128Chapter 6 Applications of the Derivativeis usually easy to do. Let’s use√3≈2. Now use the tangent line to the curve whenx= 2as an approximation to the curve, as shown in figure6.17. Sincef(x) = 2x, the slope ofthis tangent line is 4 and its equation isy= 4x-7. The tangent line is quite close tof(x),so it crosses thex-axis near the point at whichf(x) crosses, that is, near√3. It is easyto find where the tangent line crosses thex-axis: solve 0 = 4x-7 to getx= 7/4 = 1.75.This is certainly a better approximation than 2, but let us say not close enough. We canimprove it by doing the same thing again: find the tangent line atx= 1.75, find wherethis new tangent line crosses thex-axis, and use that value as a better approximation. Wecan continue this indefinitely, though it gets a bit tedious. Lets see if we can shortcut theprocess. Suppose the best approximation to the intercept we have so far isxi. To find a
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