Chapter 3

Show that the linearization of \(f(x)=(1+x)^{k}\) at \(x=0\) is \(L(x)=1+k x .\)

Flying a kite \(A\) girl flies a kite at a height of 300 \(\mathrm{ft}\) , the wind carrying the kite horizontally away from her at a rate of 25 \(\mathrm{ft} /\) sec. How fast must she let out the string when the kite is 500 ft away from her?

Find the first derivatives of the functions in Exercises \(11-18\) . $$ f(x)=\sqrt{1-\sqrt{x}} $$

In Exercises \(9-18,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x .\) $$ y=\sec (\tan x) $$

In Exercises \(13-16,\) find \(y^{\prime}\) (a) by applying the Product Rule and (b) by multiplying the factors to produce a sum of simpler terms to differentiate. $$ y=\left(x^{2}+1\right)\left(x+5+\frac{1}{x}\right) $$

In Exercises \(13-16,\) find \(d s / d t\) $$ s=\frac{1+\csc t}{1-\csc t} $$

Use the linear approximation \((1+x)^{k} \approx 1+k x\) to find an approximation for the function \(f(x)\) for values of \(x\) near zero. $$ \begin{array}{ll}{\text { a. } f(x)=(1-x)^{6}} & {\text { b. } f(x)=\frac{2}{1-x}} \\ {\text { c. } f(x)=\frac{1}{\sqrt{1+x}}} & {\text { d. } f(x)=\sqrt{2+x^{2}}} \\ {\text { e. } f(x)=(4+3 x)^{1 / 3}} & {\text { f. } f(x)=\sqrt[3]{\left(1-\frac{1}{2+x}\right)^{2}}}\end{array} $$

Boring a cylinder The mechanics at Lincoln Automotive are reboring a 6 -in.-deep cylinder to fit a new piston. The machine they are using increases the cylinder's radius one-thousandth of an inch every 3 min. How rapidly is the cylinder volume increasing when the bore (diameter) is 3.800 in.?

Find the first derivatives of the functions in Exercises \(11-18\) . $$ g(x)=2\left(2 x^{-1 / 2}+1\right)^{-1 / 3} $$

In Exercises \(9-18,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x .\) $$ y=\cot \left(\pi-\frac{1}{x}\right) $$

Find the derivatives of the functions \(s=\csc ^{5}\left(1-t+3 t^{2}\right)\)

In Exercises \(13-16,\) find \(d s / d t\) $$ s=\frac{\sin t}{1-\cos t} $$

In Exercises \(13-16,\) differentiate the functions and find the slope of the tangent line at the given value of the independent variable. $$ y=(x+1)^{3}, \quad x=-2 $$

Faster than a calculator Use the approximation \((1+x)^{k} \approx\) \(1+k x\) to estimate the following a. \((1.0002)^{50} \quad\) b. \(\sqrt[3]{1.009}\)

A growing sand pile Sand falls from a conveyor belt at the rate of 10 \(\mathrm{m}^{3} / \mathrm{min}\) onto the top of a conical pile. The height of the pile is always three-eighths of the base diameter. How fast are the (a) height and (b) radius changing when the pile is 4 \(\mathrm{m}\) high? Answer in centimeters per minute.

In Exercises \(9-18,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x .\) $$ y=\sin ^{3} x $$

In Exercises \(17-20,\) find \(d r / d \theta\) $$ r=4-\theta^{2} \sin \theta $$

Find the derivatives of the functions $$ y=\frac{2 x+5}{3 x-2} $$

In Exercises \(17-18\) , differentiate the functions. Then find an equation of the tangent line at the indicated point on the graph of the function. $$ w=g(z)=1+\sqrt{4-z}, \quad(z, w)=(3,2) $$

Find the linearization of \(f(x)=\sqrt{x+1}+\sin x\) at \(x=0 .\) How is it related to the individual linearizations of \(\sqrt{x+1}\) and \(\sin x\) at \(x=0 ?\)

A draining conical reservoir Water is flowing at the rate of 50 \(\mathrm{m}^{3} / \mathrm{min}\) from a shallow concrete conical reservoir (vertex down) of base radius 45 \(\mathrm{m}\) and height 6 \(\mathrm{m}\) . a. How fast (centimeters per minute) is the water level falling when the water is 5 \(\mathrm{m}\) deep? b. How fast is the radius of the water's surface changing then? Answer in centimeters per minute.

