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Chapter 4: Problem 29
Suppose a colony of bacteria has doubled in five hours. What is the approximate continuous growth rate of this colony of bacteria?
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(a) Find a function \(f\) such that the distance between the points (1,3) and \((x, f(x))\) equals the distance between (5,9) and \((x, f(x))\) for every real number \(x\). (b) Find a linear function \(f\) such that the graph of \(f\) contains the point (3,6) and is perpendicular to the line containing (1,3) and (5,9) . (c) Explain why the solutions to parts (a) and (b) of this problem are the same. Draw an appropriate figure to help illustrate your explanation.
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 $$ \cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y $$ for all real numbers \(x\) and \(y\).
Find a number \(r\) such that $$ \left(1+\frac{r}{10^{90}}\right)^{\left(10^{90}\right)} \approx 5 $$
Find a formula that gives the area inside a circle in terms of the diameter of the circle.
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