Chapter 3

Estimate the allowable percentage error in measuring the diameter \(D\) of a sphere if the volume is to be calculated correctly to within \(3 \%\)

In Exercises \(51-70,\) find \(d y / d t\). $$y=\left(\frac{3 t-4}{5 t+2}\right)^{-5}$$

Use a CAS to perform the following steps a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point \(P\) satisfies the equation. b. Using implicit differentiation, find a formula for the derivative \(d y / d x\) and evaluate it at the given point \(P\) c. Use the slope found in part (b) to find an equation for the tangent line to the curve at \(P .\) Then plot the implicit curve and tangent line together on a single graph. $$y^{3}+\cos x y=x^{2}, \quad P(1,0)$$

Derive the formula for the derivative with respect to \(x\) of a. \(\sec x\) b. csc \(x\) c. cot \(x\)

The curves \(y=\) \(x^{2}+a x+b\) and \(y=c x-x^{2}\) have a common tangent line at the point \((1,0) .\) Find \(a, b,\) and \(c\).

Graph \(y=3 x^{2}\) in a window that has \(-2 \leq x \leq 2,0 \leq y \leq 3\) Then, on the same screen, graph $$y=\frac{(x+h)^{3}-x^{3}}{h}$$ for \(h=2,1,0.2,\) Then try \(h=-2,-1,-0.2 .\) Explain what is going on.

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=e^{(\cos t+\ln t)}$$

Use your graphing utility. Graph \(f(x)=\sin ^{-1} x\) together with its first two derivatives. Comment on the behavior of \(f\) and the shape of its graph in relation to the signs and values of \(f^{\prime}\) and \(f^{\prime \prime}\)

The effect of flight maneuvers on the heart The amount of work done by the heart's main pumping chamber, the left ventricle, is given by the equation $$W=P V+\frac{V \delta v^{2}}{2 g}$$ where \(W\) is the work per unit time, \(P\) is the average blood pressure, \(V\) is the volume of blood pumped out during the unit of time, \(\delta\) ("delta") is the weight density of the blood, \(v\) is the average velocity of the exiting blood, and \(g\) is the acceleration of gravity. When \(P, V, \delta,\) and \(v\) remain constant, \(W\) becomes a function of \(g,\) and the equation takes the simplified form $$W=a+\frac{b}{g}(a, b \text { constant })$$. As a member of NASA's medical team, you want to know how sensitive \(W\) is to apparent changes in \(g\) caused by flight maneuvers, and this depends on the initial value of \(g\). As part of your investigation, you decide to compare the effect on \(W\) of a given change \(d g\) on the moon, where \(g=5.2 \mathrm{ft} / \mathrm{sec}^{2},\) with the effect the same change \(d g\) would have on Earth, where \(g=32 \mathrm{ft} / \mathrm{sec}^{2} .\) Use the simplified equation above to find the ratio of \(d W_{\text {moon }}\) to \(d W_{\text {Eurth }}\)

In Exercises \(51-70,\) find \(d y / d t\). $$y=\sin (\cos (2 t-5))$$

Use a CAS to perform the following steps a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point \(P\) satisfies the equation. b. Using implicit differentiation, find a formula for the derivative \(d y / d x\) and evaluate it at the given point \(P\) c. Use the slope found in part (b) to find an equation for the tangent line to the curve at \(P .\) Then plot the implicit curve and tangent line together on a single graph. $$x+\tan \left(\frac{y}{x}\right)=2, \quad P\left(1, \frac{\pi}{4}\right)$$

Find all points \((x, y)\) on the graph of \(f(x)=3 x^{2}-4 x\) with tangent lines parallel to the line \(y=8 x+5\).

Graph the derivative of \(f(x)=|x|\) Then graph \(y=(|x|-0) /(x-0)=|x| / x .\) What can you conclude?

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=e^{\sin t}\left(\ln t^{2}+1\right)$$

Use your graphing utility. Graph \(f(x)=\tan ^{-1} x\) together with its first two derivatives. Comment on the behavior of \(f\) and the shape of its graph in relation to the signs and values of \(f^{\prime}\) and \(f^{\prime \prime}\)

Drug concentration The concentration \(C\) in milligrams per milliliter (mg/ml) of a certain drug in a person's bloodstream \(t\) hrs after a pill is swallowed is modeled by $$C(t)=1+\frac{4 t}{1+t^{3}}-e^{-0.06 t}$$ Estimate the change in concentration when \(t\) changes from 20 to 30 min.

