Chapter 3

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

Sketch the graph of the circle \(x^{2}+4 x+y^{2}+3=0\) and then find equations of the two tangent lines that pass through the origin.

A projectile is fired directly upward from the ground with an initial velocity of \(v_{0}\) feet per second. Its height in \(t\) seconds is given by \(s=v_{0} t-16 t^{2}\) feet. What must its initial velocity be for the projectile to reach a maximum height of 1 mile?

Apply the Chain Rule more than once to find the indicated derivative. \(D_{t}\left[\sin ^{3}(\cos t)\right]\)

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{1}{4 x^{2}-3 x+9} $$

Einstein's Special Theory of Relativity says that an object's mass \(m\) is related to its velocity \(v\) by the formula $$ m=\frac{m_{0}}{\sqrt{1-v^{2} / c^{2}}}=m_{0}\left(1-\frac{v^{2}}{c^{2}}\right)^{-1 / 2} $$ Here \(m_{0}\) is the rest mass and \(c\) is the speed of light. Use differentials to determine the percent increase in mass of an object when its velocity increases from \(0.9 c\) to \(0.92 c\).

Find \(D_{x} y\). $$ y=x \operatorname{arcsec}\left(x^{2}+1\right) $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x}\left(e^{1 / x^{2}}+1 / e^{x^{2}}\right) $$

Find the equation of the normal line (line perpendicular to the tangent line) to the curve \(8\left(x^{2}+y^{2}\right)^{2}=100\left(x^{2}-y^{2}\right)\) at \((3,1)\).

An object thrown directly downward from the top of a cliff with an initial velocity of \(v_{0}\) feet per second falls \(s=v_{0} t+16 t^{2}\) feet in \(t\) seconds. If it strikes the ocean below in 3 seconds with a speed of 140 feet per second, how high is the cliff?

Apply the Chain Rule more than once to find the indicated derivative. \(D_{u}\left[\cos ^{4}\left(\frac{u+1}{u-1}\right)\right]\)

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{4}{2 x^{3}-3 x} $$

Find the linear approximation to the given functions at the specified points. Plot the function and its linear approximation over the indicated interval. $$ f(x)=x^{2} \text { at } a=2,[0,3] $$

Call the graph of \(y=b-a \cosh (x / a)\) an inverted catenary and imagine it to be an arch sitting on the \(x\) -axis. Show that if the width of this arch along the \(x\) -axis is \(2 a\) then each of the following is true. (a) \(b=a \cosh 1 \approx 1.54308 a\). (b) The height of the arch is approximately \(0.54308 a\). (c) The height of an arch of width 48 is approximately \(13 .\)

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ \frac{d y}{d x} \text { if } e^{x y}+x y=2 \text { Hint: Use implicit differentiation. } $$

Suppose that \(x y+y^{3}=2 .\) Then implicit differentiation twice with respect to \(x\) yields in turn: (a) \(x y^{\prime}+y+3 y^{2} y^{\prime}=0\) (b) \(x y^{\prime \prime}+y^{\prime}+y^{\prime}+3 y^{2} y^{\prime \prime}+6 y\left(y^{\prime}\right)^{2}=0 .\)

An object moves along a horizontal coordinate line in such a way that its position at time \(t\) is specified by \(s=t^{3}-3 t^{2}-24 t-6 .\) Here \(s\) is measured in centimeters and in seconds. When is the object slowing down; that is, when is its speed decreasing?

Apply the Chain Rule more than once to find the indicated derivative. \(D_{\theta}\left[\cos ^{4}\left(\sin \theta^{2}\right)\right]\)

Find the linear approximation to the given functions at the specified points. Plot the function and its linear approximation over the indicated interval. $$ g(x)=x^{2} \cos x \text { at } a=\pi / 2,[0, \pi] $$

Call the graph of \(y=b-a \cosh (x / a)\) an inverted catenary and imagine it to be an arch sitting on the \(x\) -axis. Show that if the width of this arch along the \(x\) -axis is \(2 a\) then each of the following is true. (a) \(b=a \cosh 1 \approx 1.54308 a\). (b) The height of the arch is approximately \(0.54308 a\). (c) The height of an arch of width 48 is approximately \(13 .\)

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ \frac{d y}{d x} \text { if } e^{x+y}=4+x+y $$

Apply the Chain Rule more than once to find the indicated derivative. \(D_{x}\left[x \sin ^{2}(2 x)\right]\)

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

Find the linear approximation to the given functions at the specified points. Plot the function and its linear approximation over the indicated interval. $$ h(x)=\sin x \text { at } a=0,[-\pi, \pi] $$

Find the equation of the Gateway Arch in St. Louis, Missouri, given that it is an inverted catenary (see Problem 38 ). Assume that it stands on the \(x\) -axis, that it is symmetric with respect to the \(y\) -axis, and that it is 630 feet wide at the base and 630 feet high at the center.

