9514 1404 393
Answer:
- x2 = .72413793
- x3 = .087249546
Step-by-step explanation:
Modern graphing calculators have a derivative function available, so using a calculator to find the next value of x is pretty simple.
The Newton's Method iterator for finding the next approximation to the root (x') is ...
x' = x -f(x)/f'(x) . . . . . where f'(x) is the derivative of f(x).
The attachment shows the first 3 iterations (4 approximations). We observe that the starting point is pretty far from the root, and on the wrong side of some wiggles in the function, so convergence is pretty slow.
The desired approximations are shown above and in the table below.
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<em>Additional comment</em>
To 12 significant figures, the only real root is −1.10305899649. When the calculator can interactively produce a next guess, you can type the next guess value into the iterator function even as it is showing you the next value. This lets you find the best-precision result as fast as you can type it.
For a calculator like a TI-84, the iterator function can make repeated use of "Ans" as an argument. It usually doesn't take more than 3 or 4 iterations to get a best-precision result, since the number of good decimal places is about doubled on each iteration. (Of course, you have to start with a better approximation than the one given in this problem.)
