91Ó°ÊÓ

Determine where the graph of the given function is increasing, decreasing, concave up, and concave down. Then sketch the graph (see Example 4). $$ f(x)=\sqrt{\sin x} \text { on }[0, \pi] $$

The rate of change of volume \(V\) of a melting snowball is proportional to the surface area \(S\) of the ball; that is, \(d V / d t=-k S\), where \(k\) is a positive constant. If the radius of the ball at \(t=0\) is \(r=2\) and at \(t=10\) is \(r=0.5\), show that \(r=-\frac{3}{20} t+2 .\)

Use the Fixed-Point Algorithm with \(x_{1}\) as indicated to solve the equations to five decimal places. $$ x=\frac{3}{2} \cos x ; x_{1}=1 $$

Find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. $$ F(x)=6 \sqrt{x}-4 x \text { on }[0, \infty) $$

Use the Fixed-Point Algorithm with \(x_{1}\) as indicated to solve the equations to five decimal places. $$ x=2-\sin x ; x_{1}=2 $$

Show that the rectangle with maximum perimeter that can be inscribed in a circle is a square.

Evaluate the indicated indefinite integrals. $$ \int \frac{s(s+1)^{2}}{\sqrt{s}} d s $$

Identify the critical points and find the maximum value and minimum value on the given interval. $$ h(t)=\frac{t^{5 / 3}}{2+t} ; I=[-1,8] $$

From what height must a ball be dropped in order to strike the ground with a velocity of \(-136\) feet per second?

Find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. $$ f(x)=\frac{64}{\sin x}+\frac{27}{\cos x} \text { on }(0, \pi / 2) $$