Chapter 3

Find the derivatives of the functions in Exercises \(23-50\). $$y=(4 x+3)^{4}(x+1)^{-3}$$

Verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point. $$x^{2} y^{2}=9, \quad(-1,3)$$

Find \(y^{\prime \prime}\) if a. \(y=\csc x\) b. \(y=\sec x\)

Show that the line \(y=m x+b\) is its own tangent line at any point \(\left(x_{0}, m x_{0}+b\right)\).

Find the derivatives of the function. $$y=x^{9 / 4}+e^{-2 x}$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\tan ^{-1}(\ln x)$$

Find \(d y\). $$y=\ln \left(\frac{x+1}{\sqrt{x-1}}\right)$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\frac{1}{2} \ln \frac{1+x}{1-x}$$

Give the position function \(s=f(t)\) of an object moving along the \(s\) -axis as a function of time \(t .\) Graph \(f\) together with the velocity function \(v(t)=d s / d t=f^{\prime}(t)\) and the acceleration function \(a(t)=d^{2} s / d t^{2}=f^{\prime \prime}(t) .\) Comment on the object's behavior in relation to the signs and values of \(v\) and \(a .\) Include in your commentary such topics as the following: a. When is the object momentarily at rest? b. When does it move to the left (down) or to the right (up)? c. When does it change direction? d. When does it speed up and slow down? e. When is it moving fastest (highest speed)? Slowest? f. When is it farthest from the axis origin? $$s=t^{2}-3 t+2, \quad 0 \leq t \leq 5$$

Find the derivatives of the functions in Exercises \(23-50\). $$y=(2 x-5)^{-1}\left(x^{2}-5 x\right)^{6}$$

Verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point. $$y^{2}-2 x-4 y-1=0, \quad(-2,1)$$

Find \(y^{(4)}=d^{4} y / d x^{4}\) if a. \(y=-2 \sin x\) b. \(y=9 \cos x\)

Find the slope of the tangent to the curve \(y=1 / \sqrt{x}\) at the point where \(x=4\).

Find the derivatives of the function. $$y=x^{-3 / 3}+\pi^{3 / 2}$$

In the late 1860 s, Adolf Fick, a professor of physiology in the Faculty of Medicine in Würzberg, Germany, developed one of the methods we use today for measuring how much blood your heart pumps in a minute. Your cardiac output as you read this sentence is probably about \(7 \mathrm{L} / \mathrm{min}\). At rest it is likely to be a bit under \(6 \mathrm{L} / \mathrm{min}\). If you are a trained marathon runner running a marathon, your cardiac output can be as high as \(30 \mathrm{L} / \mathrm{min}\). Your cardiac output can be calculated with the formula $$y=\frac{Q}{D}$$ where \(Q\) is the number of milliliters of \(\mathrm{CO}_{2}\) you exhale in a minute and \(D\) is the difference between the \(\mathrm{CO}_{2}\) concentration \((\mathrm{ml} / \mathrm{L})\) in the blood pumped to the lungs and the \(\mathrm{CO}_{2}\) concentration in the blood returning from the lungs. With \(Q=233 \mathrm{ml} / \mathrm{min}\) and $$\begin{aligned}D=97-56 &=41 \mathrm{ml} / \mathrm{L} \\\y &=\frac{233 \mathrm{ml} / \mathrm{min}}{41 \mathrm{ml} / \mathrm{L}} \approx 5.68 \mathrm{L} / \mathrm{min}\end{aligned}$$ fairly close to the \(6 \mathrm{L} / \mathrm{min}\) that most people have at basal (resting) conditions. (Data courtesy of J. Kenneth Herd, M.D., Quillan College of Medicine, East Tennessee State University.) Suppose that when \(Q=233\) and \(D=41,\) we also know that \(D\) is decreasing at the rate of 2 units a minute but that \(Q\) remains unchanged. What is happening to the cardiac output?

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\csc ^{-1}\left(e^{r}\right)$$

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

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\frac{1+\ln t}{1-\ln t}$$

Give the position function \(s=f(t)\) of an object moving along the \(s\) -axis as a function of time \(t .\) Graph \(f\) together with the velocity function \(v(t)=d s / d t=f^{\prime}(t)\) and the acceleration function \(a(t)=d^{2} s / d t^{2}=f^{\prime \prime}(t) .\) Comment on the object's behavior in relation to the signs and values of \(v\) and \(a .\) Include in your commentary such topics as the following: a. When is the object momentarily at rest? b. When does it move to the left (down) or to the right (up)? c. When does it change direction? d. When does it speed up and slow down? e. When is it moving fastest (highest speed)? Slowest? f. When is it farthest from the axis origin? $$s=t^{3}-6 t^{2}+7 t, \quad 0 \leq t \leq 4$$

Find the derivatives of the functions in Exercises \(23-50\). $$y=x e^{-x}+e^{x^{3}}$$

Verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point. $$6 x^{2}+3 x y+2 y^{2}+17 y-6=0, \quad(-1,0)$$

Graph the curves over the given intervals, together with their tangents at the given values of \(x\). Label each curve and tangent with its equation. $$\begin{array}{l} y=\sin x, \quad-3 \pi / 2 \leq x \leq 2 \pi \\ x=-\pi, 0,3 \pi / 2 \end{array}$$

Does the graph of $$f(x)=\left\\{\begin{array}{ll}x^{2} \sin (1 / x), & x \neq 0 \\\0, & x=0\end{array}\right.$$ have a tangent at the origin? Give reasons for your answer.

