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As another example, if you want to find out where a ball will come to rest at the bottom of a hill, you look for a point where the tangent line is horizontal.

The steepness of the tangent line—its slope—has some important properties in physics: for instance, if you’ve got a curve that represents the position of, say, a bicycle, then the slope of the tangent line to that curve at any given point tells you how fast that bicycle is going when it reaches that spot.

To draw a tangent line at any given point, it’s best to make a guess.

The line you get isn’t exactly the tangent line, but if the curve isn’t too bumpy, the two lines will be pretty close.

As you reduce the distance between the points, the guess gets closer to the tangent line.