Chapter 3

Why Calculus Works

In This Chapter

arrow Using limits to zoom in on curves

arrow Slope equals rise over run

arrow Area of a triangle equals one-half base times height

arrow The Pythagorean theorem: 9781118791295-eq0301.tif

In Chapters 1 and 2, I talk a lot about the process of zooming in on a curve till it looks straight. The mathematics of calculus works because of this basic nature of curves — that they’re locally straight — in other words, curves are straight at the microscopic level. The earth is round, but to us it looks flat because we’re sort of at the microscopic level when compared to the size of the earth. Calculus works because after you zoom in and curves look straight, you can use regular algebra and geometry with them. The zooming-in process is achieved through the mathematics of limits.

The Limit Concept: A Mathematical Microscope

The mathematics of limits is the microscope that zooms in on a curve. Here’s how a limit works. Say you want the exact slope or steepness of the parabola at the point (1, 1). ...

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