15.5: Exercises

15.1. First derivatives. Calculate the first derivative for the following functions:

(a) \(y=\sin x^{2}\)

(b) \(y=\sin ^{2} x\)

(c) \(y=\cot ^{2} \sqrt[3]{x}\)

(d) \(y=\sec \left(x-3 x^{2}\right)\)

(e) \(y=2 x^{3} \tan x\)

(f) \(y=\frac{x}{\cos x}\)

(g) \(y=x \cos x\)

(h) \(y=e^{-\sin ^{2} \frac{1}{x}}\)

(i) \(y=(2 \tan 3 x+3 \cos x)^{2}\)

(j) \(y=\cos (\sin x)+\cos x \sin x\)

15.2. Derivatives. Take the derivative of the following functions.

(a) \(f(x)=\cos \left(\ln \left(x^{4}+5 x^{2}+3\right)\right)\)

(b) \(f(x)=\sin \left(\sqrt{\cos ^{2}(x)+x^{3}}\right)\)

(c) \(f(x)=2 x^{3}+\log _{3}(x)\)

(d) \(f(x)=\left(x^{2} e^{x}+\tan (3 x)\right)^{4}\)

(e) \(f(x)=x^{2} \sqrt{\sin ^{3}(x)+\cos ^{3}(x)}\)

15.3. Point moving on a circle. A point is moving on the perimeter of a circle of radius 1 at the rate of \(0.1\) radians per second.

(a) How fast is its \(x\) coordinate changing when \(x=0.5\)?

(b) How fast is its \(y\) coordinate changing at that time?

15.4. Graphing trigonometric functions. The derivatives of the two important trig functions are \([\sin (x)]^{\prime}=\cos (x)\) and \([\cos (x)]^{\prime}=-\sin (x)\). Use these derivatives to answer the following questions.

Let \(f(x)=\sin (x)+\cos (x), 0 \leq x \leq 2 \pi\)

(a) Find all intervals where \(f(x)\) is increasing.

(b) Find all intervals where \(f(x)\) is concave up.

(c) Locate all inflection points.

(d) Graph \(f(x)\).

15.5. Tangent lines. Find all points on the graph of \(y=\tan (2 x),-\frac{\pi}{4}<x<\) \(\frac{\pi}{4}\), where the slope of the tangent line is 4.

15.6. Bird formation. A "V" shaped formation of birds forms a symmetric structure in which the distance from the leader to the last birds in the \(\mathrm{V}\) is \(r=10 \mathrm{~m}\), the distance between those trailing birds is \(D=6 \mathrm{~m}\) and the angle formed by the \(\mathrm{V}\) is \(\theta\), as shown in Figure 15.9.

Figure 15.9: Figure for Exercise 15.6; bird flock formation.

Suppose that the shape is gradually changing: the trailing birds start to get closer so that their distance apart shrinks at a constant rate \(d D / d t=-0.2 \mathrm{~m} / \mathrm{min}\) while maintaining the same distance from the leader. Assume that the structure is always in the shape of a \(\mathrm{V}\) as the other birds adjust their positions to stay aligned in the flock.

What is the rate of change of the angle \(\theta\) ?

15.7. Hot air balloon. A hot air balloon on the ground is 200 meters away from an observer. It starts rising vertically at a rate of 50 meters per minute. Find the rate of change of the angle of elevation of the observer when the balloon is 200 meters above the ground.

15.8. Cannon-ball. A cannon-ball fired by a cannon at ground level at angle \(\theta\) to the horizon \((0 \leq \theta \leq \pi / 2)\) travels a horizontal distance (called the range, \(R\) ) given by the formula below:

\[R=\frac{1}{16} v_{0}^{2} \sin \theta \cos \theta . \nonumber \]

Here \(v_{0}>0\), the initial velocity of the cannon-ball, is a fixed constant and air resistance is neglected (see Figure 15.10).

What is the maximum possible range?

Figure 15.10: Figure for Exercise 15.8; cannonball trajectory.

