Geometry for Programmers

5 The geometry of calculus

This chapter covers

• Using the derivative as a measure of how fast a function grows
• Computing the derivative of a function not only as a number, but also as a function itself
• Using derivatives to compute tangent vectors of parametric curves
• Using second derivatives to assess the curvature of a curve
• Composing smooth piecewise functions and parametric curves

People usually have a pretty good intuition about the topics we’re going to talk about. Does a function grow or decline? Is a surface smooth, or does it have some sharp edges? Is one curve curvier than another? Usually, we can answer all these questions intuitively. But we can’t program intuition. We need formal things; we need formulas; we need equations. We need ...