Proposed Outline

PreCalc

Five booklets would comprise the "text" for a PreCalculus class.

intro to functions and modelling
1. function concepts:
  1. range and domain
  2. one-to-one and onto,
  3. vertical line test
  4. the equivalent of VLT for functions given via tables, formulas and algorithms
  5. average rate of change (ARoC) of a function over an interval
  6. the difference quotient
2. combining functions
  1. basic examples
  2. sums and differences
  3. products and quotients,
  4. compositions
  5. "end to end" - piece-wise functions
3. inverse functions
  1. the horizontal line test - existence of an inverse
  2. the equivalent of HLT for functions given via tables, formulas and algorithms
  3. finding an inverse
    1. algorithm for
    2. via the "sock and shoes" principle
    3. verifying a proposed inverse.
  4. relationship between the graphs of $f(x)$ and $f^{-1}(x)$
4. some basic types of functions
  1. intro to linear functions
  2. intro to polynomial and rational functions
  3. intro to exponential functions
  4. intro to periodic functions
5. transformations: shifting and scaling
  1. vertical shifts and scalings
  2. horizontal shifts and scalings
  3. reflections: horizontal, vertical and both
  4. even and odd functions
modelling with linear, polynomial and rational functions
1. linear functions
  1. intro
    1. definition.
    2. constant ARoC
    3. slope/intercept form
    4. point/slope form
    5. standard form
  2. applications of linear functions
    1. setting zero points and changing units
      1. parsecs and light-years
      2. Celsius/Fahrenheit conversion
      3. hours on a clock vs degrees on the unit circle
      4. "Let's call the year 2000 $t=0$..."
    2. initial values and rate of change
      1. fixed costs and per/unit costs
      2. modelling the water in a reservoir
      3. direct problems
        
        "What will be the value when $t=k$?"
        
        plugging in
      4. inverse problems
        
        "At what time will the value be $v$?"
        