15.9. Leaning ladder. A ladder of length \(L\) is leaning against a wall so that its point of contact with the ground is a distance \(x\) from the wall, and its point of contact with the wall is at height \(y\). The ladder slips away from the wall at a constant rate \(C\).

(a) Find an expression for the rate of change of the height \(y\).

(b) Find an expression for the rate of change of the angle \(\theta\) formed between the ladder and the wall.

15.10. Cycloid curve. A wheel of radius 1 meter rolls on a flat surface without slipping. The wheel moves from left to right, rotating clockwise at a constant rate of 2 revolutions per second. Stuck to the rim of the wheel is a piece of gum, (labeled \(G\) ); as the wheel rolls along, the gum follows a path shown by the wide arc (called a "cycloid curve") in Figure 15.11.

Figure 15.11: Figure for Exercise 15.10; a cycloid curve.

The \((x, y)\) coordinates of the gum \((G)\) are related to the wheel’s angle of rotation \(\theta\) by the formulae

\[\begin{aligned} & x=\theta-\sin \theta \\ & y=1-\cos \theta \end{aligned} \nonumber \]

where \(0 \leq \theta \leq 2 \pi\)

(a) How fast is the gum moving horizontally at the instant that it reaches its highest point?

(b) How fast is it moving vertically at that same instant?

15.11. Zebra danio’s reaction distance. Solve Equation (15.3.3) for \(x\) and show that you get the reaction distance \(x \equiv x_{\text {react }}\) given in Eqn (15.3.4).

15.12. Bad design for a predator. Some predators are more easily detected than others. Use Equation (15.6) to find the size of predator for which the reaction distance is maximal.

15.13. Sneaking up on the prey.

(a) Use Equation (15.3.4) to show that a predator moving "slowly enough" can sneak up on the prey without being detected.

(b) What is the largest velocity for which a predator of size \(S\) is not detected by a prey that responds to a visual sighting when the rate of change of the visual angle exceeds the threshold \(K_{\text {crit }}\)?

15.14. Inverse trigonometric derivatives. Find the first derivative of the following functions.

(a) \(y=\arcsin x^{\frac{1}{3}}\)

(b) \(y=(\arcsin x)^{\frac{1}{3}}\)

(c) \(\theta=\arctan (2 r+1)\)

(d) \(y=x \operatorname{arcsec} \frac{1}{x}\)

(e) \(y=\frac{x}{a} \sqrt{a^{2}-x^{2}}-\arcsin \frac{x}{a}, a>0\)

(f) \(y=\arccos \frac{2 t}{1+t^{2}}\)

15.15. Rotating wheel. In Figure \(15.12\), the point \(P\) is connected to the point \(O\) by a rod \(3 \mathrm{~cm}\) long. The wheel rotates around \(O\) in the clockwise direction at a constant speed, making 5 revolutions per second. The point \(\mathrm{Q}\), which is connected to the point \(P\) by a rod \(5 \mathrm{~cm}\) long, moves along the horizontal line through \(O\).

Figure 15.12: Figure for Exercise 15.15; rotating wheel.

How fast and in what direction is \(Q\) moving when \(P\) lies directly above \(O\) ?

Note: recall the law of cosines: \(c^{2}=a^{2}+b^{2}-2 a b \cos \theta\).

15.16. Sailing ship. A ship sails away from a harbor at a constant speed \(v\). The total height of the ship including its mast is \(h\). See Figure \(15.13\).

Figure 15.13: Figure for Exercise 15.16; ship sailing away.

(a) At what distance away does the ship disappear below the horizon?

(b) At what rate does the top of the mast appear to drop toward the horizon just before this?

Note: in ancient times this effect lead people to conjecture that the earth is round (radius \(R\) ), a fact which you need to take into account.

15.17. Implicit differentiation. Find \(\frac{d y}{d x}\) using implicit differentiation.

(a) \(y=2 \tan (2 x+y)\)

(b) \(\sin y=-2 \cos x\)

(c) \(x \sin y+y \sin x=1\).

