91Ó°ÊÓ

Solve the given problems. The apparent power \(P_{a}\) (in \(\mathrm{W}\) ) in an electric circuit whose power is \(P\) and whose impedance phase angle is \(\phi\) is given by \(P_{a}=P \sec \phi\) Given that \(P\) is constant at \(12 \mathrm{W}\), find the time rate of change of \(P_{a}\) if \(\phi\) is changing at the rate of \(0.050 \mathrm{rad} / \mathrm{min},\) when \(\phi=40.0^{\circ}\).

Find the derivatives of the given functions. $$y=4 \cos x \csc x^{2}$$

Solve the given problems. A point on the outer edge of a \(38.0-\mathrm{cm}\) wheel can be described by the equations \(x=19.0\) cos \(6 \pi t\) and \(y=19.0\) sin \(6 \pi t\). Find the velocity of the point for \(t=0.600 \mathrm{s}\)

Evaluate each limit (if it exists). Use \(L\) Hospital's rule (if appropriate). $$\lim _{x \rightarrow 0} \frac{2 \sin x}{5 e^{x}}$$

Find the derivatives of the given functions. $$r=0.5 \ln \cos \left(\pi \theta^{2}\right)$$

Find the derivatives of the given functions. $$y=\sqrt{1+\sin 4 x}$$

Find the derivatives of the given functions. $$y=\frac{\cos ^{-1} x}{x^{3}}$$

Find the derivatives of the given functions. $$y=\left(2 e^{2 x}\right)^{3} \sin x^{2}$$

Find the derivatives of the given functions. $$y=\left(e^{3 / x} \cos x\right)^{2}$$

Solve the given problems. A machine is programmed to move an etching tool such that the position (in \(\mathrm{cm}\) ) of the tool is given by \(x=2 \cos 3 t\) and \(y=\cos 2 t\) where \(t\) is the time (in s). Find the velocity of the tool for \(t=4.1\) s. Find the acceleration of the tool of Exercise 22 for \(t=4.1\) s.