Chapter 21

Let \(f(x)=x+2 \sin x\). (a) Find all of the critical points. (b) Where is \(f(x)\) increasing? Decreasing? (c) Where does \(f(x)\) have local maxima? Local minima? (d) Does \(f(x)\) have global maxima? Global minima? If so, what are the absolute maximum and minimum values? (e) Where is \(f(x)\) concave up? Concave down? (f) Sketch a graph of \(f(x)\).

Estimate \(\lim _{x \rightarrow \pi} \frac{\sin x}{x}\) both numerically and graphically.

Using the derivatives of sine and cosine and either the Product Rule or the Quotient Rule, show that \(\frac{d}{d x} \tan x=\sec ^{2} x\).

For Problems 1 through 6, differentiate the function given. \(y=3 \tan x-4 \tan ^{-1} x\)

Consider the function \(f(x)=-\cos x+\frac{1}{2} \sin 2 x\) (a) Explain how you can tell that \(f\) is periodic with period \(2 \pi\). (b) Find and classify all the critical points of \(f\) on the interval \([0,2 \pi] .\) Do the trigonometric "algebra" on your own, then check your answers using a graphing calculator.

Let \(f(x)=\sin x .\) Use the difference quotient with \(h=0.0001\) to estimate the value of \(f^{\prime}(\pi)\), the slope of the tangent line to \(\sin x\) at \(x=\pi\).

Show that \(\frac{d}{d x} \sec x=\sec x \tan x\).

Differentiate the function given. \(f(x)=3 \arctan (2 \sqrt{x})\)

Graph fon the interval \([0,2 \pi]\) labeling the \(x\) -coordinates of all local extrema. $$ f(x)=\cos x+\sqrt{3} \sin x $$

Let \(f(x)=\sin (x)\). (a) Using a calculator, tabulate \(f\) at \(x=1.998,1.999,2.000,2.001,2.002\) (make a table with values of \(x\) and \(f(x))\). Round values of \(f\) to six decimal places. (b) Estimate \(f^{\prime}(1.999), f^{\prime}(2)\), and \(f^{\prime}(2.001)\) using the tabulated values. (c) Estimate \(f^{\prime \prime}(2)\) using the results from part (b).

Find the first and second derivatives of the following. (a) \(f(x)=5 \cos x\) (b) \(g(x)=-3 \sin (2 x)\) (c) \(h(x)=0.5 \tan x\) (d) \(j(x)=2 \sin x \cos x\)

Differentiate the function given. \(y=\sin x \cdot \arcsin x\)

Graph fon the interval \([0,2 \pi]\) labeling the \(x\) -coordinates of all local extrema. $$ f(x)=\cos x-\sin x $$

(a) What is the limit definition of \(\left.\frac{d}{d x} \cos x\right|_{x=0}\) ? (b) Numerically approximate \(\left.\frac{d}{d x} \cos x\right|_{x=0}\)

Differentiate the following. (a) \(y=\cos ^{2} x\) (b) \(y=\cos \left(x^{2}\right)\) (c) \(y=x \tan ^{2} x\) (d) \(y=\sin ^{3}\left(x^{4}\right)\) (e) \(y=7[\cos (5 x)+3]^{x}\)

Differentiate the function given. \(y=\sqrt{\tan ^{-1} x}\)

Graph fon the interval \([0,2 \pi]\) labeling the \(x\) -coordinates of all local extrema. $$ f(x)=\cos 2 x-2 \cos x $$

Numerically approximate the derivative of \(\cos x\) at \(x=\pi\).

Consider the function \(f(x)=e^{-0.3 x} \sin x\) (a) For what values of \(x\) does \(f(x)\) have its local maxima and local minima? (b) Is \(f(x)\) a periodic function? (c) Sketch the graph of \(f(x)=e^{-0.3 x} \sin x\). (d) What is the maximum value of for \(e^{-0.3 x} \sin x\) for \(x \geq 0 ?\) At what \(x\) -value is this maximum attained? Your answers must be exact, not numerical approximations from a calculator. Give justification that this value is indeed the maximum.

