91Ó°ÊÓ

Solve the given maximum and minimum problems. A de-generator with an internal resistance \(r\) develops \(V\) volts. If the variable resistance in the circuit is \(R,\) the power generated is \(P=\frac{V^{2}}{r+R} .\) What resistance \(R\) gives the maximum power?

Find those values of \(x\) for which the given functions are increasing and those values of \(x\) for which they are decreasing. $$y=x^{4}-6 x^{2}$$

In Exercises \(5-8,\) find those values of \(x\) for which the given functions are increasing and those values of \(x\) for which they are decreasing. $$y=x^{4}-6 x^{2}$$

find the equations of the lines normal to the indicated curves at the given points. sketch the curve and normal line. In Exercises 8 and \(9,\) use a calculator to view the curve and normal line. y=8-x^{3} \text { at }(-1,9)

If an airplane is moving at velocity \(v\), the drag \(D\) on the plane is \(D=a v^{2}+b / v^{2},\) where \(a\) and \(b\) are positive constants. Find the value(s) of \(v\) for which the drag is the least.

Solve the given maximum and minimum problems. If an airplane is moving at velocity \(v,\) the drag \(D\) on the plane is \(D=a v^{2}+b / v^{2},\) where \(a\) and \(b\) are positive constants. Find the value(s) of \(v\) for which the drag is the least.

Find the indicated roots of the given equations to at least four decimal places by using Newton's method. Compare with the value of the root found using a calculator. \(2 x^{3}+2 x^{2}-11 x+8=0 \quad\) (the real root)

$$\text { Solve the problems in related rates.}$$ How fast is the slope of a tangent to the curve \(y=2(1-2 x)^{2}\) changing where \(x=1.5\) if \(d x / d t=0.75\) unit \(/ \mathrm{s} ?\)

Find the differential of each of the given functions. $$y=2 \sqrt{x}-\frac{1}{8 x}$$

Solve the problems in related rates. How fast is the slope of a tangent to the curve \(y=2(1-2 x)^{2}\) changing where \(x=1.5\) if \(d x / d t=0.75\) unit \(/ \mathrm{s} ?\)