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

Unit	Topic
1	Trigonometry
2	More Trigonometry
3	Trig Applications
4	Polynomials
5	Conic Sections
6	Discrete Mathematics
7	Exponentials / Logarithms
8	Calculus: Derivatives
9	Calculus: More Derivatives
10	Calculus: Applications

Chapter Cross-Reference

Precalculus topics

DWFK: Precalculus by Demana, Waits, Foley, Kennedy, 8th ed
OSP: OpenStax Precalculus
SZ: Stitz Zeager Precalculus

Unit	Topic	DWFK	OSP	SZ
1	Trigonometry	Ch 4	Ch 5-6	Ch 10
2	More Trigonometry	Ch 5	Ch 7	Ch 10-11
3	Trig Applications	Ch 6-7	Ch 8	Ch 11, 8
4	Polynomials	Ch 1-2	Ch 3	Ch 1-5
5	Conic Sections	Ch 8	Ch 10	Ch 7
6	Discrete Mathematics	Ch 9	Ch 11	Ch 9
7	Exponentials / Logarithms	Ch 5	Ch 4	Ch 6

Calculus topics

FDWK: Calculus by Finney, Demana, Waits, Kennedy, 5th Edition
OSC1: OpenStax Calculus Volume 1

Unit	Topic	FDWK	OSC1
8	Calculus: Derivatives	Ch 2-3	Ch 2-3
9	Calculus: More Derivatives	Ch 4	Ch 3
10	Calculus: Applications	Ch 5	Ch 4

Details

Unit 1: Trigonometry

• 1.1 Unit Circle

  • learning targets:

    • find sin/cos values from special triangles

    • use the unit circle to find sin/cos values at special angles: \(\{0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3}\} + \frac{k\pi}{2}\)

  • reference:

    • OSP 5.2-5.3

    • DWFK 4.2-4.3

    • SZ 10.2

• 1.2 Trig graphs

  • learning targets:

    • graph sin/cos

    • graph tan/cot/sec/csc

    • graph transformations of the trig functions (scale x/y, shift x/y)

  • reference:

    • OSP 6.1-6.2

    • DWFK 4.4-4.5

    • SZ 10.5

• 1.3 Inverse trig functions

  • learning targets:

    • calculate inverse trig functions for values corresponding to special angles

  • reference:

    • OSP 6.3

    • DWFK 4.7

    • SZ 10.6

• 1.4 Trig word problems

  • learning targets:

    • calculate distances with trigonometry

    • calculate speeds in the x/y direction using the angle of elevation / depression

    • model physical situations with a sinusoid

  • reference:

    • OSP 5.4, 7.6

    • DWFK 4.8

    • SZ 11.1

Unit 2: More Trigonometry

• 2.1 Trig identities

  • learning targets:

    • use the basic, Pythagorean, cofunction, and odd/even identities

    • prove trigonometric identities from left to right

  • reference:

    • OSP 7.1

    • DWFK 5.1-5.2

    • SZ 10.3

• 2.2 Sum/Difference formulas

  • learning targets:

    • use the sum/difference formulas

    • derive the sum/difference formula for tan from the formulas for sin/cos

  • reference:

    • OSP 7.2

    • DWFK 5.3

    • SZ 10.4

• 2.3 Multiple angle formulas

  • learning targets:

    • derive the double angle formulas from the sum formulas

    • derive the power-reducing and half-angle formulas from the double angle formulas

    • use the double angle, power reducing, and half-angle formulas

  • reference:

    • OSP 7.3

    • DWFK 5.4

    • SZ 10.4

• 2.4 Law of Sines/Cosines

  • learning targets:

    • use the Laws of Sines/Cosines to solve triangles, including any ambiguous cases

  • reference:

    • OSP 8.1-8.2

    • DWFK 5.5-5.6

    • SZ 11.2

Unit 3: Trig Applications

• 3.1 Vectors

  • learning targets:

    • perform vector operations: addition and scalar multiplication

    • convert between component form and magnitude/direction

  • reference:

