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Home Real and Hyperreal Numbers Slope and Velocity; the Hyperreal Line Average Slope  
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Average Slope
If (x_{0}, y_{0}) and (x_{0} + Δx, y_{0} + Δy) are two points on the curve, then the "average slope" of the curve between these two points is defined as the ratio of the change in y to the change in x, This is exactly the same as the slope of the straight line through the points (x_{0}, y_{0}) and (x_{0} + Δx, y_{0} + Δy), as shown in Figure 1.4.1. Figure 1.4.1 Let us compute the average slope. The two points (x_{0}, y_{0}) and (x_{0} + Δx, y_{0} + Δy) are on the curve, so y_{0} = x_{0}^{2} y_{0} + Δy = (x_{0} + Δx)^{2}. Subtracting, Δy = (x_{0} + Δx)^{2}  x_{0}^{2}. Dividing by Δx,
This can be simplified, Thus the average slope is Notice that this computation can only be carried out when Δx ≠ 0, because at Δx = 0 the quotient Δy/Δx is undefined.


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