Differentiate the function given. \(y=x \tan ^{-1} x\)

Graph fon the interval \([0,2 \pi]\) labeling the \(x\) -coordinates of all local extrema. $$ f(x)=e^{x} \sin x $$

Evaluate the following derivatives. \(u(x)\) is a differentiable function. (a) \(\frac{d}{d x} \sin (u(x))\) (b) \(\frac{d}{d x} \cos (u(x))\) (c) \(\frac{d}{d x} u(x)(\sin x)\)

Differentiate the function given. \(y=\frac{\arctan \left(e^{x}\right)}{e}\)

Use a tangent line approximation to approximate the following. In each case, use concavity to determine whether the approximation is larger or smaller than the actual value. Then compare your results with the approximations given by a calculator or computer. (a) \(\sin 0.2\) (b) \(\sin 0.1\) (c) \(\sin 0.01\) (d) \(\sin (-0.1)\)

Evaluate the following derivatives. \(u(x)\) is a differentiable function. (a) \(\frac{d}{d x} u(x)(\cos x)\) (b) \(\frac{d}{d x} \tan (u(x))\) (c) \(\frac{d}{d x} u(x)(\tan x)\)

(a) Show that the derivative of \(\arccos x\) is \(\frac{-1}{\sqrt{1-x^{2}}}\). (b) What is the domain of \(\arccos x\) ? (c) What is the range of \(\arccos x\) ? (d) Where is the graph of \(\arccos x\) decreasing? (e) Where is the graph of \(\arccos x\) concave up? Concave down? If there is a point of in ection, where is it? (f) Graph \(f(x)=\arccos x\).

Evaluate. $$ \frac{d}{d x} \sin \left(x^{3}+\ln 3 x\right) $$

Differentiate \(f(x)=3 \cos \left(\frac{1}{x^{2}+1}\right)+x \arctan \left(\frac{1}{x}\right)\)

Verify that sec \(x\) has local minima at \(x=2 \pi k\) and local maxima at \(x=\pi+2 \pi k(k\) an integer) by identifying its critical points and using the second derivative test for maxima and minima.

Evaluate. $$ \frac{d}{d x} \cos ^{2}(\sin x) $$

Compute \(\frac{d}{d x} \frac{\sin ^{-1} x}{\cos ^{-1} x} .\) Is it the same as \(\frac{d}{d x} \tan ^{-1} x ?\)

Verify that \(\tan x\) has points of inflection at \(x=\pi k, k\) an integer, by showing that the sign of its second derivative changes at these points.

Evaluate. $$ \frac{d}{d x}\left[\frac{1}{\sin ^{3}(\cos 2 x)}\right] $$

Let \(f(x)=3 \cos x+2 \sin x\) (a) What is the period of \(f ?\) (b) What are the maximum and minimum values of \(f ?\)

Evaluate. $$ \frac{d}{d x} \sqrt{\sin \left(2 x^{3}\right)} $$

If we ignore air resistance, a baseball thrown from shoulder level at an angle of \(\theta\) radians with the ground and at an initial velocity of \(v_{0}\) meters per second will be at shoulder level again when it is \(\frac{v_{0}^{2} \sin (2 \theta)}{g}\) meters away. \(g\) is the acceleration due to gravity \((9.8\) \(\left.\mathrm{m} / \mathrm{sec}^{2}\right)\) (a) Express the maximum distance the baseball can travel (from shoulder level to shoulder level) in terms of the initial velocity. (b) The fastest baseball pitchers can throw about 100 miles per hour. How far would such a ball travel if thrown at the optimal angle? (Note: 1 mile \(=5280\) feet and 1 meter \(\approx 3.28\) feet. \()(*)\)

Evaluate. $$ \frac{d}{d x} \frac{4}{\sqrt{2-\cos (x / 7)}} $$

A policewoman is standing 80 feet away from a long, straight fence when she notices someone running along it. She points her flashlight at him and keeps it on him as he runs. When the distance between her and the runner is 100 feet he is running at 9 feet per second. At this moment, at what rate is she turning the flashlight to keep him illuminated? Include units in your answer.

Evaluate. $$ \frac{d}{d x}\left[e^{3 x} \cos ^{2}(7 x)\right] $$

Find \(d y / d x\) in terms of \(x\) and \(y\). $$ \sin (x y)+y=y \cos x $$

A lookout tower is located \(0.5\) kilometers from a line of warehouses. A searchlight on the tower is rotating at a rate of 6 revolutions per minute. How fast is the beam of light moving along the wall of warehouses when it passes by a window located 1 kilometer from the tower?

Find \(y^{\prime}\). (a) \(y=\frac{x}{\sin x}\) (b) \(y=3 \tan ^{3}\left(x^{2}\right)\) (c) \(y=\tan \left(\frac{x}{3}\right) \sec (3 x)\)

Graph \(f(x)=2^{\cos x}\). (a) Is the function periodic? If so, what is its period? (b) What is its maximum value? Its minimum value? Give exact answers.

