My Math Forum Difficult Local Linearization Problem

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 November 2nd, 2011, 05:12 PM #1 Newbie   Joined: Sep 2011 Posts: 18 Thanks: 0 Difficult Local Linearization Problem The "rule of 70" is used o estimate the number of years it will take for the money in a bank account to double based of the interest rate, i%. The rule states that the amount will take 70/i years to double. Find the local linearization of ln(x+1) and explain how this proves that the rule works. So I understand how the find the local linearization. It would be L(x)=ln(a+1)+((x-a)/(1+a)). But how does this relate to the rule? I'm completely lost after this point. Any help?
 November 3rd, 2011, 06:12 PM #2 Senior Member   Joined: Jul 2011 Posts: 118 Thanks: 0 Re: Difficult Local Linearization Problem $f(x+\Delta x)\simeq f(x)+f'(x)\cdot\Delta x (\ln x)'=\frac{1}{x} \ln (1+x)\simeq \ln 1+\frac{1}{1}\cdot x=x M$$1+\frac{i}{100}$$^y=2M $$1+\frac{i}{100}$$^y=2 \ln$$1+\frac{i}{100}$$^y=\ln 2 y\ln$$1+\frac{i}{100}$$=\ln 2 y\ln$$1+\frac{i}{100}$$\simeq y\cdot\frac{i}{100} \ln 2\simeq 0.69 y\cdot\frac{i}{100}=0.69 yi=69 y=\frac{69}{i}\simeq \frac{70}{i}$
 November 3rd, 2011, 09:08 PM #3 Newbie   Joined: Sep 2011 Posts: 18 Thanks: 0 Re: Difficult Local Linearization Problem Thank you!!

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