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Chapter 1: Problem 30
Find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? $$ f(x)=\left\\{\begin{array}{ll} -2 x+3, & x<1 \\ x^{2}, & x \geq 1 \end{array}\right. $$
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Rate of Change A patrol car is parked 50 feet from a long warehouse (see figure). The revolving light on top of the car turns at a rate of \(\frac{1}{2}\) revolution per second. The rate \(r\) at which the light beam moves along the wall is \(r=50 \pi \sec ^{2} \theta \mathrm{ft} / \mathrm{sec}\) (a) Find \(r\) when \(\theta\) is \(\pi / 6\). (b) Find \(r\) when \(\theta\) is \(\pi / 3\). (c) Find the limit of \(r\) as \(\theta \rightarrow(\pi / 2)^{-}\)
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. $$ f(x)=\frac{e^{-x}+1}{e^{x}-1} $$
Prove that if \(f\) has an inverse function, then \(\left(f^{-1}\right)^{-1}=f\).
Prove that if \(f\) is continuous and has no zeros on \([a, b],\) then either \(f(x)>0\) for all \(x\) in \([a, b]\) or \(f(x)<0\) for all \(x\) in \([a, b]\)
$$ \lim _{x \rightarrow 2} f(x)=3, \text { where } f(x)=\left\\{\begin{array}{ll} 3, & x \leq 2 \\ 0, & x>2 \end{array}\right. $$
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