Question

Let f(x) be a polynomial and let r be a root of f(x). If x_1 is sufficiently close to r then x_2 = i(x_1) is closer, x_3 = i(x_2) is closer still, etc. Here i(x) = x - f(x)/f'(x) is what we called the improvement function

a. Let f(x)=x^2-10. Compute i(x) in simplified form (i.e. everything in one big fraction involving x). Let r = sqrt(10) and x_1=3. Show a hand computation of x_2 and then x_3, expressing both your answers first as fractions and then (using a calculator if you like) as decimals.

b. A calculator says that sqrt(10) is 3.162277660168379331... Using your answer to a, compute x_3-sqrt(10) to five significant figures, writing your answer without using scientific notation. (If you're using Mathematica, things like N[Pi,30] to get 30 digits of accuracy would be useful).   

c Hand-draw (copying from a calculator or Mathematica would be fine) a reasonably accurate picture of the situation. Your picture must show the graph of f(x), the x-values x_1, x_2, x_3, and r, and the two tangent lines relevant to your computation. If you think it's better, you can draw two different pictures at different scales.  

Answer 1

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