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Precalculus

Overview

This course builds on a strong algebra foundation to explore further implications of the function concept, starting with graph transformations of the functions we already know. We draw on geometry knowledge to develop the circular functions, which enjoy a beautiful connection to the exponential function by way of the complex numbers and infinite series. Armed with these tools, we then consider polar coordinates, vector-valued functions (AKA parametric equations), and the algebra of vectors in Cartesian coordinates.

Selected learning outcomes

analyze 2D and 3D geometric constructions using the language of vector algebra.
build and interpret models of periodic phenomena using the circular functions.
solve trigonometric equations and prove trigonometric identities.
develop sequences and series to model discrete-time phenomena.
apply laws of exponents and logarithms to solve transcendental equations.
graph by hand rational and polynomial equations.
find (complex) roots of algebraic equations and (real) solution sets of algebraic inequalities.

Connections to other courses

Strengthened algebraic manipulation emerges by proving identities and simplifying transcendental expressions. This fluency will remove one of the major obstacles to success in calculus.
Deepened mathematical appreciation follows from revisiting the same result in different settings, e.g. proving the addition formulas for sine and cosine by using a coordinate transformation, or by multiplying two exponential functions of imaginary arguments and interpreting the result in both polar and rectangular coordinates.
Expanded mathematical vocabulary will allow succinct discourse about periodic phenomena (with circular functions) and theorems in Euclidean geometry (with vectors).

Non-traditional sequencing of topics

Changes to a course always stem from the desire to improve student learning. Sometimes my students give feedback directly, and other times I use assessment data to determine what isn't working. I planned out my first precalculus course for a private high school using as a template the sequencing my own high school teachers followed. That semester, the early introduction of vectors and dot product suffered more from my inexperience in the classroom than from any mismatch between the learning outcomes and the students' background. In subsequent semesters I stuck with the textbook sequencing more rigidly, with student outcomes much better but possibly more fragmented than in the capstone course I imagined myself teaching. By the spring of 2014 I had worked with enough college algebra students to attract a quorum for that experimental precalculus course. That semester, variations on the themes of vectors and multidimensional thinking unfolded in the following unorthodox sequence:

vectors in rectangular coordinates, dot product
cross product, equations of lines and planes, law of cosines
law of sines, area formula, solving triangles
unit circle, radian measure, circular functions
graphs of the circular functions, fundamental identities
inverse circular functions, angle sum and difference formulas
further identities, solving trigonometric equations
sequences and series of numbers, sigma notation
exponential functions, one-to-one functions and their inverses
logarithmic functions
exponential and logarithmic equations
rational functions and graphs, dividing polynomials long-hand and synthetically
finding polynomial roots, Rouche's theorem

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