Why have we been telling you that radians are more appropriate than degrees when using calculus? Suppose \(x\) is measured in degrees. Then \(\cos x^{\circ}=\cos \left(\frac{x^{\circ} \pi \text { radians }}{180^{\circ}}\right)=\) \(\cos \left(\frac{\pi x}{180}\right)\) where the argument is now in radians. Find the derivative. Is the derivative \(-\sin x^{\circ} ?\)

Let \(f(x)=-\cos x\) and \(g(x)=\sin x\). (a) What is the maximum distance between these two curves on the interval \(\left[-\frac{\pi}{4}, \frac{3 \pi}{4}\right] ?\) (b) What is the point of intersection of the tangent lines to these curves at the points from part (a) where the curves are farthest apart? Does this answer surprise you? Explain.

A wheel of radius 5 meters is oriented vertically and spinning counterclockwise at a rate of 7 revolutions per minute. If the origin is placed at the center of the wheel a point on the rim has a horizontal position of \((7,0)\) at time \(t=0\). What is the horizontal component of the point's velocity at \(t=2 ?\)

In this problem you will show that \(y=C_{1} \sin k x+C_{2} \cos k x\) is a solution to the differential equation $$y^{\prime \prime}=-k^{2} y .$$ Recall that a differential equation is an equation involving a derivative and a function is a solution to the differential equation if it satisfies the differential equation. (a) Show that \(y_{1}=\sin k x\) is a solution to \(y^{\prime \prime}=-k^{2} y .\) To do this, first find \(y_{1}^{\prime \prime} .\) Then write $$y_{1}^{\prime \prime} \stackrel{?}{=}-k^{2} y_{1}$$ and verify that the two sides are indeed equal. (b) Show that \(y_{2}=C_{1} \sin k x\) is a solution to \(y^{\prime \prime}=-k^{2} y\). (c) Show that if \(y_{1}\) and \(y_{2}\) are solutions to \(y^{\prime \prime}=-k^{2} y\), then \(y_{3}=C_{1} y_{1}+C_{2} y_{2}\) is a solution to \(y^{\prime \prime}=-k^{2} y\) as well. Conclude that \(y=C_{1} \sin k x+C_{2} \cos k x\) is a solution to the differential equation \(y^{\prime \prime}=-k^{2} y\).

For each of the functions below, determine whether the function is a solution to differential equation (i), differential equation (ii), or neither. Differential equations (i) and (ii) are given below. \(\begin{array}{ll}\text { i. } y^{\prime \prime}=16 y & \text { ii. } y^{\prime \prime}=-16 y\end{array}\) (a) \(y_{1}(t)=\sin 16 t\) (b) \(y_{2}(t)=e^{4 t}\) (c) \(y_{3}(t)=3 \cos 4 t\) (d) \(y_{4}(t)=\sin 4 t+1\) (e) \(y_{5}(t)=e^{-16 t}\) (f) \(y_{6}(t)=-3 e^{-4 t}\) (g) \(y_{7}(t)=e^{4 t}+3\) (h) \(y_{8}(t)=-\sin 4 t\)

Each of the functions below is a solution to one of the differential equations below. i. \(y^{\prime \prime}=9\) ii. \(y^{\prime \prime}=9 y\) iii. \(y^{\prime \prime}=-9 y\) For each function, determine which of the three differential equations it satisfies. (a) \(y_{1}(t)=5 \sin 3 t\) (b) \(y_{2}(t)=e^{3 t}\) (c) \(y_{3}(t)=2 \cos 3 t\) (d) \(y_{4}(t)=4.5 t^{2}+3 t+8\) (e) \(y_{5}(t)=4 e^{-3 t}\) (f) \(y_{6}(t)=4.5 t^{2}-t+2\)

After having done the previous problem, make up a solution to each of the three differential equations below. i. \(y^{\prime \prime}=9\) ii. \(y^{\prime \prime}=9 y \quad\) iii \(y^{\prime \prime}=-9 y\) Your answers must be different from the solutions given in the preceding problem, but you can use those answers for inspiration. Check that your answers are right by "plugging them back" into the differential equation. For instance, if you guess that \(y=e^{3 t}+1\) is a solution to the differential equation \(y^{\prime \prime}=9 y\), test it out as follows. First calculate \(y^{\prime \prime}\). Since \(y^{\prime}=e^{3 t} \cdot 3, y^{\prime \prime}=9 e^{3 t}\). Now see if it satisfies the differential equation. $$\begin{aligned} y^{\prime \prime} & \stackrel{?}{=} 9 y \\ 9 e^{3 t} &=9\left(e^{3 t}+1\right) \\ 9 e^{3 t} & \neq 9 e^{3 t}+9 \end{aligned}$$ So \(y=e^{3 t}+1\) is not a solution to \(y^{\prime \prime}=9 y\).