Find the exact length of the curve. Use a graph to determine the parameter interval. \(r=\cos^2(\frac{\theta}{2})\)

Change from rectangular to spherical coordinates. (Let \(\rho\geq0,\ 0\leq\theta\leq 2\pi\), and \(0\leq\phi\leq\pi.)\) a) \((0,\ -8,\ 0)(\rho,\theta, \phi)=?\) b) \((-1,\ 1,\ -2)(\rho, \theta, \phi)=?\)

Let \(\displaystyle{A}={\left\lbrace{x}:{x}{A}={\left\lbrace{x}:{x}\ {i}{s}\ {a}\ {n}{a}{t}{u}{r}{a}{l}\ nu{m}{b}{e}{r}{\quad\text{and}\quad}{a}\ {f}{a}{c}to{r}{\ o}{f}\ {18}\right\rbrace}\right.}\) \(\displaystyle{B}={\left\lbrace{x}:{x}{B}={\left\lbrace{x}:{x}\ {i}{s}\ {a}\ {n}{a}{t}{u}{r}{a}{l}\ nu{m}{b}{e}{r}{\quad\text{and}\quad}\ le{s}{s}\ {t}{h}{a}{n}\ {6}\right\rbrace}\right.}\) Find \(\displaystyle{A}\cup{B}{\quad\text{and}\quad}{A}\cap{B}.\)

Find the gradient at the point (3,-3,2) of the scaler field given by \(f=xy-4ze+35\)