Find the first derivatives of the functions in Exercises \(11-18\) . $$ k(\theta)=(\sin (\theta+5))^{5 / 4} $$

In Exercises \(9-18,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x .\) $$ y=5 \cos ^{-4} x $$

In Exercises \(17-20,\) find \(d r / d \theta\) $$ r=\theta \sin \theta+\cos \theta $$

In Exercises \(17-18\) , differentiate the functions. Then find an equation of the tangent line at the indicated point on the graph of the function. $$ y=f(x)=\frac{8}{\sqrt{x-2}}, \quad(x, y)=(6,4) $$

In Exercises \(19-30,\) find \(d y\) $$ y=x^{3}-3 \sqrt{x} $$

Use implicit differentiation to find \(d y / d x\) in Exercises \(19-32\) $$ x^{2} y+x y^{2}=6 $$

A draining hemispherical reservoir Water is flowing at the rate of 6 \(\mathrm{m}^{3} / \mathrm{min}\) from a reservoir shaped like a hemispherical bowl of radius \(13 \mathrm{m},\) shown here in profile. Answer the following questions, given that the volume of water in a hemispherical bowl of radius \(R\) is \(V=(\pi / 3) y^{2}(3 R-y)\) when the water is \(y\) meters deep. a. At what rate is the water level changing when the water is 8 \(\mathrm{m}\) deep? b. What is the radius \(r\) of the water's surface when the water is \(y \mathrm{m}\) deep? c. At what rate is the radius \(r\) changing when the water is 8 \(\mathrm{m}\) deep?

Find the derivatives of the functions in Exercises \(19-38\) $$ p=\sqrt{3-t} $$

In Exercises \(17-20,\) find \(d r / d \theta\) $$ r=\sec \theta \csc \theta $$

Find the derivatives of the functions $$ g(x)=\frac{x^{2}-4}{x+0.5} $$

In Exercises \(19-22,\) find the values of the derivatives. $$ \left.\frac{d s}{d t}\right|_{t=-1} \quad \text { if } \quad s=1-3 t^{2} $$

In Exercises \(19-30,\) find \(d y\) $$ y=x \sqrt{1-x^{2}} $$

Use implicit differentiation to find \(d y / d x\) in Exercises \(19-32\) $$ x^{3}+y^{3}=18 x y $$

A growing raindrop Suppose that a drop of mist is a perfect sphere and that, through condensation, the drop picks up moisture at a rate proportional to its surface area. Show that under these circumstances the drop's radius increases at a constant rate.

Find the derivatives of the functions in Exercises \(19-38\) $$ q=\sqrt{2 r-r^{2}} $$

In Exercises \(17-20,\) find \(d r / d \theta\) $$ r=(1+\sec \theta) \sin \theta $$

Find the derivatives of the functions. $$ f(t)=\frac{t^{2}-1}{t^{2}+t-2} $$

In Exercises \(19-22,\) find the values of the derivatives. $$ \left.\frac{d y}{d x}\right|_{x=\sqrt{3}} \text { if } \quad y=1-\frac{1}{x} $$

In Exercises \(19-30,\) find \(d y\) $$ y=\frac{2 x}{1+x^{2}} $$

Use implicit differentiation to find \(d y / d x\) in Exercises \(19-32\) $$ 2 x y+y^{2}=x+y $$

The radius of an inflating balloon A spherical balloon is inflated with helium at the rate of 100\(\pi \mathrm{ft}^{3} / \mathrm{min}\) . How fast is the balloon's radius increasing at the instant the radius is 5 \(\mathrm{ft} ?\) How fast is the surface area increasing?

In Exercises \(21-24,\) find \(d p / d q\) $$ p=5+\frac{1}{\cot q} $$

Find the derivatives of the functions in Exercises \(v=(1-t)\left(1+t^{2}\right)^{-1}\)

In Exercises \(19-22,\) find the values of the derivatives. $$ \left.\frac{d r}{d \theta}\right|_{\theta=0} \quad \text { if } \quad r=\frac{2}{\sqrt{4-\theta}} $$

In Exercises \(19-30,\) find \(d y\) $$ y=\frac{2 \sqrt{x}}{3(1+\sqrt{x})} $$

Use implicit differentiation to find \(d y / d x\) in Exercises \(19-32\) $$ x^{3}-x y+y^{3}=1 $$

Find the derivatives of the functions in Exercises \(19-38\) $$ s=\sin \left(\frac{3 \pi t}{2}\right)+\cos \left(\frac{3 \pi t}{2}\right) $$

In Exercises \(21-24,\) find \(d p / d q\) $$ p=(1+\csc q) \cos q $$

In Exercises \(19-22,\) find the values of the derivatives. $$ \left.\frac{d w}{d z}\right|_{z=4} \quad \text { if } \quad w=z+\sqrt{z} $$