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x}\left(6^{2 x}\right) $$

Apply the Chain Rule more than once to find the indicated derivative. \(\frac{d}{d x}\\{\sin [\cos (\sin 2 x)]\\}\)

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{2 x^{2}-1}{3 x+5} $$

Find the linear approximation to the given functions at the specified points. Plot the function and its linear approximation over the indicated interval. $$ F(x)=3 x+4 \text { at } a=3,[0,6] $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x}\left(3^{2 x^{2}-3 x}\right) $$

Use implicit differentiation twice to find \(y^{\prime \prime}\) at \((3,4)\) if \(x^{2}+y^{2}=25\)

Apply the Chain Rule more than once to find the indicated derivative. \(\frac{d}{d t}\left\\{\cos ^{2}[\cos (\cos t)]\right\\}\)

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{5 x-4}{3 x^{2}+1} $$

The graph of a function \(y=f(x)\) is given. Use this graph to sketch the graph of \(y=f^{\prime}(x)\).

Find the linear approximation to the given functions at the specified points. Plot the function and its linear approximation over the indicated interval. $$ f(x)=\sqrt{1-x^{2}} \text { at } a=0,[-1,1] $$

The structural steel work of a new office building is finished. Across the street, 60 feet from the ground floor of the freight elevator shaft in the building, a spectator is standing and watching the freight elevator ascend at a constant rate of 15 feet per second. How fast is the angle of elevation of the spectator's line of sight to the elevator increasing 6 seconds after his line of sight passes the horizontal?

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x} \log _{3} e^{x} $$

Show that the normal line to \(x^{3}+y^{3}=3 x y\) at \(\left(\frac{3}{2}, \frac{3}{2}\right)\) passes through the origin.

Let \(f(x)=x[\sin x-\cos (x / 2)]\). (a) Draw the graphs of \(f(x), f^{\prime}(x), f^{\prime \prime}(x)\), and \(f^{\prime \prime \prime}(x)\) on \([0,6]\) using the same axes. (b) Evaluate \(f^{\prime \prime \prime}(2.13)\).

Find the linear approximation to the given functions at the specified points. Plot the function and its linear approximation over the indicated interval. $$ g(x)=\sin ^{-1} x \text { at } a=0,[-1,1] $$

An airplane is flying at a constant altitude of 2 miles and a constant speed of 600 miles per hour on a straight course that will take it directly over an observer on the ground. How fast is the angle of elevation of the observer's line of sight increasing when the distance from her to the plane is 3 miles? Give your result in radians per minute.

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x} \log _{10}\left(x^{3}+9\right) $$

Show that the hyperbolas \(x y=1\) and \(x^{2}-y^{2}=1\) intersect at right angles.

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{5 x^{2}+2 x-6}{3 x-1} $$

Find the linear approximation to the given functions at the specified points. Plot the function and its linear approximation over the indicated interval. $$ h(x)=x \sec x \text { at } a=0,(-\pi / 2, \pi / 2) $$

A revolving beacon light is located on an island and is 2 miles away from the nearest point \(P\) of the straight shoreline of the mainland. The beacon throws a spot of light that moves along the shoreline as the beacon revolves. If the speed of the spot of light on the shoreline is \(5 \pi\) miles per minute when the spot is 1 mile from \(P\), how fast is the beacon revolving?

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{z}\left[3^{z} \ln (z+5)\right] $$

Show that the graphs of \(2 x^{2}+y^{2}=6\) and \(y^{2}=4 x\) intersect at right angles.

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{x^{2}-2 x+5}{x^{2}+2 x-3} $$

Find the linear approximation to the given functions at the specified points. Plot the function and its linear approximation over the indicated interval. $$ G(x)=x+\sin 2 x, \text { at } a=\pi / 2,[0, \pi] $$