    • OSP 8.8

    • DWFK 6.1

    • SZ 11.8

• 3.2 Dot Product

  • learning targets:

    • compute dot product of two vectors

    • compute the dot product with unit vectors to find the projection of a vector on axes

    • determine whether two vectors are orthogonal, parallel, or neither

    • find the angle between two vectors

  • reference:

    • OSP 8.8

    • DWFK 6.2

    • SZ 11.9

• 3.3 Parametric Equations

  • learning targets:

    • parametrize a line (segment)

    • parametrize a circle / circular motion

    • solve projectile motion problems

  • reference:

    • OSP 8.6-8.7

    • DWFK 6.3

    • SZ 11.10

• 3.4 Polar Coordinates

  • learning targets:

    • convert expressions between rectangular and polar coordinates

    • convert equations between rectangular and polar coordinates

  • reference:

    • OSP 8.3

    • DWFK 6.4

    • SZ 11.4

• 3.5 Polar Graphs

  • learning targets:

    • analyze graphs:

      • max r value (and corresponding \(\theta\))

      • symmetry

  • reference:

    • OSP 8.4

    • DWFK 6.5

    • SZ 11.5

• 3.6 Linear systems, matrices

  • learning targets:

    • solve linear systems of equations by substitution, elimination

    • solve linear systems of equations by diagonalization of matrices

  • reference:

    • OSP 9.1-9.2, 9.6

    • DWFK 7.1, 7.3

    • SZ 8.2

• 3.7 Matrix algebra

  • learning targets:

    • perform matrix operations: addition, scalar multiplication

    • perform matrix multiplication (composition)

  • reference:

    • OSP 9.5

    • DWFK 7.2

    • SZ 8.3, 8.4

• 3.8 Matrix inverses and determinants

  • learning targets:

    • use the determinant of a matrix to determine whether the matrix is invertible

    • compute matrix inverses

    • solve linear systems with matrix inverses

  • reference:

    • OSP 9.7, 9.8

    • DWFK 7.2

    • SZ 8.5

Unit 4: Polynomials

• 4.1 Function properties

  • learning targets:

    • find the domain and range of a function

    • find intervals where a function is increasing/decreasing

    • identify discontinuities (types and locations) of a function

    • determine whether a function is bounded (above/below)

    • determine the local/global (relative/absolute) min/max of a function

  • reference:

    • OSP 1.1-1.3

    • DWFK 1.2-1.3

    • SZ 1.6

• 4.2 Operations, transformations, composition, inverses

  • learning targets:

    • perform operations on functions

    • determine the composition of two functions

    • determine the inverse of a function

  • reference:

    • OSP 1.4-1.5

    • DWFK 1.4-1.6

    • SZ 1.5, 1.7, 5.1, 5.2

• 4.3 Polynomials 1

  • learning targets:

    • describe the end behavior of a polynomial using limit notation

    • determine the zeros (roots) of a polynomial from its factors

    • graph a polynomial (factored)

    • apply the Remainder theorem

    • apply the Factor theorem

  • reference:

    • OSP 3.2-3.6

    • DWFK 2.1-2.2

    • SZ 3.1, 3.2

• 4.4 Polynomials 2, Complex numbers

  • learning targets:

    • perform synthetic division

    • determine and verify potential rational zeros

  • reference:

    • OSP 3.5-3.6, 3.1

    • DWFK 2.3-2.4, P.6

    • SZ 3.3, 3.4

• 4.5 Complex numbers, Fundamental theorem of Algebra

  • learning targets:

    • perform operations on complex numbers:
      \(+, \cdot, \overline{z}, |z|\)

    • find complete factorization of a polynomial over the real numbers

    • find complete factorization of a polynomial over the complex numbers

  • reference:

    • OSP 3.6

    • DWFK P.6, 2.5

    • SZ 3.4

• 4.6 Rational functions

  • learning targets:

