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smi98485_ch06b.qxd 5/17/01 1:36 PM Page 521 Section 6.6 6.6 HISTORICAL NOTES Leonhard Euler (1707–1783) A Swiss mathematician regarded as the most prolific mathematician of all time. Euler’s complete works fill over 100 large volumes, with much of his work being completed in the last 20 years of his life after going blind. Euler made important and lasting contributions in numerous research fields, including calculus, number theory, calculus of variations, complex variables, graph theory and differential geometry. Mathematics author George Simmons calls Euler, “the Shakespeare of mathematics— universal, richly detailed and inexhaustible.” Euler’s Method 521 EULER’S METHOD In section 6.5, we saw how to solve some simple first-order differential equations, namely, those that are separable. While there are numerous other special cases of differential equations whose solutions are known (you will encounter many of these in any beginning course in differential equations), the vast majority cannot be solved exactly. For instance, the equation y = x 2 + y2 + 1 is not separable and cannot be solved using our current techniques. Nevertheless, some information about the solution(s) can be determined. In particular, since y = x 2 + y 2 + 1 > 0, we can conclude that every solution is an increasing function. This type of information is called qualitative, since it tells us about some quality of the solution without providing any specific quantitative information. In this section, we examine first-order differential equations in a more general setting. We consider the more general first-order equation of the form y = f (x, y). (6.1) While we cannot solve all such equations, it turns out that there are many numerical methods available for approximating the solution of such problems. We will study one simple method here, called Euler’s method. We begin by observing that any solution of equation (6.1) is a function y = y(x) whose slope at any particular point (x, y) is given by f (x, y). In order to get a handle on what a solution curve looks like, we draw a short line segment through each of a sequence of points (x, y), with slope f (x, y), respectively. This collection of line segments is called the direction field or slope field of the differential equation. Notice that if a particular solution curve passes through a given point (x, y), then its slope at that point is f (x, y). Thus, the direction field gives an indication of the behavior of the family of solutions of a differential equation. Example 6.1 Constructing a Direction Field Construct the direction field for y = 1 y. 2 (6.2) Solution All that needs to be done is to plot a number of points and then draw through each point (x, y), a short line segment with slope f (x, y). For example, at the point (0, 1), draw a short line segment with slope y (0) = f (0, 1) = 12 (1) = 12 . Draw similar segments at 25 to 30 points. This is a bit tedious to do by hand, but a good graphing utility can do this for you with minimal effort. See Figure 6.26a (on the following page) for the direction field for equation (6.2). Notice that equation (6.2) is separable. We leave it as an exercise to produce the general solution 1 y = A e2x. We plot a number of the curves in this family of solutions in Figure 6.26b (on the following page) using the same graphing window we used for Figure 6.26a. Notice that if you connected some of the line segments in Figure 6.26a, you would obtain a close approximation to the exponential curves depicted in Figure 6.26b. What is significant about this smi98485_ch06b.qxd 522 5/17/01 1:36 PM Page 522 Chapter 6 Exponentials, Logarithms and Other Transcendental Functions y y 4 2 4 4 x 2 2 4 2 x 4 2 4 4 4 Figure 6.26b Several solutions of y = 12 y. Figure 6.26a Direction field for y = 12 y. is that Figure 6.26a was constructed using elementary algebra, without first solving the differential equation. That is, by constructing the direction field, we can obtain a reasonably good picture of how the solution curves behave. Since most differential equations are not solvable exactly, this is an extremely useful observation. This is what we