Chapter 3

Estimating volume Estimate the volume of material in a cylindrical shell with length 30 in., radius 6 in., and shell thickness 0.5 in. (IMAGE CAN'T COPY)

In Exercises \(51-70,\) find \(d y / d t\). $$y=(1+\cos 2 t)^{-4}$$

How do \(d y / d x\) and \(d x / d y\) seem to be related? Explain the relationship geometrically in terms of the graphs. $$x y^{3}+x^{2} y=6$$

Find the limits. $$\lim _{t \rightarrow 0} \tan \left(1-\frac{\sin t}{t}\right)$$

Suppose \(u\) and \(v\) are functions of \(x\) that are differentiable at \(x=0\) and that $$u(0)=5, \quad u^{\prime}(0)=-3, \quad v(0)=-1, \quad v^{\prime}(0)=2$$ Find the values of the following derivatives at \(x=0\) $$\text { a. } \frac{d}{d x}(u v)$$ $$\text { b. } \frac{d}{d x}\left(\frac{u}{v}\right)$$ $$\text { c. } \frac{d}{d x}\left(\frac{v}{u}\right)$$ $$\text { d. } \frac{d}{d x}(7 v-2 u)$$

Does the parabola \(y=2 x^{2}-13 x+5\) have a tangent whose slope is \(-1 ?\) If so, find an equation for the line and the point of tangency. If not, why not?

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\sqrt[3]{\frac{x(x+1)(x-2)}{\left(x^{2}+1\right)(2 x+3)}}$$

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

How do \(d y / d x\) and \(d x / d y\) seem to be related? Explain the relationship geometrically in terms of the graphs. $$x^{3}+y^{2}=\sin ^{2} y$$

Find the limits. $$\lim _{\theta \rightarrow 0} \cos \left(\frac{\pi \theta}{\sin \theta}\right)$$

Suppose \(u\) and \(v\) are differentiable functions of \(x\) and that $$u(1)=2, \quad u^{\prime}(1)=0, \quad v(1)=5, \quad v^{\prime}(1)=-1$$ Find the values of the following derivatives at \(x=1\) $$\text { a. } \frac{d}{d x}(u v)$$ $$\text { b. } \frac{d}{d x}\left(\frac{u}{v}\right)$$ $$\text { c. } \frac{d}{d x}\left(\frac{v}{u}\right)$$ $$\text { d. } \frac{d}{d x}(7 v-2 u)$$.

Does any tangent to the curve \(y=\sqrt{x}\) cross the \(x\) -axis at \(x=-1 ?\) If so, find an cquation for the line and the point of tangency. If not. why not?

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

Find the domain and range of each composite Iunction. Then graph the composites on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see. a. \(y=\tan ^{-1}(\tan x)\) b. \(y=\tan \left(\tan ^{-1} x\right)\)

The radius \(r\) of a circle is measured with an error of at most \(2 \%\) What is the maximum corresponding percentage error in computing the circle's a. circumference? b. area?

In Exercises \(51-70,\) find \(d y / d t\). $$y=(t \tan t)^{10}$$

How do \(d y / d x\) and \(d x / d y\) seem to be related? Explain the relationship geometrically in terms of the graphs. Derivative of arcsine Assume that \(y=\sin ^{-1} x\) is a differentiable function of \(x\). By differentiating the equation \(x=\sin y\) implicitly, show that \(d y / d x=1 / \sqrt{1-x^{2}}\)

The equations give the position \(s=f(t)\) of a body moving on a coordinate line ( \(s\) in meters, \(t\) in seconds). Find the body's velocity, speed, acceleration, and jerk at time \(t=\pi / 4 \mathrm{sec}\). $$s=2-2 \sin t$$

a. Normal to a curve Find an equation for the line perpendicular to the tangent to the curve \(y=x^{3}-4 x+1\) at the point (2,1) b. Smallest slope What is the smallest slope on the curve? At what point on the curve does the curve have this slope? c. Tangents having specified slope Find equations for the tangents to the curve at the points where the slope of the curve is 8 .

Does knowing that a function \(f(x)\) is differentiable at \(x=x_{0}\) tell you anything about the differentiability of the function \(-f\) at \(x=x_{0} ?\) Give reasons for your answer.

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln \left(3 \theta e^{-\theta}\right)$$

Find the domain and range of each composite Iunction. Then graph the composites on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see. a. \(y=\sin ^{-1}(\sin x)\) b. \(y=\sin \left(\sin ^{-1} x\right)\)

The edge \(x\) of a cube is measured with an error of at most \(0.5 \%\) What is the maximum corresponding percentage error in computing the cube's a. surface area? b. volume?

