91Ó°ÊÓ

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.

Solve the given problems by finding the appropriate derivative. By Newton's method, find the value of \(x\) for which \(y=e^{\cos x}\) is minimum for \(0

Find the derivatives of the given functions. $$y=\frac{1}{2} \sin 2 x \sec x$$

Evaluate each limit (if it exists). Use \(L\) Hospital's rule (if appropriate). $$\lim _{x \rightarrow \infty} \frac{e^{2 x}-1}{4 x+1}$$

Find the derivatives of the given functions. $$y=\frac{2 \csc 3 x}{x^{2}}$$

Solve the given problems by finding the appropriate derivative. The electric current \(i\) (in A) through an inductor of \(0.50 \mathrm{H}\) as a function of time \(t\) (in s) is \(i=e^{-5.0 t} \sin 120 \pi t .\) The voltage across the inductor is given by \(V_{L}=L(d i / d t),\) where \(L\) is the inductance (in \(\mathrm{H}\) ). Find the voltage across the inductor for \(t=1.0 \mathrm{ms}\).

Find the derivatives of the given functions. $$r=\frac{\sin (3 t-\pi / 3)}{2 t}$$

Evaluate each limit (if it exists). Use \(L\) Hospital's rule (if appropriate). $$\lim _{x \rightarrow \infty} \frac{1+e^{2 x}}{2+\ln x}$$

Find the derivatives of the given functions. $$y=\sin \ln x$$

Find the derivatives of the given functions. $$u=4 \sqrt{\ln 2 t+e^{2 t}}$$