mean by qualitative information about the solution: we get a graphical idea of how solutions behave, but no details, such as the value of a solution at a specific point. We’ll see later in this section that we can obtain approximate values of the solution of an IVP numerically. ■ As we have already seen, differential equations are used to describe a wide variety of phenomena in science and engineering. Among many other applications, differential equations are used to find flow lines or equipotential lines for electromagnetic fields. In such cases, it is very helpful to visualize solutions graphically, so as to gain an intuitive understanding of the behavior of such solutions and the physical phenomenon they are modeling. Example 6.2 Using a Direction Field to Visualize the Behavior of Solutions Construct the direction field for y = x + e−y . Solution There’s really no trick to this; just draw a number of line segments with the correct slope. Again, we let our CAS do this for us and obtained the direction field in Figure 6.27a. Unlike example 6.1, you do not know how to solve this differential equation y 4 2 4 x 2 2 4 2 4 Figure 6.27a Direction field for y = x + e− y . smi98485_ch06b.qxd 5/17/01 1:36 PM Page 523 Section 6.6 y 4 2 4 2 x 2 Figure 6.27b Solution of y = x + e− y passing through (−4, 2). Euler’s Method 523 exactly. Even so, you should be able to clearly see from the direction field how solutions behave. For example, solutions that start out in the second quadrant initially decrease very rapidly, may dip into the third quadrant and then get pulled into the first quadrant and increase quite rapidly toward infinity. This is quite a bit of information to determine using little more than elementary algebra. In Figure 6.27b, we have plotted the solution of the differential equation that also satisfies the initial condition y(−4) = 2. We’ll see how to generate such an approximate solution later in this section. Note how well this corresponds with what you get by connecting a few of the line segments in Figure 6.27a. ■ You have already seen (in sections 6.4 and 6.5) how differential equation models can provide important information about how populations change over time. A model that includes a critical threshold is P (t) = −2[1 − P(t)][2 − P(t)]P(t), where P(t) represents the size of a population at time t. A simple context in which to understand a critical threshold is with the problem of the sudden infestations of pests. For instance, suppose that you have some method for removing ants from your home. As long as the reproductive rate of the ants is lower than your removal rate, you will keep the ant population under control. However, as soon as the ant reproductive rate becomes larger than your removal rate (i.e., crosses the critical threshold), you won’t be able to keep up with the extra ants and you will suddenly be faced with a big ant problem. We see this type of behavior in the following example. Example 6.3 Population Growth with a Critical Threshold Draw the direction field for P (t) = −2[1 − P(t)][2 − P(t)]P(t) and discuss the eventual size of the population. Solution The direction field is particularly easy to sketch here since the right-hand side depends on P, but not on t. If P(t) = 0, then P (t) = 0, also, so that the direction field is horizontal. The same is true for P(t) = 1 and P(t) = 2. If 0 < P(t) < 1, then P (t) < 0 and the solution decreases. If 1 < P(t) < 2, then P (t) > 0 and the solution increases. Finally, if P(t) > 2, then P (t) < 0 and the solution decreases. Putting all of these pieces together, we get the direction field seen in Figure 6.28. The constant P 3 2 1 3 2 1 t 1 2 Figure 6.28 Direction field for P (t) = −2[1 − P(t)][2 − P(t)]P(t). 