm 2
    • solving equations
    • base change formula
    • applications: population growth / radioactive decay

Unit 8: Derivatives

  • 8.1 Limits and End Behavior
    • FDWK 2.1-2.2, OSC1 2.2-2.3
    • calculating limits
    • infinite limits at asymptotes
    • end behavior
    • special limit: $\lim_{x \to 0}\frac{\sin x}{x} = 1$
  • 8.2 Continuity
    • FDWK 2.3, OSC1 2.4
    • types of discontinuities (removable, jump, infinite)
    • definition of continuity
  • 8.3 Derivative Limit Definition
    • FDWK 3.1-3.2, OSC1 3.1
    • standard definition (h->0)
    • alternate definition (x->a)
  • 8.4 Differentiation Rules
    • FDWK 3.3, OSC1 3.2-3.3
    • $\frac{d}{dx}(const) = 0$
    • power rule
    • sum rule
    • product rule / quotient rule
  • 8.5 Rates of change
    • FDWK 2.4, OSC1 3.4
    • applications:
      • projectiles
      • circular motion
      • population growth
  • 8.6 Derivatives of Trig Functions
    • FDWK 3.5, OSC1 3.5
    • derivatives of sin/cos from special limits
    • derive formulas for the other trig functions

Unit 9: More Derivatives

  • 9.1 Chain rule
    • FDWK 4.1, OSC1 3.6
    • function composition and the chain rule
  • 9.2 Implicit differentiation
    • FDWK 4.2, OSC1 3.8
    • chain rule -> implicit differentiation
    • general power rule
  • 9.3 Derivatives of exponential / logarithm
    • FDWK 4.4, OSC1 3.9
    • special limit: $\lim_{h \to 0} \frac{a^h-1}{h} = \ln a$
    • derivative of $e^x$, $a^x$
    • derivative of $\ln$, $\log_a$
  • 9.4 Derivatives of inverse trig functions
    • FDWK 4.3, OSC1 3.7
    • derivation of formulas using implicit differentiation + reference triangle

Unit 10: Applications

  • 10.1 Extreme values
    • FDWK 5.1, OSC1 4.3
    • extreme value theorem
  • 10.2 Mean Value Theorem
    • FDWK 5.2, OSC1 4.4
    • Rolle’s Theorem
    • Mean Value Theorem
    • Cor 1: f’ = 0 -> const
    • Cor 2: f’ = g’ -> f = g + const (-> antiderivatives)
    • Cor 3: f’>0 -> f increasing
  • 10.3 Extreme value tests
    • FDWK 5.3, OSC1 4.5
    • 1st derivative test (including one-sided)
    • concavity / inflection points
    • 2nd derivative test
  • 10.4 Optimization
    • FDWK 5.4, OSC1 4.7
    • applications: optimizing functions, with constraints
  • 10.5 Linearization / Newton’s Method
    • FDWK 5.5, OSC1 4.2, 4.9
    • linear approximation of functions
    • 2nd order approximation -> 2nd derivative test
    • Newton’s Method for finding roots