    • graph rational functions by hand

    • describe behavior near asymptotes using limit notation

    • describe end behavior using limit notation

    • describe asymptotic end behavior

  • reference:

    • OSP 3.7

    • DWFK 2.6

    • SZ 4.2

Unit 5: Conic Sections

• 5.1 Parabolas

  • learning targets:

    • find the focus and directrix of a parabola from its standard form equation

    • find the equation of a parabola from its focus and directrix

    • graph any parabola

  • reference:

    • OSP 10.1

    • DWFK 8.1

    • SZ 7.3

• 5.2 Ellipses

  • learning targets:

    • graph any ellipse

    • find the center, foci, major/minor axes, and eccentricity of an ellipse from its standard form equation

    • find the standard form equation of an ellipse given some of its properties

  • reference:

    • OSP 10.2

    • DWFK 8.1

    • SZ 7.4

• 5.3 Hyperbolas

  • learning targets:

    • graph any hyperbola

    • find the center, foci, major/minor axes, eccentricity, and asymptotes of a hyperbola from its standard form equation

    • find the standard form equation of a hyperbola given some of its properties

  • reference:

    • OSP 10.3

    • DWFK 8.1

    • SZ 7.5

Unit 6: Discrete mathematics

• 6.1 Combinatorics

  • learning targets:

    • count events with independent choices using the multiplicative rule

    • compute permutations \(_n P_r\)

    • compute combinations \(_n C_r = \begin{pmatrix} n \\ r \end{pmatrix} \)

    • compute distinguishible permutations

  • reference:

    • OSP 11.5

    • LDM 1.1, 1.3

    • DWFK 9.1

• 6.2 Binomial Theorem

  • learning targets:

    • use Pascal’s triangle and the Binomial Theorem to find the standard form of a power of a binomial

    • find a single term of the expanded polynomial

  • reference:

    • OSP 11.6

    • DWFK 9.2

    • SZ 9.4

• 6.3 Probability

  • learning targets:

    • compute probabilities using the basic definition

    • calculate the probability of independent events using multiplication

    • calculate conditional probability \(P(A|B)\)

    • determine whether two events are independent

  • reference:

    • OSP 11.7

    • DWFK 9.3

• 6.4 More Probability

  • learning targets:

    • calculate probabilities using the binomial distribution

    • calculate probabilities using the hypergeometric distribution

  • reference:

    • OSP 11.7

    • DWFK 9.3

• 6.5 Sequences

  • learning targets:

    • determine the term of a sequence from an explicit formula

    • find a recursive definition for an arithmentic or geometric sequence

    • find an explicit definition for an arithmentic or geometric sequence

    • determine the limit of a sequence

  • reference:

    • OSP 11.1-11.3

    • DWFK 9.4

    • SZ 9.1

• 6.6 Series

  • learning targets:

    • use summation notation to describe a sum

    • calculate finite sums from summation notation

    • calculate finite sums of arithmetic sequences

    • calculate finite sums of geometric sequences

    • calculate sum of an infinite geometric series

  • reference:

    • OSP 11.4

    • DWFK 9.5

    • SZ 9.2

Unit 7: Exponential and logarithm

• 7.1 Exponential / logarithm 1

  • learning targets:

    • graph \(2^x\), \(10^x\), \(e^x\)

    • graph \(\log_2 x\), \(\log_{10} x\), and \(\log_e x = \ln x\)

  • reference:

    • OSP 4.1-4.7

    • DWFK 3.1-3.4

    • SZ 6.1-6.5

• 7.2 Exponential / logarithm 2

  • learning targets:

    • solve equations involving exponentials or logarithms

    • apply the base change formula

    • apply exponential/logarithmic models to problems involving population growth or radioactive decay

Unit 8: Derivatives

• 8.1 Limits and End Behavior

  • learning targets:

    • calculate limits of functions, and describe using limit notation

    • describe function behavior at asymptotes using limit notation

    • describe end behavior of functions using limit notation

    • apply the special limit: \(\lim_{x \to 0}\frac{\sin x}{x} = 1\)

