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Chapter 4: Problem 9
Find a number \(x\) such that \(\ln x=-2\).
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Find the equation of the circle centered at the origin in the \(x y\) -plane that has circumference \(9 .\)
Show that the area inside a circle with circumference \(c\) is \(\frac{c^{2}}{4 \pi}\).
(a) Using a calculator or computer, verify that $$ 2^{t}-1 \approx 0.693147 t $$ for some small numbers \(t\) (for example, try \(t=0.001\) and then smaller values of \(t\) ). (b) Explain why \(2^{t}=e^{t \ln 2}\) for every number \(t\). (c) Explain why the approximation in part (a) follows from the approximation \(e^{t} \approx 1+t\)
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 $$ \cosh x \approx \sinh h \approx \frac{e^{x}}{2} $$
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 the range of \(\sinh\) is the set of real numbers.
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