Find the derivatives of the function. $$s=2 t^{3 / 2}+3 e^{2}$$

A particle moves along the parabola \(y=x^{2}\) in the first quadrant in such a way that its \(x\) -coordinate (measured in meters) increases at a steady \(10 \mathrm{m} / \mathrm{sec} .\) How fast is the angle of inclination \(\theta\) of the line joining the particle to the origin changing when \(x=3 \mathrm{m} ?\)

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\cos ^{-1}\left(e^{-t}\right)$$

Find \(d y\). $$y=\cot ^{-1}\left(\frac{1}{x^{2}}\right)+\cos ^{-1} 2 x$$

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

Give the position function \(s=f(t)\) of an object moving along the \(s\) -axis as a function of time \(t .\) Graph \(f\) together with the velocity function \(v(t)=d s / d t=f^{\prime}(t)\) and the acceleration function \(a(t)=d^{2} s / d t^{2}=f^{\prime \prime}(t) .\) Comment on the object's behavior in relation to the signs and values of \(v\) and \(a .\) Include in your commentary such topics as the following: a. When is the object momentarily at rest? b. When does it move to the left (down) or to the right (up)? c. When does it change direction? d. When does it speed up and slow down? e. When is it moving fastest (highest speed)? Slowest? f. When is it farthest from the axis origin? $$s=4-7 t+6 t^{2}-t^{3}, \quad 0 \leq t \leq 4$$

Find the derivatives of the functions in Exercises \(23-50\). $$y=(1+2 x) e^{-2 x}$$

Verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point. $$x^{2}-\sqrt{3} x y+2 y^{2}=5, \quad(\sqrt{3}, 2)$$

Graph the curves over the given intervals, together with their tangents at the given values of \(x\). Label each curve and tangent with its equation. $$\begin{array}{l} y=\tan x,-\pi / 2

Does the graph of $$g(x)=\left\\{\begin{array}{ll}x \sin (1 / x), & x \neq 0 \\\0, & x=0\end{array}\right.$$ have a tangent at the origin? Give reasons for your answer.

Find the derivatives of the function. $$w=\frac{1}{z^{1 / 4}}+\frac{\pi}{\sqrt{z}}$$

The coordinates of a particle in the metric xy-plane are differentiable functions of time \(t\) with \(d x / d t=\) \(-1 \mathrm{m} / \mathrm{sec}\) and \(d y / d t=-5 \mathrm{m} / \mathrm{sec} .\) How fast is the particle's distance from the origin changing as it passes through the point (5,12)\(?\)

Find the derivative of \(y\) with respect to the appropriate variable. $$y=s \sqrt{1-s^{2}}+\cos ^{-1} s$$

Find \(d y\). $$y=\sec ^{-1}\left(e^{-x}\right)$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln (\sec (\ln \theta))$$

Find the derivatives of the functions in Exercises \(23-50\). $$y=\left(x^{2}-2 x+2\right) e^{5 x / 2}$$

Verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point. $$2 x y+\pi \sin y=2 \pi, \quad(1, \pi / 2)$$

Graph the curves over the given intervals, together with their tangents at the given values of \(x\). Label each curve and tangent with its equation. $$\begin{array}{l} y=\sec x,-\pi / 2

Does the graph of $$f(x)=\left\\{\begin{aligned}-1, & x<0 \\\0, & x=0 \\\1, & x>0\end{aligned}\right.$$ have a vertical tangent at the origin? Give reasons for your answer.

Find the derivatives of the function. $$y=\sqrt[7]{x^{2}}-x^{e}$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\sqrt{s^{2}-1}-\sec ^{-1} s$$

Find \(d y\). $$y=e^{\tan ^{-1} \sqrt{x^{2}+1}}$$

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

Find the derivatives of the functions in Exercises \(23-50\). $$y=\left(9 x^{2}-6 x+2\right) e^{x^{3}}$$

Verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point. $$x \sin 2 y=y \cos 2 x, \quad(\pi / 4, \pi / 2)$$

Graph the curves over the given intervals, together with their tangents at the given values of \(x\). Label each curve and tangent with its equation. $$\begin{aligned} &y=1+\cos x, \quad-3 \pi / 2 \leq x \leq 2 \pi\\\ &x=-\pi / 3,3 \pi / 2 \end{aligned}$$

Does the graph of $$U(x)=\left\\{\begin{array}{ll}0, & x<0 \\\1, & x \geq 0\end{array}\right.$$ have a vertical tangent at the point (0,1)\(?\) Give reasons for your answer.