  • reference:

    • OSC1 2.2-2.3

    • FDWK 2.1-2.2

• 8.2 Continuity

  • learning targets:

    • determine locations and types of discontinuities (removable, jump, infinite)

    • describe the behavior of a function near discontinuities using limit notation

  • reference:

    • OSC1 2.4

    • FDWK 2.3

• 8.3 Derivative Limit Definition

  • learning targets:

    • find the derivative of a function using the standard definition (\(h\to 0\))

    • find the derivative of a function using the alternate definition \((x \to a)\)

  • reference:

    • OSC1 3.1

    • FDWK 3.1-3.2

• 8.4 Differentiation Rules

  • learning targets:

    • apply the power rule

    • apply the sum rule

    • apply the product rule / quotient rule

    • find the derivative of any polynomial or rational function

  • reference:

    • OSC1 3.2-3.3

    • FDWK 3.3

• 8.5 Rates of change

  • learning targets:

    • use calculus to solve projectile motion problems

    • use calculus to solve circular motion problems

    • use calculus to solve population growth problems

  • reference:

    • OSC1 3.4

    • FDWK 2.4

• 8.6 Derivatives of Trig Functions

  • learning targets:

    • find the derivatives of sin/cos using the limit definition, plus special limits

    • derive formulas for derivatives of the other trig functions

  • reference:

    • OSC1 3.5

    • FDWK 3.5

Unit 9: More Derivatives

• 9.1 Chain rule

  • learning targets:

    • apply the chain rule to find the derivative of a composition of functions

  • reference:

    • OSC1 3.6

    • FDWK 4.1

• 9.2 Implicit differentiation

  • learning targets:

    • use implicit differentiation to find the derivative of an implicitly defined function

    • apply the general power rule

  • reference:

    • OSC1 3.8

    • FDWK 4.2

• 9.3 Derivatives of exponential / logarithm

  • learning targets:

    • apply the special limit: \(\lim_{h \to 0} \frac{a^h-1}{h} = \ln a\)

    • find the derivative of functions involving \(e^x\), \(a^x\)

    • find the derivative of functions involving \(\ln x\), \(\log_a x\)

  • reference:

    • OSC1 3.9

    • FDWK 4.4

• 9.4 Derivatives of inverse trig functions

  • learning targets:

    • derive formulas for inverse trig functions using implicit differentiation (plus a reference triangle)

  • reference:

    • OSC1 3.7

    • FDWK 4.3

Unit 10: Applications

• 10.1 Extreme values

  • learning targets:

    • identify whether the Extreme Value Theorem applies to a given function on a given interval

    • identify the extreme values of a function

  • reference:

    • OSC1 4.3

    • FDWK 5.1

• 10.2 Mean Value Theorem

  • learning targets:

    • apply Rolle’s Theorem

    • state the Mean Value Theorem

    • state Corollary 1 (if \(f' = 0\) then \(f\) is constant)

    • state Corollary 2 (if \(f' = g'\) then \(f = g + \text{constant}\)

    • state Corollary 3: (if \(f'>0\) then \(f\) is increasing)

    • find all antiderivatives of a function

  • reference:

    • OSC1 4.4

    • FDWK 5.2

• 10.3 Extreme value tests

  • learning targets:

    • apply the 1st derivative test (including one-sided) to verify a max/min

    • determine intervals of concavity and inflection points

    • apply the 2nd derivative test to verify a max/min

  • reference:

    • OSC1 4.5

    • FDWK 5.3

• 10.4 Optimization

  • learning targets:

    • apply calculus to find the max/min of a function

    • find the max/min of constrained functions

  • reference:

    • OSC1 4.7

    • FDWK 5.4

• 10.5 Linearization / Newton’s Method

  • learning targets:

    • find the linear approximation of functions

    • apply Newton’s Method for finding roots

  • reference:

    • OSC1 4.2, 4.9

    • FDWK 5.5