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Actual Slope

Reasoning in a nonrigorous way, the actual slope of the curve at the point (x0, y0) can be found thus. Let Δx be very small (but not zero). Then the point (x0 + Δx, y0 + Δy) is close to (x0, y0), so the average slope between these two points is close to the slope of the curve at (x0, y0);

[slope at (x0, y0)] is close to 2x0 + Δx.

We neglect the term Δx because it is very small, and we are left with

[slope at (x0, y0)] = 2x0.

For example, at the point (0, 0) the slope is zero, at the point (1, 1) the slope is 2, and at the point ( -3, 9) the slope is -6. (See Figure 1.4.2.)


Figure 1.4.2

Last Update: 2010-11-25