3 smi98485_ch06b.qxd 524 5/17/01 1:36 PM Page 524 Chapter 6 Exponentials, Logarithms and Other Transcendental Functions solutions P(t) = 0, P(t) = 1 and P(t) = 2 are called equilibrium solutions. P(t) = 1 is called an unstable equilibrium, since populations that start near 1 don’t remain close to 1. P(t) = 0 and P(t) = 2 are called stable equilibria, since populations either rise to 2 or drop to 0 (extinction), depending on which side of the critical threshold P(t) = 1 they are on. (Look again at Figure 6.28.) ■ In cases where you are interested in finding a particular solution, the numerous arrows of a direction field can be distracting. Euler’s method, developed below, enables you to approximate a single solution curve. The method is quite simple, based almost entirely on the idea of a direction field. However, Euler’s method does not provide particularly accurate approximations. More accurate but more complicated methods will be explored in the exercises. Consider the IVP y = f (x, y), y(x0 ) = y0 . y y y(x) (x1, y(x1)) (x1, y1) x0 x1 x Figure 6.29 Tangent line approximation. We must emphasize once again that, assuming there is a solution y = y(x), the differential equation tells us that the slope of the tangent line to the solution curve at any point (x, y) is given by f (x, y). Remember that the tangent line to a curve stays close to that curve near the point of tangency. This idea was the key to both Newton’s method and differentials. Notice that we already know one point on the graph of y = y(x), namely, the initial point (x0 , y0 ). Referring to Figure 6.29, if we would like to approximate the value of the solution at x = x1 [i.e., y(x1 )], and if x1 is not too far from x0 , then we could follow the tangent line at (x0 , y0 ) to the point corresponding to x = x1 and use the y-value at that point (call it y1 ) as an approximation to y(x1 ). Again, this is virtually the same thinking we employed when we devised Newton’s method and differential (tangent line) approximations. It’s a simple matter to obtain the equation of the tangent line at x = x0 : y = y0 + y (x0 )(x − x0 ). Thus, an approximation to the value of the solution at x = x1 is the y-coordinate of the point on the tangent line corresponding to x = x1 , that is, y(x1 ) ≈ y1 = y0 + y (x0 )(x1 − x0 ). (6.3) You have only to glance at Figure 6.29 to realize that this approximation is valid only when x1 is close to x0 . In solving an IVP, though, we are usually interested in finding the value of the solution on an interval [a, b] of the x-axis. Since this can only be done when we can find an exact solution, we settle for finding an approximate solution at a sequence of points in the interval [a, b]. Still, this tangent line approximation is valid only for points very close to the initial point. We can approximate the solution at other points of interest, as follows. First, we partition the interval [a, b] into n equal-sized pieces (a regular partition; where did you see this notion before?): a = x0 < x1 < x2 < · · · < xn = b, where xi+1 − xi = h, for all i = 0, 1, . . . , n − 1. We call h the step size. From the tangent line approximation (6.3), we already have y(x1 ) ≈ y1 = y0 + y (x0 )(x1 − x0 ) = y0 + h f (x0 , y0 ), smi98485_ch06b.qxd 5/17/01 1:36 PM Page 525 Section 6.6 Euler’s Method 525 where we have replaced (x1 − x0 ) by the step size, h and used the differential equation to write y (x0 ) = f (x0 , y0 ). Next, we would like to approximate the value of y(x1 ). Certainly, we could use the tangent line at the point (x1 , y(x1 )) to produce a tangent line approximation, but we don’t even know the y-coordinate of the point of tangency, y(x1 ). We do, however, have an approximation for this, produced in the preceding step. So, we make the further approximation y(x2 ) ≈ y(x1 ) + y (x1 )(x2 − x1 ) = y(x1 ) + h f (x1 , y(x1 )), where we have used the differential equation to replace y (x1 ) by f (x1 , y(x1 )) and used the fact that x2 − x1 = h. Finally, we approximate y(x1 ) by the approximation obtained in the previous step, y1 , to obtain y(x2 ) ≈ y(x1 ) + h f (x1 , y(x1 )) ≈ y1 + h f (x1 , y1 ) = y2 . Continuing in this way, we obtain the sequence of approximate values y(xi+1 ) ≈ yi+1 = yi + h f (xi , yi ), for i = 0, 1, 2, . . . . Euler’s method (6.4) This tangent line method of approximation is called Euler’s method. Example 6.4 Using Euler’s Method Use Euler’s method to approximate the solution of the IVP y = y, y(0) = 1. Solution Of course, this is just about the simplest differential equation imaginable. You can probably solve it by inspection, but if not, notice that it’s separable and that the solution of the IVP is