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Chapter 4: Problem 15
A ship sails north for 2 miles and then west for 5 miles. How far is the ship from its starting point?
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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 $$ (\cosh x+\sinh x)^{t}=\cosh (t x)+\sinh (t x) $$ for all real numbers \(x\) and \(t\).
Estimate the indicated value without using a calculator. $$ \ln 0.993 $$
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 $$ \cosh x \approx \sinh h \approx \frac{e^{x}}{2} $$
Suppose \(b\) is a small positive number. Estimate the slope of the line containing the points \(\left(e^{3}, 5+b\right)\) and \(\left(e^{3+b}, 5\right)\)
Show that \(\sinh x \approx x\) if \(x\) is close to 0 [The definition of sinh was given before Exercise 52 in Section \(4.3 .\)
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