In Exercises \(51-70,\) find \(d y / d t\). $$y=\cos \left(5 \sin \left(\frac{t}{3}\right)\right)$$

Use a CAS to perform the following steps a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point \(P\) satisfies the equation. b. Using implicit differentiation, find a formula for the derivative \(d y / d x\) and evaluate it at the given point \(P\) c. Use the slope found in part (b) to find an equation for the tangent line to the curve at \(P .\) Then plot the implicit curve and tangent line together on a single graph. $$x y^{3}+\tan (x+y)=1, \quad P\left(\frac{\pi}{4}, 0\right)$$

Assume that a particle's position on the \(x\) -axis is given by $$x=3 \cos t+4 \sin t,$$ where \(x\) is measured in feet and \(t\) is measured in seconds. a. Find the particle's position when \(t=0, t=\pi / 2,\) and \(t=\pi\). b. Find the particle's velocity when \(t=0, t=\pi / 2,\) and \(t=\pi\).

Find all points \((x, y)\) on the graph of \(g(x)=\frac{1}{3} x^{3}-\frac{3}{2} x^{2}+1\) with tangent lines parallel to the line \(8 x-2 y=1\).

The sum of the first eight terms of the Weierstrass function \(f(x)=\sum_{n=0}^{\infty}(2 / 3)^{n} \cos \left(9^{n} \pi x\right)\) is $$\begin{aligned}g(x) &=\cos (\pi x)+(2 / 3)^{1} \cos (9 \pi x)+(2 / 3)^{2} \cos \left(9^{2} \pi x\right) \\\&+(2 / 3)^{3} \cos \left(9^{3} \pi x\right)+\cdots+(2 / 3)^{7} \cos \left(9^{7} \pi x\right)\end{aligned}$$ Graph this sum. Zoom in several times. How wiggly and bumpy is this graph? Specify a viewing window in which the displayed portion of the graph is smooth.

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

Unclogging arteries The formula \(V=k r^{4},\) discovered by the physiologist Jean Poiseuille (1797-1869), allows us to predict how much the radius of a partially clogged artery has to be expanded in order to restore normal blood flow. The formula says that the volume \(V\) of blood flowing through the artery in a unit of time at a fixed pressure is a constant \(k\) times the radius of the artery to the fourth power. How will a \(10 \%\) increase in \(r\) affect \(V ?\)

In Exercises \(51-70,\) find \(d y / d t\). $$y=\left(1+\tan ^{4}\left(\frac{t}{12}\right)\right)^{3}$$

Graph \(y=\cos x\) for \(-\pi \leq x \leq 2 \pi .\) On the same screen, graph $$y=\frac{\sin (x+h)-\sin x}{h}$$ for \(h=1,0.5,0.3,\) and \(0.1 .\) Then, in a new window, try \(h=-1,-0.5,\) and \(-0.3 .\) What happens as \(h \rightarrow 0^{+} ?\) As \(h \rightarrow 0^{-} ?\) What phenomenon is being illustrated here?

Find all points \((x, y)\) on the graph of \(y=x /(x-2)\) with tangent lines perpendicular to the line \(y=2 x+3\).

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

Measuring acceleration of gravity When the length \(L\) of a clock pendulum is held constant by controlling its temperature, the pendulum's period \(T\) depends on the acceleration of gravity \(g\). The period will therefore vary slightly as the clock is moved from place to place on the earth's surface, depending on the change in g. By keeping track of \(\Delta T\), we can estimate the variation in \(g\) from the equation \(T=2 \pi(L / g)^{1 / 2}\) that relates \(T, g,\) and \(L\)a. With \(L\) held constant and \(g\) as the independent variable, calculate \(d T\) and use it to answer parts (b) and (c). b. If \(g\) increases, will \(T\) increase or decrease? Will a pendulum clock speed up or slow down? Explain. c. A clock with a \(100-\mathrm{cm}\) pendulum is moved from a location where \(g=980 \mathrm{cm} / \mathrm{sec}^{2}\) to a new location. This increases the period by \(d T=0.001 \mathrm{sec} .\) Find \(d g\) and estimate the value of \(g\) at the new location.

In Exercises \(51-70,\) find \(d y / d t\). $$y=\frac{1}{6}\left(1+\cos ^{2}(7 t)\right)^{3}$$

Use a CAS to perform the following steps a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point \(P\) satisfies the equation. b. Using implicit differentiation, find a formula for the derivative \(d y / d x\) and evaluate it at the given point \(P\) c. Use the slope found in part (b) to find an equation for the tangent line to the curve at \(P .\) Then plot the implicit curve and tangent line together on a single graph. $$x \sqrt{1+2 y}+y=x^{2}, \quad P(1,0)$$

Graph \(y=-\sin x\) for \(-\pi \leq x \leq 2 \pi .\) On the same screen, graph $$y=\frac{\cos (x+h)-\cos x}{h}$$ for \(h=1,0.5,0.3,\) and \(0.1 .\) Then, in a new window, try \(h=-1,-0.5,\) and \(-0.3 .\) What happens as \(h \rightarrow 0^{+} ?\) As \(h \rightarrow 0^{-} ?\) What phenomenon is being illustrated here?