y = y(x) = e x . We will use this exact solution as a basis for comparison for the performance of Euler’s method. From (6.4) with f (x, y) = y and taking h = 1, we have y(x1 ) ≈ y1 = y0 + h f (x0 , y0 ) = y0 + hy0 = 1 + 1(1) = 2. Likewise, for further approximations, we have y y(x2 ) ≈ y2 = y1 + h f (x1 , y1 ) = y1 + hy1 = 2 + 1(2) = 4, 8 7 y(x3 ) ≈ y3 = y2 + h f (x2 , y2 ) = y2 + hy2 = 4 + 1(4) = 8 6 5 4 3 2 1 x 1 2 3 Figure 6.30 Exact solution versus the approximate solution (dashed line). and so on. In this way, we construct a sequence of approximate values of the solution function. Remember: we are applying this approximate method to a problem with a known solution, so that we might compare the approximate and the exact values. In Figure 6.30, we have plotted the exact solution (solid line) against the approximate solution obtained from Euler’s method (dashed line). Notice how the error grows as x gets farther and farther from the initial point. This is characteristic of Euler’s method (and other similar methods). This growth in error becomes even more apparent if we look smi98485_ch06b.qxd 526 5/17/01 1:36 PM Page 526 Chapter 6 Exponentials, Logarithms and Other Transcendental Functions at a table of values of the approximate and exact solutions together. We display these in the table that follows where we have used h = 0.1 (values are displayed to seven digits). x Euler Exact Error = Exact − Euler 0.1 1.1 1.1051709 0.0051709 0.2 1.21 1.2214028 0.0114028 0.3 1.331 1.3498588 0.0188588 0.4 1.4641 1.4918247 0.0277247 0.5 1.61051 1.6487213 0.0382113 0.6 1.71561 1.8221188 0.1065088 0.7 1.9487171 2.0137527 0.0650356 0.8 2.1435888 2.2255409 0.0819521 0.9 2.3579477 2.4596031 0.1016554 1.0 2.5937425 2.7182818 0.1245393 As you might expect from our development of Euler’s method, the smaller we make h, the more accurate the approximation at a given point tends to be. As well, the smaller the value of h, the more steps it takes to reach a given value of x. In the following table, we display the Euler’s method approximation, the error and the number of steps needed to reach x = 1.0. Here, the exact value of the solution is y = e1 ≈ 2.718281828459. h Euler Error = Exact − Euler Number of Steps 1.0 2 0.7182818 1 0.5 2.25 0.4682818 2 0.25 2.4414063 0.2768756 4 0.125 2.5657845 0.1524973 8 0.0625 2.6379285 0.0803533 16 0.03125 2.6769901 0.0412917 32 0.015625 2.697345 0.0209369 64 0.0078125 2.707739 0.0105428 128 0.00390625 2.7129916 0.0052902 256 From the table, you should observe that each time the step size h is cut in half, the error is also cut roughly in half. This increased accuracy though, comes at the cost of the additional steps of Euler’s method required to reach a given point (doubled each time h is halved). ■ The point of having a numerical method, of course, is to find meaningful approximations to the solution of problems that we do not know how to solve exactly. Our final example is of this type. smi98485_ch06b.qxd 5/17/01 1:36 PM Page 527 Section 6.6 Example 6.5 Euler’s Method 527 Finding an Approximate Solution Find an approximate solution for the IVP 1 y = x 2 + y 2 , y(−1) = − . 2 y 1 1 1 2 x 1 Figure 6.31 Direction field for y = x 2 + y 2 . Solution First, let’s take a look at the direction field, so that we can see how solutions to this differential equation should behave (see Figure 6.31). Using Euler’s method, with h = 0.1, we get y(x1 ) ≈ y1 = y0 + h f (x0 , y0 ) = y0 + h x02 + y02 1 2 1 2 = −0.375 = − + 0.1 (−1) + − 2 2 and y(x2 ) ≈ y2 = y1 + h f (x1 , y1 ) = y1 + h x12 + y12 = −0.375 + 0.1[(−0.9)2 + (−0.375)2 ] = −0.2799375 and so on. Continuing in this way, we generate the table of values that follows. x Euler x Euler x Euler −0.9 −0.375 0.1 −0.0575822 1.1 0.3369751 −0.8 −0.2799375 0.2 −0.0562506 1.2 0.4693303 −0.7 −0.208101 0.3 −0.0519342 1.3 0.6353574 −0.6 −0.1547704 0.4 −0.0426645 1.4 0.8447253 −0.5 −0.116375 0.5 −0.0264825 1.5 1.1120813 −0.4 −0.0900207 0.6 −0.0014123 1.6 1.4607538 −0.3 −0.0732103 0.7 0.0345879 1.7 1.9301340 −0.2 −0.0636743 0.8 0.0837075 1.8 2.5916757 −0.1 −0.0592689 0.9 0.1484082 1.9 3.587354 0.0 −0.0579176 1.0 0.2316107 2.0 5.235265 smi98485_ch06b.qxd 528 5/17/01 1:36 PM Page 528 Chapter 6 