Precalculus: free lessons and course outline

Precalculus is the course that decides how the first month of calculus goes. Most students who struggle in calculus are not struggling with calculus; they are struggling with the trigonometry, the algebra of rational expressions, the logarithms and the function notation that calculus assumes is automatic. The eleven units are ordered so that each one supplies what the next needs. Unit 1 sets up functions, transformations, composition, inverses and rate of change, because every later unit is an application of those ideas to a new family. Units 2 and 3 cover the algebraic and exponential families with their behavior at the edges. Units 4 through 7 build trigonometry from the unit circle, through graphs, identities and equations, to triangles, so that sine and cosine are understood as functions of an angle and not only as ratios in a right triangle. Unit 8 covers vectors, Unit 9 polar, parametric and complex forms, which are three ways of describing the same plane, and Unit 10 the conic sections and the sequences and series that calculus will later sum. Unit 11 introduces limits, continuity and the difference quotient, which is the course ending where calculus begins. Every lesson opens with the method, names the specific error that costs the marks, works an example in full, and then gives ten practice problems with complete solutions.