Chapter 3

Complete each part to establish that the reflection of a point \(P(a, b)\) about the line \(y=x\) is the point \(Q(b, a)\) (a) Prove that if \(P\) is not on the line \(y=x,\) then \(P\) and \(Q\) are distinct, and the line \(\overleftrightarrow{P Q}\) is perpendicular to the line \(y=x\). (b) Prove that if \(P\) is not on the line \(y=x,\) the midpoint of segment \(P Q\) is on the line \(y=x\) (c) Carefully explain what it means geometrically to reflect \(P\) about the line \(y=x\) (d) Use the results of parts (a)-(c) to prove that \(Q\) is the reflection of \(P\) about the line \(y=x\)

Find \(d y / d x\). $$y=\sqrt{\ln x}$$

Find \(d y / d x\) by implicit differentiation. $$\tan ^{3}\left(x y^{2}+y\right)=x$$

Find the limits. $$\lim _{x \rightarrow 0^{+}} \frac{\sin x}{x^{2}}$$

A stone dropped into a still pond sends out a circular ripple whose radius increases at a constant rate of \(3 \mathrm{ft} / \mathrm{s}\). How rapidly is the area enclosed by the ripple increasing at the end of \(10 \mathrm{s} ?\)

Confirm that the stated formula is the local linear approximation of \(f\) at \(x_{0}=1,\) where \(\Delta x=x-1\). $$f(x)=\sqrt{x} ; \sqrt{1+\Delta x} \approx 1+\frac{1}{2} \Delta x$$

Find \(d y / d x\). $$y=\ln \sqrt{x}$$

Find \(d y / d x\) by implicit differentiation. $$\frac{x y^{3}}{1+\sec y}=1+y^{4}$$

Find the limits. $$\lim _{x \rightarrow+\infty} \frac{\ln x}{x}$$

Oil spilled from a ruptured tanker spreads in a circle whose area increases at a constant rate of \(6 \mathrm{mi}^{2} / \mathrm{h}\). How fast is the radius of the spill increasing when the area is \(9 \mathrm{mi}^{2} ?\)

Suppose that \(f\) and \(g\) are increasing functions. Determine which of the functions \(f(x)+g(x), f(x) g(x),\) and \(f(g(x))\) must also be increasing.

Find \(d y / d x\). $$y=x \ln x$$

Find \(d^{2} y / d x^{2}\) by implicit differentiation. $$2 x^{2}-3 y^{2}=4$$

Find the limits. $$\lim _{x \rightarrow+\infty} \frac{e^{3 x}}{x^{2}}$$

A spherical balloon is inflated so that its volume is increasing at the rate of \(3 \mathrm{ft}^{3} / \mathrm{min} .\) How fast is the diameter of the balloon increasing when the radius is \(1 \mathrm{ft} ?\)

Confirm that the stated formula is the local linear approximation of \(f\) at \(x_{0}=1,\) where \(\Delta x=x-1\). $$f(x)=(4+x)^{3} ;(5+\Delta x)^{3} \approx 125+75 \Delta x$$

Suppose that \(f\) and \(g\) are one-to-one functions. Determine which of the functions \(f(x)+g(x), f(x) g(x),\) and \(f(g(x))\) must also be one-to-one.

Find \(d y / d x\). $$y=x^{3} \ln x$$

Find \(d^{2} y / d x^{2}\) by implicit differentiation. $$x^{3}+y^{3}=1$$

A spherical balloon is to be deflated so that its radius decreases at a constant rate of \(15 \mathrm{cm} / \mathrm{min.}\). At what rate must air be removed when the radius is \(9 \mathrm{cm} ?\)

Confirm that the stated formula is the local linear approximation of \(f\) at \(x_{0}=1,\) where \(\Delta x=x-1\). $$\tan ^{-1} x ; \tan ^{-1}(1+\Delta x) \approx \frac{\pi}{4}+\frac{1}{2} \Delta x$$

Find \(d y / d x\). $$y=e^{7 x}$$

Find \(d y / d x\). $$y=x^{2} \log _{2}(3-2 x)$$

Find \(d^{2} y / d x^{2}\) by implicit differentiation. $$x^{3} y^{3}-4=0$$

Find the limits. $$\lim _{x \rightarrow 0^{+}} \frac{1-\ln x}{e^{1 / x}}$$

A 17 ft ladder is leaning against a wall. If the bottom of the ladder is pulled along the ground away from the wall at a constant rate of \(5 \mathrm{ft} / \mathrm{s}\), how fast will the top of the ladder be moving down the wall when it is 8 ft above the ground?