Exponentials, Logarithms and Other Transcendental Functions y y 1.5 1.5 1.0 1.0 0.5 0.5 1.0 0.5 1.0 1.5 2.0 x 1.0 0.5 0.5 0.5 1.0 1.0 1.5 1.5 1.0 1.5 2.0 x Figure 6.32b Approximate solution superimposed on the direction field. Figure 6.32a Approximate solution of y = x 2 +y 2 , passing through −1, − 12 . In Figure 6.32a, we have displayed a graph of the data in the preceding table. Take particular note of how well this corresponds with the direction field in Figure 6.31. To make this correspondence more apparent, we have drawn a graph of the approximate solution superimposed on the direction field in Figure 6.32b. Since this corresponds so well with the behavior you expect from the direction field, you should expect that there are no gross errors in this approximate solution. (Certainly, there is always some level of round-off and other numerical errors.) ■ EXERCISES 6.6 1. For Euler’s method, explain why using a smaller step size should produce a better approximation. 2. Look back at the direction field in Figure 6.31 and the Euler’s method solution in Figure 6.32. Describe how the direction field gives you a more accurate sense of the exact solution. Given this, explain why Euler’s method is important. (Hint: How would you get a table of approximate values of the solution from a direction field?) 3. 4. In exercises 5–16, construct four of the direction field arrows by hand and use your CAS or calculator to do the rest. Describe the general pattern of solutions. 5. y = x + 4y 6. y = x 2 + y 2 7. y = 2y − y 2 8. y = y3 − 1 9. y = 2x y − y 2 10. y = y3 − x In the situation of example 6.3, if you only needed to know the stability of an equilibrium solution, explain why a qualitative method is preferred over trying to solve the differential equation. Describe one situation in which you would need to solve the equation. 11. y = cos y + 1/x 12. y = sin y − x 2 13. y = 1 − y 2 − e−x 14. y = 1 − x 2 + e− y 15. y = 1 − 16. y = Imagine superimposing solution curves over Figure 6.27a. Explain why the Euler’s method approximation takes you from one solution curve to a nearby one. Use one of the examples in this section to describe how such a small error could lead to very large errors in approximations for large values of x. In exercises 17–24, use Euler’s method with h = 0.1 and h = 0.05 to approximate y(1) and y(2). Show the first two steps by hand. y2 x y3 −1 x 17. y = 2x y, y(0) = 1 18. y = x/y, y(0) = 2 19. y = 4y − y 2 , y(0) = 1 20. y = x/y 2 , y(0) = 2 smi98485_ch06b.qxd 5/17/01 1:36 PM Page 529 Section 6.6 21. y = 1 − y + e−x , y(0) = 3 √ 22. y = sin y − x 2 , y(0) = 1 23. y = 25. Find the exact solutions in exercises 17 and 18 and compare y(1) and y(2) to the Euler’s method approximations. 26. Find the exact solutions in exercises 19 and 20 and compare y(1) and y(2) to the Euler’s method approximations. 27. Sketch the direction fields for exercises 21 and 22, highlight the curve corresponding to the given initial condition and compare the Euler’s method approximations to the location of the curve at x = 1 and x = 2. 28. Sketch the direction fields for exercises 23 and 24, highlight the curve corresponding to the given initial condition and compare the Euler’s method approximations to the location of the curve at x = 1 and x = 2. x + y, y(0) = 1 24. y = x 2 + y 2 , y(0) = 4 29. y = 2y − y 2 30. y = y −1 31. y = y2 − y4 32. y = e−y − 1 33. y = (1 − y) 1 + y 2 34. y = 35. 37. 38. t →∞ 36. Many species of trees are plagued by sudden infestations of worms. Let x(t) be the population of a species of worm on a 2x Show that f (x) = 2 + e2x is a solution of the initial value 2−e problem in exercise 37. Compute f (0.1), f (0.2), f (0.3), f (0.4) and f (0.5) and compare to the approximations in exercise 37. Graph the solution of y = y 2 − 1, y(0) = 3, given in exercise 38. Find an equation of the vertical asymptote. Explain why Euler’s method would be “unaware” of the existence of this asymptote and would therefore provide very unreliable approximations. 40. In exercises 37–39, suppose that x represents time (in hours) and y represents the force (in Newtons) exerted on an arm of a robot. Explain what happens to the arm. Given this, explain why the negative function values in exercise 38 are irrelevant and, in some sense, the Euler’s method approximations in exercise 37 give useful information. 