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Home Real and Hyperreal Numbers Slope and Velocity; the Hyperreal Line Actual Slope  
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Actual Slope
Reasoning in a nonrigorous way, the actual slope of the curve at the point (x_{0}, y_{0}) can be found thus. Let Δx be very small (but not zero). Then the point (x_{0} + Δx, y_{0} + Δy) is close to (x_{0}, y_{0}), so the average slope between these two points is close to the slope of the curve at (x_{0}, y_{0}); [slope at (x_{0}, y_{0})] is close to 2x_{0} + Δx. We neglect the term Δx because it is very small, and we are left with [slope at (x_{0}, y_{0})] = 2x_{0}. 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


Home Real and Hyperreal Numbers Slope and Velocity; the Hyperreal Line Actual Slope 