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Chapter 4: Problem 12
Find a number \(w\) such that \(\ln (3 w-2)=5\).
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Estimate the indicated value without using a calculator. $$ \ln 3.0012-\ln 3 $$
Suppose a colony of bacteria has doubled in two hours. What is the approximate continuous growth rate of this colony of bacteria?
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 $$ \cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y $$ for all real numbers \(x\) and \(y\).
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 if \(x\) is very large, then $$ \cosh x \approx \sinh h \approx \frac{e^{x}}{2} $$
Using a calculator, discover a formula for a good approximation for $$ \ln (2+t)-\ln 2 $$ for small values of \(t\) (for example, try \(t=0.04\), \(t=0.02, t=0.01,\) and then smaller values of \(t\) ). Then explain why your formula is indeed a good approximation.
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