91Ó°ÊÓ

Find the local extrema of \(f(x)=8 \sec x+\csc x\) on the interval \((0, \pi / 2),\) and describe where \(f(x)\) is increasing or is decreasing on that interval.

Evaluate the integral. \(\int \frac{e^{x}}{\sqrt{4-e^{x}}} d x\)

Find the area of the region bounded by the graphs of \(y=x /\left(x^{4}+1\right), x=1,\) and \(y=0\).

Find the arc length of the graph of \(y=\cosh x\) from (0,1) to \((1, \cosh 1)\)

Find the points on the graph of \(y=\sinh x\) at which the tangent line has slope 2 .

Damped oscillations are oscillations of decreasing magnitude that occur when frictional forces are considered. Shown in the figure on the next page is a graph of the damped oscillations given by \(f(x)=e^{-x / 2} \sin 2 x\). (a) Find the \(x\) -coordinates of the extrema of \(f\) for \(0 \leq x \leq 2 \pi\) (b) Approximate the \(x\) -coordinates in part (a) to two decimal places.

The region bounded by the graphs of \(y=\cosh x\), \(x=-1, x=1,\) and \(y=0\) is revolved about the \(x\) -axis. Find the volume of the resulting solid.

Find the arc length of the graph of \(y=\ln \tanh \frac{1}{2} x\) from \(x=1\) to \(x=2\).

Use differentials to approximate the are length of the graph of \(y=\tan ^{-1} x\) from \(A(0,0)\) to \(B\left(0.1, \tan ^{-1} 0.1\right)\)

An airplane at a constant altitude of 5 miles and a speed of \(500 \mathrm{mi} / \mathrm{hr}\) is flying in a direction away from an observer on the ground. Use inverse trigonometric functions to find the rate at which the angle of elevation is changing when the airplane flies over a point 2 miles from the observer.