        solving equations
  3. linear regression
    1. History: C. F. Gauss and re-locating the asteroid Ceres
    2. Using a calculator or a spreadsheet to determine a line of regression
    3. Meaning of the correlation coefficient
    4. Appropriateness of a linear model...
    5. Approximate solutions using the line of regression
2. polynomial functions
  1. intro
    1. what is a polynomial?
      1. power functions
      2. power functions vs polynomials
    2. terminology: terms, coefficients, leading term, constant term, zeros.
  2. quadratic polynomials
    1. intro
      1. projectile motion
      2. parabolic mirrors
      3. arches
      4. conic sections
    2. standard form
    3. factored form
    4. vertex form
    5. the quadratic formula
    6. reducible and irreducible quadratics
    7. completing the square
  3. cubic polynomials
    1. shifted and scaled version of $y=x^3$.
    2. cubics with 1, 2 or 3 zeros
      1. geometric features: y-intercept, zeros, extrema and points of inflection
    3. History - the cubic formula
      1. cubic discriminant
      2. Del Ferro, Fiore, Tartaglia, Cardano, Bombelli and Viète.
  4. general polynomials
    1. quartic, quintic and higher degree polynomials
    2. beyond the FOIL rule -- multiplying polynomials with the table method
      1. application: raising $(x+h)$ to various powers
    3. long division and synthetic division of polynomials
    4. the fundamental theorem of algebra
    5. graphing: local and eventual behavior
    6. factoring (completely) over the complex numbers
    7. History: the impossibility of solving quintic and higher degree polynomials "in radicals"
3. rational functions
  1. definition
  2. domain
  3. graphing
    1. local behavior
      1. zeros at zeros of the numerator
      2. vertical asymptotes at zeros of the denominator
      3. removable singularities (cancelling a factor from num. and denom.)
    2. eventual behavior
      1. zero or horizontal asymptotes
      2. slant asymptotes
      3. long division and general asymptotes
modelling with exponential and logarithmic functions
1. exponentials
  1. motivating examples
    1. how many times can we fold a sheet of paper?
    2. grains of rice on a chessboard
    3. carbon dating
  2. contrasting power functions and exponentials
  3. analogy between linear functions (and addition) and exponential functions (and multiplication)
  4. standard form: $A(t) = A_0 b^t$.
  5. the annual interest variant: $A(t) = A_0 (1+r)^t$
  6. compound interest
    1. semi-annually, quarterly, monthly, daily, secondly...
    2. the formula in terms of P, r and n.
  7. the limiting case: continuous interest and Euler's number
    1. $Pe^{rt}$
    2. History - Leonhard Euler
  8. the same formula in Finance and in Science
    1. $Pe^{rt}$ - $P$ is principal, $r$ is interest rate
    2. $A_0e^{kt} - $A_0$ is initial amount, $k$ is the "growth constant"
    3. conversion between $b^t$ and $e^{kt}$
  9. graphs of various exponential functions
    1. common features
    2. how they differ (horizontal scaling)
      1. $k$ and/or $r$ as a horizontal scale factor
  10. applications
    1. direct problems
    2. approximate answers to inverse problems
2. logarithms
  1. History: Napier in 1614 (tables) slide rules evolved 1620-1632 (Gunter and Oughtred)
    1. Originally developed before exponentials and used (because of their property of turning multiplication into addition) as an aid to calculation.
  2. common logarithms
  3. definition ($\log_b{(x)}$ as the inverse of $b^x$)
    1. informal version: logarithms as powers
  4. properties
    1. $\log{(ab)} = \log{(a)} + \log{(b)}$
    2. $\log{(a^b)} = b \cdot \log{(a)}$
  5. features of the graphs
    1. relationship between logs with different bases (vertical scaling)
    2. the "natural" logarithm
  6. the change of base formula
  7. logarithmic models
    1. sound intensity
    2. chemical pH
    3. earthquake intensity
    4. astronomical quantities
    5. using semilog and log-log graph paper
3. using logarithms to solve inverse problems with exponential models
4. using exponentials to solve inverse problems with logarithmic models
periodic functions and trigonometry
1. motivating examples
  1. average temperature throughout the year
  2. position of the piston in an internal combustion engine.
2. motion at one degree per minute (speed 1 in these units) around a circle of radius 1 centered at (0,0).
  1. sine and cosine functions giving the y and x coordinates (respectively) of the point.
  2. conventions about the unit circle $\theta = 0$ is at the 3 o'clock position ((1,0)), and $\theta$ increases going in the counterclockwise direction.
  3. the points at 0, 90, 180 and 270
3. shape of the graph of $y=\sin{(x)}$
  1. filling in more points on the graph - the 45-45-90 and the 30-60-90 triangles
  2. the unit circle with 16 points labelled with coordinates and angles
  3. plotting
4. periodic functions
  1. definition
  2. additional examples of periodic functions
    1. $\tan{(x)}$ - interpretation as a slope
    2. square-wave
    3. sawtooth
5. right triangle trigonometry
  1. terminology
    1. standard labelling of triangles, hypotenuse, "legs"
    2. naming relative to an angle - opposite and adjacent sides.
  2. the dreaded SOHCAHTOA mnemonic
    1. relation between the unit circle ($\sin{(x)}$ is the $y$-coordinate of the point...) and the triangle interpretations ($\sin{(x)}$ as "opp" over "hyp")
  3. solving right triangles
    1. basic tools
      1. the triangle inequality
      2. the sum of the angles is 180 degrees
      3. the Pythagorean theorem
    2. trigonometric tools
      1. inverses for the "big three" trig functions
        
        appropriate restrictions to obtain invertibility
  4. Examples
    1. abstract examples
    2. triangles embedded in word problems
    3. navigation problems - the compass rose (NNW etc) and headings in degrees
      1. the unfortunate failure of the Earth's surface to be flat...spherical triangles
    4. increased accuracy - minutes and seconds vs decimal degrees
  5. solving general triangles
    1. mnemonics for "determined" triangles (SSS, SSA -not necessarily unique, SAS, ASA)
      1. mechanical visualizations of which triangles are "determined"
    2. the law of sines
      1. proof
      2. applications - opposite pairs
      3. unknown sides vs. unknown angles - the reciprocal variant
      4. caution concerning obtuse angles
    3. the law of cosines
      1. proof
      2. applications - lack of an opposite pair
      3. unknown angles - the angle variant
    4. examples (tons of 'em)
      1. abstract
      2. embedded
  6. radian measure
    1. what makes "radians" natural? - dimensionless quantities
    2. History: degrees, minutes and seconds - angles, time and ancient Babylon
systems of equations

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Proposed Outline

PreCalc

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