91Ó°ÊÓ

Use trigonometric identities to compute the indefinite integrals. $$ \int \frac{\sin (x)}{\cos ^{2}(x)} d x $$

In each of Exercises \(41-48\), use the given information to find \(F(c)\). $$ F^{\prime}(x)=3 x^{2}, \quad F(2)=-4, \quad c=3 $$

A piece of paper is circular in shape, with radius 6 inches. A sector is cut from the circle and the two straight edges taped together to form a cone (see Figure 15). What angle for the sector will maximize the volume inside the cone?

In each of Exercises \(41-48\), use the given information to find \(F(c)\). $$ F^{\prime}(x)=4 \sin (x), \quad F(\pi / 3)=3, \quad c=\pi $$

A particle moves along the curve \(y=\sqrt{x}\) in an \(x y\) -plane in which each horizontal unit and each vertical unit represents \(1 \mathrm{~cm}\). At the moment the particle passes through the point \((4,2),\) its \(x\) -coordinate increases at the rate \(7 \mathrm{~cm} / \mathrm{s}\) At what rate does the distance between the particle and the point (0,5) change?

The derivative \(f^{\prime}\) of a function \(f\) is given. Determine and classify all local extrema of \(f\). $$ f^{\prime}(x)=(x-1)(x-2)^{2}(x-3)^{3}(x-4)^{4} $$

Put the fractions over a common denominator and use l'Hôpital's Rule to evaluate the limit, if it exists. \(\lim _{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{\ln (x+1)}\right)\)

Use trigonometric identities to compute the indefinite integrals. \(\text { Evaluate } \int \frac{1}{2^{x}} d x\)

Two \(10000 bonds with the same maturity are offered. Determine which is the better investment by calculating the effective yield of each. Price =\$ 8681,\) coupon rate \(=5 \%\) or Price \(=\$ 10674\) coupon rate \(=7 \%, n=20\)

Put the fractions over a common denominator and use l'Hôpital's Rule to evaluate the limit, if it exists. \(\lim _{x \rightarrow 0}\left(\frac{x}{1-\cos (x)}-\frac{2}{x}\right)\)