Approximations explained with calculus 1) The mechanic's rule for approximating va is to first obtain an approximation x1 Then get successive approximations by xn+1 =;(xn +; a) Use the rule to approximate V11 and v13 to three decimal places. b) Use a single iteration of the rule and give your answer as a mixed number for the following cases Estimating v27 with initial guess of 5 Estimating V82 with initial guess of 9 Estimating v97 with initial guess of 10 c) With some practice that first iteration should be able to be done mentally. Give an explanation of what one iteration of the rule involves.

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Approximations explained with calculus

1) The mechanic’s rule for approximating \(\sqrt{a}\) is to first obtain an approximation \(x_1\). Then get successive approximations by \(x_{n+1} = \frac{1}{2} \left( x_n + \frac{a}{x_n} \right)\).

a) Use the rule to approximate \(\sqrt{11}\) and \(\sqrt{13}\) to three decimal places.

b) Use a single iteration of the rule and give your answer as a mixed number for the following cases:
   - Estimating \(\sqrt{27}\) with initial guess of 5
   - Estimating \(\sqrt{82}\) with initial guess of 9
   - Estimating \(\sqrt{97}\) with initial guess of 10

c) With some practice that first iteration should be able to be done mentally. Give an explanation of what one iteration of the rule involves.
Transcribed Image Text:Approximations explained with calculus 1) The mechanic’s rule for approximating \(\sqrt{a}\) is to first obtain an approximation \(x_1\). Then get successive approximations by \(x_{n+1} = \frac{1}{2} \left( x_n + \frac{a}{x_n} \right)\). a) Use the rule to approximate \(\sqrt{11}\) and \(\sqrt{13}\) to three decimal places. b) Use a single iteration of the rule and give your answer as a mixed number for the following cases: - Estimating \(\sqrt{27}\) with initial guess of 5 - Estimating \(\sqrt{82}\) with initial guess of 9 - Estimating \(\sqrt{97}\) with initial guess of 10 c) With some practice that first iteration should be able to be done mentally. Give an explanation of what one iteration of the rule involves.
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