Chapter 1: Problem 71
Solve the inequality for \(x\). $$ -2<\ln x<0 $$
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In Exercises 115 and \(116,\) find the point of intersection of the graphs of the functions. $$ \begin{array}{l} y=\arccos x \\ y=\arctan x \end{array} $$
The signum function is defined by \(\operatorname{sgn}(x)=\left\\{\begin{array}{ll}-1, & x<0 \\ 0, & x=0 \\ 1, & x>0\end{array}\right.\) Sketch a graph of \(\operatorname{sgn}(x)\) and find the following (if possible). (a) \(\lim _{x \rightarrow 0^{-}} \operatorname{sgn}(x)\) (b) \(\lim _{x \rightarrow 0^{+}} \operatorname{sgn}(x)\) (c) \(\lim _{x \rightarrow 0} \operatorname{sgn}(x)\)
Use the Intermediate Value Theorem to show that for all spheres with radii in the interval [1,5] , there is one with a volume of 275 cubic centimeters.
$$ \begin{aligned} &\text { Prove that if } f \text { and } g \text { are one-to-one functions, then }\\\ &(f \circ g)^{-1}(x)=\left(g^{-1} \circ f^{-1}\right)(x). \end{aligned} $$
Use a graphing utility to graph the function on the interval \([-4,4] .\) Does the graph of the function appear continuous on this interval? Is the function continuous on [-4,4]\(?\) Write a short paragraph about the importance of examining a function analytically as well as graphically. $$ f(x)=\frac{e^{-x}+1}{e^{x}-1} $$
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