41. There are several ways of deriving the Euler’s method formula. One benefit of having an alternative derivation is that it may inspire an improvement of the method. Here, we use an alternative form of Euler’s method to derive a method known as the Improved Euler’s method. Start with the differential equation y (x) = f (x, y(x)) and integrate both sides x = xn to x = xn+1 . Show that y(xn+1 ) = y(xn ) + from x n+1 f (x, y(x)) dx. Given y(xn ), then, you can estimate xn x y(xn+1 ) by estimating the integral x nn+1 f (x, y(x)) dx. One such estimate is a Riemann sum using left-endpoint evaluation, given by f (xn , y(xn )) x. Show that with this estimate you get Euler’s method. There are numerous ways of getting better estimates of the integral. One is to use the Trapezoidal Rule, x n+1 f (xn , y(xn )) + f (xn+1 , y(xn+1 )) f (x, y(x)) dx ≈ x. 2 xn 1 − y2 amount of activated gene. Suppose that a patch of zebra skin extends from x = 0 to x = 4π with an initial activated-gene distribution g(0) = 32 + 32 sin x at location x. If black corresponds to an eventual activated-gene level of 0 and white corresponds to an eventual activated-gene level of b, show what the zebra stripes will look like. Apply Euler’s method with h = 0.1 to the initial value problem y = y 2 − 1, y(0) = 3, and estimate y(0.5). Repeat with h = 0.05 and h = 0.01. In general, Euler’s method is more accurate with smaller h-values. Conjecture how the exact solution behaves in this example. (This is explored further in the next three exercises.) 39. 3 Zebra stripes and patterns on butterfly wings are thought to be the result of gene-activated chemical processes. Suppose g(t) is the amount of gene that is activated at time t. The dif3g 2 has been used to model ferential equation g = −g + 1 + g2 the process. Show that there are three equilibrium solutions: 0 and two positive solutions a and b, with a < b. Show that g > 0 if a < g < b and g < 0 if 0 < g < a or g > b. Explain why lim g(t) could be 0 or b, depending on the initial 529 particular tree. For some species, a model for population change is x = 0.1x(1 − x/k) − x 2 /(1 + x 2 ) for some positive constant k. If k = 10, show that there is only one positive equilibrium solution. If k = 50, show that there are three positive equilibrium solutions. Sketch the direction field for k = 50. Explain why the middle equilibrium value is called a threshold. An outbreak of worms corresponds to crossing the threshold for a large value of k (k is determined by the resources available to the worms). In exercises 29–34, find the equilibrium solutions and determine which are stable and which are unstable. Euler’s Method The drawback with this estimate is that you know y(xn ) but you do not know y(xn+1 ). Briefly explain why this statement is correct. The way out is to use Euler’s method; you do not know y(xn+1 ) but you can approximate it by y(xn+1 ) ≈ y(xn ) + h f (xn , y(xn )). Put all of this together to get the smi98485_ch06b.qxd 530 5/17/01 1:36 PM Page 530 Chapter 6 Exponentials, Logarithms and Other Transcendental Functions for positive constants a, b and c. First, look carefully at the equations. The term bx(t)y(t) is included to represent the effects of encounters between the species. This effect is negative on species X and positive on species Y. If b = 0, the species don’t interact at all. In this case, show that species Y dies out (with death rate c) and species X thrives (with growth rate a). Given all of this, explain why X must be the prey and Y the predator. Next, you should find the equilibrium point for coexistence. That is, find positive values x̄ and ȳ such that both x (t) = 0 and y (t) = 0. For this problem, think of X as an insect that damages farmers’ crops and Y as a natural predator (e.g., a bat). A farmer might decide to use a pesticide to reduce the damage caused by the X’s. Briefly explain why the effects of the pesticide might be to decrease the value of a and increase the value of c. Now, determine how these changes affect the equilibrium values. Show that the pest population X actually increases and the predator population Y decreases. Explain, in terms of the interaction between predator and prey, how this could happen. The moral is that the long-range effects of pesticides can be the exact opposite of the short-range (and desired) effects. Improved Euler’s method: yn+1 = yn + h [ f (xn , yn ) + f (xn + h, yn + h f (xn , yn ))]. 