Ahead of the Curve

This 70-page manual focuses on 33 mathematical equations and their curves, different from what the student usually sees in a typical calculus course.  Curves with strange names like cissoids, lemniscates, quartics, nephroids, catenaries, ovals, foliums, conchoids, and devil curves, to name a few, are analyzed.

Students are asked to use their implcit differentiation skills to find tangent lines including vertical and horizontal tangents, areas enclosed by the curves, volume when the curve is rotated about an axis, and for BC students, arc length of the curve. Formulas for the curves are given in Cartesian, parametric, and polar forms and students will see why these alternate forms make certain integration problems easier. They will also appreciate the mathematicians who first discovered these curves, all well before the age of calculators and computers.

The workbook encourages students to use computer graphing apps such as Grapher and Desmos, as many of the curves cannot be graphed on their calculators because they cannot easily be written in closed form (y as a function of x). Download links to view these curves in Mac Grapher and Desmos are given as well.

The calculus necessary for this workbook is no more advanced than what is used in AP calculus. But the algebra is more complicated and this workbook is terrific in enhancing the algebraic skills that sadly is missing in even the better calculus students. 

Teachers will enjoy having this collection of curves to challenge students. They are perfect as emergency lesson plans and teachers can distribute them as extra credit problems as well.

As usual, the student version of Ahead of the Curve is absolutely free.