Chapter 3

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?

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.

Find the indicated derivative. \(D_{x}\left(\sqrt{e^{x^{2}}}+e^{\sqrt{x^{2}}}\right)\)

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

\(\lim _{h \rightarrow 0} \frac{\cos (x+h)-\cos x}{h}\)

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{1}{4 x^{2}-3 x+9} $$

In Problems 33-40, apply the Chain Rule more than once to find the indicated derivative. $$ D_{t}\left[\sin ^{3}(\cos t)\right] $$

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

Find the indicated derivative. \(D_{x}\left(e^{1 / x^{2}}+1 / e^{x^{2}}\right)\)

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

\(\lim _{h \rightarrow 0} \frac{\tan (t+h)-\tan t}{h}\)

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{4}{2 x^{3}-3 x} $$

In Problems 33-40, apply the Chain Rule more than once to find the indicated derivative. $$ D_{u}\left[\cos ^{4}\left(\frac{u+1}{u-1}\right)\right] $$

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 \(t\) in seconds. When is the object slowing down; that is, when is its speed decreasing?

. 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) \(x y^{\prime \prime}+y^{\prime}+y^{\prime}+3 y^{2} y^{\prime \prime}+6 y\left(y^{\prime}\right)^{2}=0\).

Find the indicated derivative. \(\frac{d y}{d x}\) if \(e^{x y}+x y=2\)

Draw the graphs of \(y=\sinh x, y=\ln \left(x+\sqrt{x^{2}+1}\right)\), and \(y=x\) using the same axes and scaled so that \(-3 \leq x \leq 3\) and \(-3 \leq y \leq 3\). What does this demonstrate?

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{x-1}{x+1} $$

In Problems 33-40, 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 indicated derivative. \(\frac{d y}{d x}\) if \(e^{x+y}=4+x+y\)

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 .

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{2 x-1}{x-1} $$

In Problems 33-40, apply the Chain Rule more than once to find the indicated derivative. $$ D_{x}\left[x \sin ^{2}(2 x)\right] $$

Leibniz obtained a general formula for \(D_{x}^{n}(u v)\), where \(u\) and \(v\) are both functions of \(x\). See if you can find it. Hint: Begin by considering the cases \(n=1, n=2\), and \(n=3\).

Find the indicated derivative. \(D_{x}\left(6^{2 x}\right)\)

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.

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{2 x^{2}-1}{3 x+5} $$

In Problems 33-40, apply the Chain Rule more than once to find the indicated derivative. $$ \frac{d}{d x}\\{\sin [\cos (\sin 2 x)]\\} $$

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

Find the indicated derivative. \(D_{x}\left(3^{2 x^{2}-3 x}\right)\)

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{5 x-4}{3 x^{2}+1} $$

In Problems 33-40, apply the Chain Rule more than once to find the indicated derivative. $$ \frac{d}{d t}\left\\{\cos ^{2}[\cos (\cos t)]\right\\} $$

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 \(f^{\prime \prime \prime}(2.13)\).

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.

Find the indicated derivative. \(D_{x} \log _{3} e^{x}\)

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?

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$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{2 x^{2}-3 x+1}{2 x+1} $$

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

Find the indicated derivative. \(D_{x} \log _{10}\left(x^{3}+9\right)\)

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.

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{5 x^{2}+2 x-6}{3 x-1} $$

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

Find the indicated derivative. \(D_{z}\left[3^{z} \ln (z+5)\right]\)

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?

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{x^{2}-x+1}{x^{2}+1} $$

Find the indicated derivative. \(D_{\theta} \sqrt{\log _{10}\left(3^{\theta^{2}-\theta}\right)}\)

. A man on a dock is pulling in a rope attached to a rowboat at a rate of 5 feet per second. If the man's hands are 8 feet higher than the point where the rope is attached to the boat, how fast is the angle of depression of the rope changing when there are still 17 feet of rope out?

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{x^{2}-2 x+5}{x^{2}+2 x-3} $$

Find the indicated derivative. \(D_{x}\left(10^{\left(x^{2}\right)}+\left(x^{2}\right)^{10}\right)\)