91Ó°ÊÓ

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. $$y=\left(3 x-\cos ^{2} x\right)^{4}$$

Find the derivatives of the given functions. $$y=\ln (x \sqrt{x+1})$$

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 \(t=4.1 \mathrm{s}\)

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

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. $$y=\frac{3 \csc ^{2} x}{x}$$

In Exercises \(3-36,\) 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. $$u=4 \sqrt{\ln 2 t+e^{2 t}}$$

Solve the given problems by finding the appropriate derivative. The electric current \(i\) (in \(\mathrm{A}\) ) through an inductor of \(0.50 \mathrm{H}\) as a function of time \(t\) (in s) is \(i=e^{-5.0 t} \sin 120 \pi t .\) Find the voltage across the inductor for \(t=1.0 \mathrm{ms}\). (See Exercise 31 on page 773.)