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Chapter 4: Problem 8
Find a number \(c\) such that \(\ln c=5\).
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How long does it take for money to triple when compounded continuously at \(5 \%\) per year?
The functions cosh and \(\sinh\) are defined by $$ \cosh x=\frac{e^{x}+e^{-x}}{2} \text { and } \sinh x=\frac{e^{x}-e^{-x}}{2} $$ for every real number \(x .\) For reasons that do not concern us here, these functions are called the hyperbolic cosine and hyperbolic sine; they are useful in engineering. Show that $$ (\cosh x)^{2}-(\sinh x)^{2}=1 $$ for every real number \(x\).
Suppose a bank account that compounds interest continuously grows from \(\$ 100\) to \(\$ 110\) in two years. What annual interest rate is the bank paying?
Estimate the value of $$ \left(1-\frac{2}{8^{99}}\right)^{\left(8^{99}\right)} $$
Suppose one bank account pays \(5 \%\) annual interest compounded once per year, and a second bank account pays 5\% annual interest compounded continuously. If both bank accounts start with the same initial amount, how long will it take for the second bank account to contain twice the amount of the first bank account?
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