91Ó°ÊÓ

Wheat is poured through a chute at the rate of \(10 \mathrm{ft}^{3} / \mathrm{min}\) and falls in a conical pile whose bottom radius is always half the altitude. How fast will the circumference of the base be increasing when the pile is \(8 \mathrm{ft}\) high?

Use implicit differentiation to find the specified derivative. \(a^{4}-t^{4}=6 a^{2} t ; d a / d t\)

Find the limits. $$ \lim _{x \rightarrow 0}\left(e^{x}+x\right)^{1 / x} $$

Find \(f^{\prime}(x)\) by Formula (7) and then by logarithmic differentiation. $$ f(x)=\pi^{\sin x} $$

An aircraft is climbing at a \(30^{\circ}\) angle to the horizontal. How fast is the aircraft gaining altitude if its speed is \(500 \mathrm{mi} / \mathrm{h}\) ?

A boat is pulled into a dock by means of a rope attached to a pulley on the dock (see the accompanying figure). The rope is attached to the bow of the boat at a point \(10 \mathrm{ft}\) below the pulley. If the rope is pulled through the pulley at a rate of \(20 \mathrm{ft} / \mathrm{min}\), at what rate will the boat be approaching the dock when \(125 \mathrm{ft}\) of rope is out?

Find the limits. $$ \lim _{x \rightarrow+\infty}(1+a / x)^{b x} $$

Find \(f^{\prime}(x)\) by Formula (7) and then by logarithmic differentiation. $$ f(x)=\pi^{x \tan x} $$

Use implicit differentiation to find the specified derivative. \(\sqrt{u}+\sqrt{v}=5 ; \quad d u / d v\)

Use an appropriate local linear approximation to estimate the value of the given quantity. $$ \tan 0.2 $$