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Slope of a Nonlinear Curve

We now move from the simple world of linear relationships (straight lines) to the more complex world of nonlinear relationships. The slope of a straight line is the same at all its points. The slope of a line representing a nonlinear relationship changes from one point to another. Such lines are referred to as curves. (It is also permissible to refer to a straight line as a "curve.") Consider the downsloping curve in Figure A1-4. Its slope is negative throughout, but the curve flattens as we move down along it. Thus, its slope constantly changes; the curve has a different slope at each point.

To measure the slope at a specific point, we draw a straight line tangent to the curve at that point. A line is tangent at a point if it touches, but does not intersect, the curve at that point. Thus line aa is tangent to the curve in Figure A1-4 at point A. The slope of the curve at that point is equal to the slope of the tangent line. Specifically, the total vertical change (drop) in the tangent line aa is -20 and the total horizontal change (run) is +5. Because the slope of the tangent line aa is -20/ +5, or -4, the slope of the curve at point A is also -4.

Line bb in Figure A1-4 is tangent to the curve at point B. Following the same procedure, we find the slope at B to be -5/+15, or -1/ . Thus, in this flatter part of the curve, the slope is less negative. (Key Appendix Question 6)