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