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Your No-Nonsense Guide to Calculus — Finding Slopes

A quick overview without jargon

Two puzzles kept ancient smart people tossing and turning at night: 1) The area under a curve; 2) The slope of a curve at a point on the curve.

In this article, we’ll take a long-range view of both problems. Then we’ll home in on the calculus approach to finding the slope of a curve.

Here’s the visual:

Two problems of antiquity: the area under a curve, and its slope at a point.

Let’s begin with a trivial case.

A trivial example. Find the area of the triangle and the slope of its hypotenuse.

In the above diagram, our curve is the straight line itself. We are interested in the area of the triangle beneath it.

Here’s the slope of the straight line.

Give yourself an pat on the back. You’ve just done some calculus.

We want to generalize this process. We want to turn our curve formula into two other formulas. One formula will give us the slope; the other the area.