121616949-math.142

# 121616949-math.142 - 128 Chapter 6 Applications of the...

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128 Chapter 6 Applications of the Derivative is usually easy to do. Let’s use 3 2. Now use the tangent line to the curve when x = 2 as an approximation to the curve, as shown in figure 6.17 . Since f ( x ) = 2 x , the slope of this tangent line is 4 and its equation is y = 4 x - 7. The tangent line is quite close to f ( x ), so it crosses the x -axis near the point at which f ( x ) crosses, that is, near 3. It is easy to find where the tangent line crosses the x -axis: solve 0 = 4 x - 7 to get x = 7 / 4 = 1 . 75. This is certainly a better approximation than 2, but let us say not close enough. We can improve it by doing the same thing again: find the tangent line at x = 1 . 75, find where this new tangent line crosses the x -axis, and use that value as a better approximation. We can continue this indefinitely, though it gets a bit tedious. Lets see if we can shortcut the process. Suppose the best approximation to the intercept we have so far is x i . To find a
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