15.18. Equation of a tangent line. Use implicit differentiation to find the equation of the tangent line to the following curve at the point \((1,1)\) :

\[x \sin \left(x y-y^{2}\right)=x^{2}-1 \nonumber \]

15.19. Implicit differentiation and arcsin. The function \(y=\arcsin (a x)\) is a so-called inverse trigonometric function. It expresses the same relationship as does the equation \(a x=\sin (y)\).

Note: this function is defined only for values of \(x\) between \(1 / a\) and \(-1 / a\).

Use implicit differentiation to find \(y^{\prime}\).

15.20. Best view.Your room has a window whose height is \(1.5 \mathrm{~m}\). The bottom edge of the window is \(10 \mathrm{~cm}\) above your eye level, as depicted in Figure \(15.14\). How far away from the window should you stand to get the best view?

Figure 15.14: Figure for Exercise 15.20; distance from a window.

Note: "best view" means the largest visual angle, i.e. angle between the lines of sight to the bottom and to the top of the window.

15.21. Fireworks. You are directly below English Bay during a summer fireworks event and looking straight up. A single fireworks explosion occurs directly overhead at a height of \(500 \mathrm{~m}\) as depicted in Figure \(15.15\). The rate of change of the radius of the flare is \(100 \mathrm{~m} / \mathrm{sec}\).

Figure 15.15: Figure for Exercise 15.21; fireworks overhead.

Assuming that the flare is a circular disk parallel to the ground (with its centre directly overhead), what is the rate of change of the visual angle at the eye of an observer on the ground at the instant that the radius of the disk is \(r=100\) meters?

Note: the visual angle is the angle between the vertical direction and the line between the edge of the disk and the observer.

15.22. Differential equations and their solutions. Match the differential equations given in parts (i-iv) with the functions in (a-f) which are solutions for them. Differential equations:

(i) \(d^{2} y / d t^{2}=4 y\)

(ii) \(d^{2} y / d t^{2}=-4 y\)

(iii) \(d y / d t=4 y\)

(iv) \(d y / d t=-4 y\)

Solutions:

(a) \(y(t)=4 \cos (t)\)

(b) \(y(t)=2 \cos (2 t)\)

(c) \(y(t)=4 e^{-2 t}\)

(d) \(y(t)=5 e^{2 t}\)

(e) \(y(t)=\sin (2 t)-\cos (2 t)\)

(f) \(y(t)=2 e^{-4 t}\).

Note: each differential equation may have more than one solution

15.23. Periodic motion.

(a) Show that the function \(y(t)=A \cos (w t)\) satisfies the differential equation

\[\frac{d^{2} y}{d t^{2}}=-w^{2} y \nonumber \]

where \(w>0\) is a constant, and \(A\) is an arbitrary constant.

Note: \(w\) corresponds to the frequency and \(A\) to the amplitude of an oscillation represented by the cosine function.

(b) It can be shown using Newton’s laws of motion that the motion of a pendulum is governed by a differential equation of the form

\[\frac{d^{2} y}{d t^{2}}=-\frac{g}{L} \sin (y), \nonumber \]

where \(L\) is the length of the string, \(g\) is the acceleration due to gravity (both positive constants), and \(y(t)\) is displacement of the pendulum from the vertical.

What property of the sine function is used when this equation is approximated by the Linear Pendulum Equation:

\[\frac{d^{2} y}{d t^{2}}=-\frac{g}{L} y . \nonumber \]

(c) Based on this Linear Pendulum Equation, what function would represent the oscillations? What would be the frequency of the oscillations?

(d) What happens to the frequency of the oscillations if the length of the string is doubled?

15.24. Jack and Jill. Jack and Jill have an on-again off-again love affair. The sum of their love for one another is given by the function

\[y(t)=\sin (2 t)+\cos (2 t) . \nonumber \]

(a) Find the times when their total love is at a maximum.

(b) Find the times when they dislike each other the most.

15.25. Differential equations and critical points. Let

\[y=f(t)=e^{-t} \sin t,-\infty<t<\infty . \nonumber \]

(a) Show that \(y\) satisfies the differential equation \(y^{\prime \prime}+2 y^{\prime}+2 y=0\).

(b) Find all critical points of \(f(t)\).