Chapter 4: Problem 19
In Exercises \(19-30\) , find the extreme values of the function and where they occur. \(y=2 x^{2}-8 x+9\)
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Multiple Choice A cylindrical rubber cord is stretched at a constant rate of 2 \(\mathrm{cm}\) per second. Assuming its volume does no change, how fast is its radius shrinking when its length is 100 \(\mathrm{c}\) and its radius is 1 \(\mathrm{cm} ?\) $$\begin{array}{ll}{\text { (A) } 0 \mathrm{cm} / \mathrm{sec}} & {\text { (B) } 0.01 \mathrm{cm} / \mathrm{sec}} 67 {\text{ (C) } 0.02 \mathrm{cm} / \mathrm{sec}}$\\\ {\text { (D) } 2 \mathrm{cm} / \mathrm{sec}} & {\text { (E) } 3.979 \mathrm{cm} / \mathrm{sec}}\end{array}
$$ \begin{array}{l}{\text { Multiple Choice Which of the following functions is an }} \\ {\text { antiderivative of } \frac{1}{\sqrt{x}} ? \quad \mathrm{}}\end{array} $$ $$ (\mathbf{A})-\frac{1}{\sqrt{2 x^{3}}}(\mathbf{B})-\frac{2}{\sqrt{x}} \quad(\mathbf{C}) \frac{\sqrt{x}}{2}(\mathbf{D}) \sqrt{x}+5(\mathbf{E}) 2 \sqrt{x}-10 $$
The domain of f^{\prime}\( is \)[0,1) \cup(1,2) \cup(2,3]
Linearization Show that the approximation of tan \(x\) by its linearization at the origin must improve as \(x \rightarrow 0\) by showing that $$\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$$
In Exercises 62 and \(63,\) feel free to use a CAS (computer algebra system), if you have one, to solve the problem. Logistic Functions Let \(f(x)=c /\left(1+a e^{-h x}\right)\) with \(a>0\) \(a b c \neq 0\) (a) Show that \(f\) is increasing on the interval \((-\infty, \infty)\) if \(a b c>0\) and decreasing if \(a b c<0\) . (b) Show that the point of inflection of \(f\) occurs at \(x=(\ln |a|) / b\)
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