91Ó°ÊÓ

Jewelry A jeweler resizes a ring so that its inner circumference is 6 centimeters. (a) What is the radius of the ring? (b) If the ring's inner circumference can vary between 5.5 centimeters and 6.5 centimeters, how can the radius vary? (c) Use the \(\varepsilon-\delta\) definition of a limit to describe this situation. Identify \(\varepsilon\) and \(\delta\).

\( \text { True or False? } \) Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Prove that the figure formed by connecting consecutive midpoints of the sides of any quadrilateral is a parallelogram.

An open box of maximum volume is to be made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides.

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 \([a, b]\)

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) \(\lim _{x \rightarrow 0} \operatorname{sgn}(x)\)

Find the point of intersection of the graphs of the functions. $$ \begin{array}{l} y=\arcsin x \\ y=\arccos x \end{array} $$

Prove that \(\arctan x+\arctan y=\arctan \frac{x+y}{1-x y}, x y \neq 1\). Use this formula to show that \(\arctan \frac{1}{2}+\arctan \frac{1}{3}=\frac{\pi}{4}\)