Use Newton\'s Method to approximate the positive solution to the equation ln(x+4

Question

Use Newton's Method to approximate the positive solution to the equation ln(x+4)=x to three decimal places

Explanation / Answer

Solution: Let f(x) = 5x + lnx - 10000. We need to approximate the root(s) of the equation f(x) = 0. The function f is only dened for positive x. Note that the function is steadily increasing, since f0(x) = 5+1=x > 0 for all positive x. It follows that the function can be 0 for at most one value of x. It is easy to verify that f(1) < 0 and f(2000) > 0, and therefore the equation has a root in the interval (1; 2000). 4 The Newton Method iteration is easy to set up. We get xn+1 = xn - 5xn + lnxn - 10000 5 + 1=xn : We could simplify the right hand side somewhat. This is probably not worthwhile. Now we need to choose x0. The idea is that even when x is large, ln x is by comparison quite small. So as a rst approximation we can forget about the ln x term, and decide that f(x) is approximately 5x-10000. Thus the root of our original equation must be near x = 2000. Shall we choose x0 = 2000? It is sensible to do so. But we can do better. Note that ln(2000) is about 7:6. So we can take 5x0 10000 - 7:6. Let x0 = 1998:48. A quick computation gives x1 = 1998:479972. This agrees with x0 to 4 decimal places, so the answer, correct to 4 decimal places, should be 1998:4800. If we feel like it, we can show by the usual sign change" procedure that this answer is indeed correct to 4 places.

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