Chapter 3

A steel ball will drop \(16 t^{2}\) feet in \(t\) seconds. Such a ball is dropped from a height of 64 feet at a horizontal distance 10 feet from a 48-foot street light. How fast is the ball's shadow moving when the ball hits the ground?

Find \(d y / d x\). \(y=\sqrt[4]{3 x^{2}-4 x}\)

Find the indicated derivative. $$ D_{t}\left(\frac{3 t-2}{t+5}\right)^{3} $$

Find \(D_{x} y\) using the rules of this section. $$ y=x\left(x^{2}+1\right) $$

Use \(f^{\prime}(x)=\lim _{t \rightarrow x}[f(t)-f(x)] /[t-x]\) to find \(f^{\prime}(x)\) (see Example 5). $$ f(x)=x^{2}-3 x $$

The interior of an open cylindrical tank is 12 feet in diameter and 8 feet deep. The bottom is copper and the sides are steel. Use differentials to find approximately how many gallons of waterproofing paint are needed to apply a \(0.05\) -inch coat to the steel part of the inside of the tank \((1\) gallon \(\approx 231\) cubic inches \() .\)

Find \(D_{x} y\). $$ y=\operatorname{coth}^{-1}(\tanh x) $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ h^{\prime}(x) \text { if } h(x)=\ln \left(x+\sqrt{x^{2}-1}\right) $$

Find \(d y / d x\). \(y=\left(x^{3}-2 x\right)^{1 / 3}\)

Find the indicated derivative. $$ D_{s}\left(\frac{s^{2}-9}{s+4}\right) $$

A Ferris wheel of radius 20 feet is rotating counterclockwise with an angular velocity of 1 radian per second. One seat on the rim is at \((20,0)\) at time \(t=0\). (a) What are its coordinates at \(t=\pi / 6 ?\) (b) How fast is it rising (vertically) at \(t=\pi / 6 ?\) (c) How fast is it rising when it is rising at the fastest rate?

Find \(D_{x} y\) using the rules of this section. $$ y=3 x\left(x^{3}-1\right) $$

Use \(f^{\prime}(x)=\lim _{t \rightarrow x}[f(t)-f(x)] /[t-x]\) to find \(f^{\prime}(x)\) (see Example 5). $$ f(x)=x^{3}+5 x $$

Assuming that the equator is a circle whose radius is approximately 4000 miles, how much longer than the equator would a concentric, coplanar circle be if each point on it were 2 feet above the equator? Use differentials.

Find \(D_{x} y\). $$ y=\sin ^{-1}\left(2 x^{2}\right) $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ f^{\prime}(81) \text { if } f(x)=\ln \sqrt[3]{x} $$

Find \(d y / d x\). \(y=\frac{1}{\left(x^{3}+2 x\right)^{2 / 3}}\)

An object is moving along a horizontal coordinate line according to the formula \(s=f(t)\), where \(s\), the directed distance from the origin, is in feet and \(t\) is in seconds. In each case, answer the following questions (see Examples 2 and 3\()\). (a) What are \(v(t)\) and \(a(t)\), the velocity and acceleration, at time \(t\) ? (b) When is the object moving to the right? (c) When is it moving to the left? (d) When is its acceleration negative? (e) Draw a schematic diagram that shows the motion of the object. $$ s=t^{3}-9 t^{2}+24 t $$

Find the indicated derivative. $$ \frac{d}{d t}\left(\frac{(3 t-2)^{3}}{t+5}\right) $$

$$ \begin{array}{l} \text {. Find the equation of the tangent line to } y=\tan x \text { at }\\\ x=0 \end{array} $$

Find \(D_{x} y\) using the rules of this section. $$ y=(2 x+1)^{2} $$

Use \(f^{\prime}(x)=\lim _{t \rightarrow x}[f(t)-f(x)] /[t-x]\) to find \(f^{\prime}(x)\) (see Example 5). $$ f(x)=\frac{x}{x-5} $$

The period of a simple pendulum of length \(L\) feet is given by \(T=2 \pi \sqrt{L / g}\) seconds. We assume that \(g\), the acceleration due to gravity on (or very near) the surface of the earth, is 32 feet per second per second. If the pendulum is that of a clock that keeps good time when \(L=4\) feet, how much time will the clock gain in 24 hours if the length of the pendulum is decreased to \(3.97\) feet?