Confirm that the stated formula is the local linear approximation of \(f\) at \(x_{0}=1,\) where \(\Delta x=x-1\). $$\sin ^{-1}\left(\frac{x}{2}\right) ; \sin ^{-1}\left(\frac{1}{2}+\frac{1}{2} \Delta x\right) \approx \frac{\pi}{6}+\frac{1}{\sqrt{3}} \Delta x$$

Find \(d y / d x\). $$y=e^{-5 x^{2}}$$

Find \(d y / d x\). $$y=x\left[\log _{2}\left(x^{2}-2 x\right)\right]^{3}$$

Find \(d^{2} y / d x^{2}\) by implicit differentiation. $$x y+y^{2}=2$$

Find the limits. $$\lim _{x \rightarrow+\infty} \frac{x^{100}}{e^{x}}$$

A 13 ft ladder is leaning against a wall. If the top of the ladder slips down the wall at a rate of \(2 \mathrm{ft} / \mathrm{s}\), how fast will the foot be moving away from the wall when the top is \(5 \mathrm{ft}\) above the ground?

Confirm that the formula is the local linear approximation at \(x_{0}=0,\) and use a graphing utility to estimate an interval of \(x\) -values on which the error is at most ±0.1. $$\sqrt{x+3} \approx \sqrt{3}+\frac{1}{2 \sqrt{3}} x$$

Find \(d y / d x\). $$y=x^{3} e^{x}$$

Find \(d y / d x\). $$y=\frac{x^{2}}{1+\log x}$$

Find \(d^{2} y / d x^{2}\) by implicit differentiation. $$y+\sin y=x$$

Find the limits. $$\lim _{x \rightarrow 0^{+}} \frac{\ln (\sin x)}{\ln (\tan x)}$$

A \(10 \mathrm{ft}\) plank is leaning against a wall. If at a certain instant the bottom of the plank is 2 ft from the wall and is being pushed toward the wall at the rate of 6 in/s, how fast is the acute angle that the plank makes with the ground increasing?

Confirm that the formula is the local linear approximation at \(x_{0}=0,\) and use a graphing utility to estimate an interval of \(x\) -values on which the error is at most ±0.1. $$\frac{1}{\sqrt{9-x}} \approx \frac{1}{3}+\frac{1}{54} x$$

Find \(d y / d x\). $$y=e^{1 / x}$$

Find \(d^{2} y / d x^{2}\) by implicit differentiation. $$x \cos y=y$$

Find \(d y / d x\). $$y=\frac{\log x}{1+\log x}$$

Find the limits. $$\lim _{x \rightarrow 0} \frac{\sin ^{-1} 2 x}{x}$$

A softball diamond is a square whose sides are 60 ft long. Suppose that a player running from first to second base has a speed of \(25 \mathrm{ft} / \mathrm{s}\) at the instant when she is \(10 \mathrm{ft}\) from second base. At what rate is the player's distance from home plate changing at that instant?

Confirm that the formula is the local linear approximation at \(x_{0}=0,\) and use a graphing utility to estimate an interval of \(x\) -values on which the error is at most ±0.1. $$\tan 2 x \approx 2 x$$

Find \(d y / d x\). $$y=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$$

Find the slope of the tangent line to the curve at the given points in two ways: first by solving for \(y\) in terms of \(x\) and differentiating and then by implicit differentiation. $$x^{2}+y^{2}=1 ;(1 / 2, \sqrt{3} / 2),(1 / 2,-\sqrt{3} / 2)$$

Find \(d y / d x\). $$y=\ln (\ln x)$$

Find the limits. $$\lim _{x \rightarrow 0} \frac{x-\tan ^{-1} x}{x^{3}}$$

A rocket, rising vertically, is tracked by a radar station that is on the ground 5 mi from the launchpad. How fast is the rocket rising when it is 4 mi high and its distance from the radar station is increasing at a rate of \(2000 \mathrm{mi} / \mathrm{h} ?\)