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

The equations give the position \(s=f(t)\) of a body moving on a coordinate line ( \(s\) in meters, \(t\) in seconds). Find the body's velocity, speed, acceleration, and jerk at time \(t=\pi / 4 \mathrm{sec}\). $$s=\sin t+\cos t$$

a.Find equations for the horizontal tangents to the curve \(y=x^{3}-3 x-2 .\) Also find equations for the lines that are perpendicular to these tangents at the points of tangency. b.What is the smallest slope on the curve? At what point on the curve does the curve have this slope? Find an equation for the line that is perpendicular to the curve's tangent at this point.

Does knowing that a function \(g(t)\) is differentiable at \(t=7\) tell you anything about the differentiability of the function \(3 g\) at \(t=7 ?\) Give reasons for your answer.

Find the domain and range of each composite Iunction. Then graph the composites on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see. a. \(y=\cos ^{-1}(\cos x)\) b. \(y=\cos \left(\cos ^{-1} x\right)\)

Tolerance The height and radius of a right circular cylinder are equal, so the cylinder's volume is \(V=\pi h^{3} .\) The volume is to be calculated with an error of no more than \(1 \%\) of the true value. Find approximately the greatest error that can be tolerated in the measurement of \(h,\) expressed as a percentage of \(h\)

In Exercises \(51-70,\) find \(d y / d t\). $$y=e^{\cos ^{2}(\pi t-1)}$$

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. $$x^{3}-x y+y^{3}=7, \quad P(2,1)$$

Is there a value of \(c\) that will make $$f(x)=\left\\{\begin{array}{ll} \frac{\sin ^{2} 3 x}{x^{2}}, & x \neq 0 \\ c, & x=0 \end{array}\right.$$ continuous at \(x=0 ?\) Give reasons for your answer.

Suppose that functions \(g(t)\) and \(h(t)\) are defined for all values of \(t\) and \(g(0)=h(0)=0 .\) Can \(\lim _{t \rightarrow 0}(g(t)) /(h(t))\) exist? If it does exist, must it equal zero? Give reasons for your answers.

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

Use your graphing utility. Graph \(y=\sec \left(\sec ^{-1} x\right)=\sec \left(\cos ^{-1}(1 / x)\right) .\) Explain what you see.

a. About how accurately must the interior diameter of a 10-m-high cylindrical storage tank be measured to calculate the tank's volume to within \(1 \%\) of its true value? b. About how accurately must the tank's exterior diameter be measured to calculate the amount of paint it will take to paint the side of the tank to within \(5 \%\) of the true amount?

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

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. $$x^{5}+y^{3} x+y x^{2}+y^{4}=4, \quad P(1,1)$$

Is there a value of \(b\) that will make $$g(x)=\left\\{\begin{array}{ll} x+b, & x<0 \\ \cos x, & x \geq 0 \end{array}\right.$$ continuous at \(x=0 ?\) Differentiable at \(x=0 ?\) Give reasons for your answers.

a. Let \(f(x)\) be a function satisfying \(|f(x)| \leq x^{2}\) for \(-1 \leq x \leq 1\) Show that \(f\) is differentiable at \(x=0\) and find \(f^{\prime}(0)\) b. Show that $$f(x)=\left\\{\begin{array}{ll}x^{2} \sin \frac{1}{x}, & x \neq 0 \\\0, & x=0\end{array}\right.$$ is differentiable at \(x=0\) and find \(f^{\prime}(0)\)

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

Use your graphing utility. Graph Newton's serpentine, \(y=4 x /\left(x^{2}+1\right)\) Then graph \(y=2 \sin \left(2 \tan ^{-1} x\right)\) in the same graphing window. What do you see? Explain.

The diameter of a sphere is measured as \(100 \pm 1 \mathrm{cm}\) and the volume is calculated from this measurement. Estimate the percentage error in the volume calculation.

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

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. $$y^{2}+y=\frac{2+x}{1-x}, \quad P(0,1)$$

By computing the first few derivatives and looking for a pattern, find \(d^{999} / d x^{999}(\cos x)\).

The curve \(y=\) \(a x^{2}+b x+c\) passes through the point (1,2) and is tangent to the line \(y=x\) at the origin. Find \(a, b,\) and \(c\).

Graph \(y=1 /(2 \sqrt{x})\) in a window that has \(0 \leq x \leq 2 .\) Then, on the same screen, graph $$y=\frac{\sqrt{x+h}-\sqrt{x}}{h}$$ for \(h=1,0.5,0.1 .\) Then try \(h=-1,-0.5,-0.1 .\) Explain what is going on.

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

Use your graphing utility. Graph the rational function \(y=\left(2-x^{2}\right) / x^{2} .\) Then graph \(y=\) \(\cos \left(2 \sec ^{-1} x\right)\) in the same graphing window. What do you see? Explain.