Find \(D_{x} y\). $$ y=\arccos \left(e^{x}\right) $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ f^{\prime}\left(\frac{\pi}{4}\right) \text { if } f(x)=\ln (\cos x) $$

Find \(d y / d x\). \(y=(3 x-9)^{-5 / 3}\)

Find the indicated derivative. $$ \frac{d}{d \theta}\left(\sin ^{3} \theta\right) $$

Find all points on the graph of \(y=\tan ^{2} x\) where the tangent line is horizontal.

Find \(D_{x} y\) using the rules of this section. $$ y=(-3 x+2)^{2} $$

Use \(f^{\prime}(x)=\lim _{t \rightarrow x}[f(t)-f(x)] /[t-x]\) to find \(f^{\prime}(x)\) (see Example 5). $$ f(x)=\frac{x+3}{x} $$

Find \(D_{x} y\). $$ y=x^{3} \tan ^{-1}\left(e^{x}\right) $$

An 18 -foot ladder leans against a 12 -foot vertical wall, its top extending over the wall. The bottom end of the ladder is pulled along the ground away from the wall at 2 feet per second. (a) Find the vertical velocity of the top end when the ladder makes an angle of \(60^{\circ}\) with the ground. (b) Find the vertical acceleration at the same instant.

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x} e^{x+2} $$

Find \(d y / d x\). \(y=\sqrt{x^{2}+\sin x}\)

Find the indicated derivative. $$ \frac{d y}{d x}, \text { where } y=\left(\frac{\sin x}{\cos 2 x}\right)^{3} $$

. Find all points on the graph of \(y=9 \sin x \cos x\) where the tangent line is horizontal.

Find \(D_{x} y\) using the rules of this section. $$ y=\left(x^{2}+2\right)\left(x^{3}+1\right) $$

The given limit is a derivative, but of what function and at what point? (See Example 6.) $$ \lim _{h \rightarrow 0} \frac{2(5+h)^{3}-2(5)^{3}}{h} $$

A cylindrical roller is exactly 12 inches long and its diameter is measured as \(6 \pm 0.005\) inches. Calculate its volume with an estimate for the absolute error and the relative error.

Find \(D_{x} y\). $$ y=e^{x} \arcsin x^{2} $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x} e^{2 x^{2}-x} $$

Find \(d y / d x\). \(y=\sqrt{x^{2} \cos x}\)

An object is moving along a horizontal coordinate line according to the formula \(s=f(t)\), where \(s\), the directed distance from the origin, is in feet and \(t\) is in seconds. In each case, answer the following questions (see Examples 2 and 3\()\). (a) What are \(v(t)\) and \(a(t)\), the velocity and acceleration, at time \(t\) ? (b) When is the object moving to the right? (c) When is it moving to the left? (d) When is its acceleration negative? (e) Draw a schematic diagram that shows the motion of the object. $$ s=t+\frac{4}{t}, t>0 $$

Find the indicated derivative. $$ \frac{d y}{d t}, \text { where } y=\left[\sin t \tan \left(t^{2}+1\right)\right] $$

Let \(f(x)=x-\sin x .\) Find all points on the graph of \(y=f(x)\) where the tangent line is horizontal. Find all points on the graph of \(y=f(x)\) where the tangent line has slope 2 .

Find \(D_{x} y\) using the rules of this section. $$ y=\left(x^{4}-1\right)\left(x^{2}+1\right) $$

The rate of change of electric charge with respect to time is called current. Suppose that \(\frac{1}{3} t^{3}+t\) coulombs of charge flow through a wire in \(t\) seconds. Find the current in amperes (coulombs per second) after 3 seconds. When will a 20-ampere fuse in the line blow?

The angle \(\theta\) between the two equal sides of an isosceles triangle measures \(0.53 \pm 0.005\) radian. The two equal sides are exactly 151 centimeters long. Calculate the length of the third side with an estimate for the absolute error and the relative error.

Find \(D_{x} y\). $$ y=\left(\tan ^{-1} x\right)^{3} $$

A snowball melts at a rate proportional to its surface area. (a) Show that its radius shrinks at a constant rate. (b) If it melts to \(\frac{8}{27}\) its original volume in one hour, how long will it take to melt completely?