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Chapter 3: Problem 56
Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=8 \cdot 7^{x} $$
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Show that \(\sinh\) is a one-to-one function and that its inverse is given by the formula $$ (\sinh )^{-1}(y)=\ln \left(y+\sqrt{y^{2}+1}\right) $$ for every real number \(y\).
Estimate the indicated value without using a calculator. \(e^{-0.00046}\)
Estimate the indicated value without using a calculator. \(\ln 1.003\)
Show that $$ (\cosh x)^{2}-(\sinh x)^{2}=1 $$ for every real number \(x\).
What is wrong with the following apparent paradox: You have two parents, four grandparents, eight greatgrandparents, and so in. Going back \(n\) generations, you should have \(2^{n}\) ancestors. Assuming three generations per century, if we go back 2000 years (which equals 20 centuries and thus 60 generations), then you should have \(2^{60}\) ancestors from 2000 years ago. However, \(2^{60}=\left(2^{10}\right)^{6} \approx\left(10^{3}\right)^{6}=10^{18},\) which equals a billion billion, which is far more than the total number of people who have ever lived.
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