Find all points \((x, y)\) on the graph of \(f(x)=x^{2}\) with tangent lines passing through the point (3,8). (GRAPH CAN'T COPY)

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

a. Let \(Q(x)=b_{0}+b_{1}(x-a)+b_{2}(x-a)^{2}\) be a quadratic approximation to \(f(x)\) at \(x=a\) with the properties: i) \(Q(a)=f(a)\) ii) \(Q^{\prime}(a)=f^{\prime}(a)\) iii) \(Q^{\prime \prime}(a)=f^{\prime \prime}(a)\) Determine the coefficients \(b_{0}, b_{1},\) and \(b_{2}\) b. Find the quadratic approximation to \(f(x)=1 /(1-x)\) at \(x=0\) c. Graph \(f(x)=1 /(1-x)\) and its quadratic approximation at \(x=0 .\) Then zoom in on the two graphs at the point (0,1) Comment on what you see. d. Find the quadratic approximation to \(g(x)=1 / x\) at \(x=1\) Graph \(g\) and its quadratic approximation together. Comment on what you see. e. Find the quadratic approximation to \(h(x)=\sqrt{1+x}\) at \(x=0 .\) Graph \(h\) and its quadratic approximation together. Comment on what you see. f. What are the linearizations of \(f, g,\) and \(h\) at the respective points in parts (b), (d), and (e)?

In Exercises \(51-70,\) find \(d y / d t\). $$y=\sqrt{1+\cos \left(t^{2}\right)}$$

a. Find an equation for the line that is tangent to the curve \(y=x^{3}-x\) at the point (-1,0) b. Graph the curve and tangent line together. The tangent intersects the curve at another point. Use Zoom and Trace to estimate the point's coordinates. c. Confirm your estimates of the coordinates of the second intersection point by solving the equations for the curve and tangent simultaneously (Solver key).

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

In Exercises \(51-70,\) find \(d y / d t\). $$y=4 \sin (\sqrt{1+\sqrt{t}})$$

A caution about centered difference quotients (Contimuation of Exercise \(65 .\) ) The quotient $$\frac{f(x+h)-f(x-h)}{2 h}$$ may have a limit as \(h \rightarrow 0\) when \(f\) has no derivative at \(x .\) As a case in point, take \(f(x)=|x|\) and calculate $$\lim _{h \rightarrow 0} \frac{|0+h|-|0-h|}{2 h}.$$ As you will see, the limit exists even though \(f(x)=|x|\) has no derivative at \(x=0 .\) Moral: Before using a centered difference quotient, be sure the derivative exists.

a. Find an equation for the line that is tangent to the curve \(y=x^{3}-6 x^{2}+5 x\) at the origin. b. Graph the curve and tangent together. The tangent intersects the curve at another point. Use Zoom and Trace to estimate the point's coordinates. c. Confirm your estimates of the coordinates of the second intersection point by solving the equations for the curve and tangent simultaneously (Solver key).

Find the derivative of \(y\) with respect to the given independent variable. $$y=2^{x}$$

a. Find the linearization of \(f(x)=2^{x}\) at \(x=0 .\) Then round its coefficients to two decimal places. b. Graph the linearization and function together for \(-3 \leq x \leq 3\) and \(-1 \leq x \leq 1\)

In Exercises \(51-70,\) find \(d y / d t\). $$y=\tan ^{2}\left(\sin ^{3} t\right)$$

Slopes on the graph of the tangent function Graph \(y=\tan x\) and its derivative together on \((-\pi / 2, \pi / 2) .\) Does the graph of the tangent function appear to have a smallest slope? A largest slope? Is the slope ever negative? Give reasons for your answers.

Evaluate each limit by first converting each to a derivative at a particular \(x\) -value. $$\lim _{x \rightarrow 1} \frac{x^{50}-1}{x-1}$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=3^{-x}$$

a. Find the linearization of \(f(x)=\log _{3} x\) at \(x=3 .\) Then round its coefficients to two decimal places. b. Graph the linearization and function together in the window \(0 \leq x \leq 8\) and \(2 \leq x \leq 4\)

In Exercises \(51-70,\) find \(d y / d t\). $$y=\cos ^{4}\left(\sec ^{2} 3 t\right)$$

Exploring \((\sin k x) / x \quad\) Graph \(y=(\sin x) / x, y=(\sin 2 x) / x,\) and \(y=(\sin 4 x) / x\) together over the interval \(-2 \leq x \leq 2 .\) Where does each graph appear to cross the \(y\) -axis? Do the graphs really intersect the axis? What would you expect the graphs of \(y=(\sin 5 x) / x\) and \(y=(\sin (-3 x)) / x\) to do as \(x \rightarrow 0 ?\) Why? What about the graph of \(y=(\sin k x) / x\) for other values of \(k ?\) Give reasons for your answers.

Evaluate each limit by first converting each to a derivative at a particular \(x\) -value. $$\lim _{x \rightarrow-1} \frac{x^{2 / 9}-1}{x+1}$$