Chapter 4 | Applications of Derivatives
479
(0)+4=4 (4)+4=6 (6)+4=7 (7)+4=7.5
x 1 = 1 2 x 2 = 1 2 x 3 = 1 2 x 4 = 1 2 x 5 = 1 2 x 6 = 1 2 x 7 = 1 2 x 8 = 1 2 x 9 = 1 2
(7.5) + 4 = 7.75
(7.75) + 4 = 7.875 (7.875) + 4 = 7.9375
(7.9375) + 4 = 7.96875 (7.96875) + 4 = 7.984375.
From this list, we conjecture that the values x n approach 8. Figure 4.82 provides a graphical argument that the values approach 8 as n →∞. Starting at the point ( x 0 , x 0 ), we draw a vertical line to the point ⎛ ⎝ x 0 , F ( x 0 ) ⎞ ⎠ . The next number in our list is x 1 = F ( x 0 ). We use x 1 to calculate x 2 . Therefore, we draw a horizontal line connecting ( x 0 , x 1 ) to the point ( x 1 , x 1 ) on the line y = x , and then draw a vertical line connecting ( x 1 , x 1 ) to the point ⎛ ⎝ x 1 , F ( x 1 ) ⎞ ⎠ . The output F ( x 1 ) becomes x 2 . Continuing in this way, we could create an infinite number of line segments. These line segments are trapped between the lines F ( x ) = x 2 +4 and y = x . The line segments get closer to the intersection point of these two lines, which occurs when x = F ( x ). Solving the equation x = x 2 +4, we conclude they intersect at x =8. Therefore, our graphical evidence agrees with our numerical evidence that the list of numbers x 0 , x 1 , x 2 ,… approaches x * =8 as n →∞.
Figure 4.82 This iterative process approaches the value x * =8.
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