2 Use the Improved Euler’s method for the IVP y = y, y(0) = 1 with h = 0.1 to compute y1 , y2 and y3 . Compare to the exact values and the Euler’s method approximations given in example 6.4. 42. In sections 6.4 and 6.5, you explored some differential equation models of population growth. An obvious flaw in those models was the consideration of a single species in isolation. In this exercise, you will investigate a predator-prey model. In this case, there are two species, X and Y, with populations x(t) and y(t), respectively. The general form of the model is x (t) = ax(t) − bx(t)y(t) y (t) = bx(t)y(t) − cy(t) 6.7 In this section, we expand the set of functions available to you by defining inverses to the trigonometric functions. To get started, let’s again look at a graph of y = sin x (see Figure 6.33). Notice that we cannot define an inverse function, since sin x is not one-toone. We can remedy this at only a portion of the domain. If we restrict the by looking domain to the interval − π2 , π2 , then y = sin x is one-to-one there (see Figure 6.34) and hence, has an inverse. We thus define the inverse sine function by y 1 x p q THE INVERSE TRIGONOMETRIC FUNCTIONS q y = sin−1 x p if and only if sin y = x and −π2 ≤ y ≤ π . 2 (7.1) to think of this definition as follows. If y = sin−1 x, then y is the angle It is convenient π π between − 2 and 2 for which sin y = x. You should note that we could have selected any interval on which sin x is one-to-one, but − π2 , π2 is the most convenient. To see that these are indeed inverse functions, you should observe that 1 Figure 6.33 y = sin x. sin(sin−1 x) = x, for all x ∈ [−1, 1] y and 1 x q q 1 Figure 6.34 y = sin x on − π2 , π2 . π π sin−1 (sin x) = x, for all x ∈ − , . 2 2 (7.2) −1 Read equation (7.2) very carefully. It does not that say sin (sin x) = x for all x, but π π rather, only for those in the restricted domain, − 2 , 2 . So, while it might be tempting to write sin−1 (sin π) = π , this is incorrect, as sin−1 (sin π) = sin−1 (0) = 0. smi98485_ch06b.qxd 5/17/01 1:36 PM Page 531 Section 6.7 531 The Inverse Trigonometric Functions Remark 7.1 Mathematicians often use the notation arcsin x in place of sin−1 x. People will read sin−1 x interchangeably as “inverse sine of x’’ or “arcsine of x.’’ Example 7.1 √ Evaluate sin−1 23 . Evaluating an Inverse Sine sin θ = Solution We look for the angle θ in the interval − π2 , π2 for which π √3 √ Note that since sin 3 = 2 and π3 ∈ − π2 , π2 , we have that sin−1 23 = π3 . √ 3 . 2 ■ Example 7.2 Evaluate sin−1 − 12 . Solution Evaluating an Inverse Sine with a Negative Argument Here, note that sin − π6 = − 12 and − π6 ∈ − π2 , π2 . Thus, 1 π −1 − sin =− . 2 6 ■ Judging by the preceding two examples, you might think that (7.1) is a roundabout way of defining a function. If so, you’ve got the idea exactly. In fact, we want to emphasize that what we know about the inverse sine function is principally through reference to the sine function. We will not have any other definition of arcsine, nor are there any algebraic formulas for this function. (These things are true of most inverse functions.) Further, you should recall from our discussion in section 6.2 that we can draw a graph of y = sin−1 x simply by reflecting the graph of y = sin x on the interval − π2 , π2 (from Figure 6.34) through the line y = x (see Figure 6.35). Turning to y = cos x, can you think of how to restrict the domain to make the function one-to-one? Notice that restricting the domain to the interval − π2 , π2 , as we did for the inverse sine function will not work here. (Why not?) The simplest way to do this is to restrict its domain to the interval [0, π] (see Figure 6.36). Consequently, we define the inverse cosine function by y q x 1 1 q Figure 6.35 y = sin−1 x. y = cos−1 x if and only if cos y = x and 0 ≤ y ≤ π. (7.3) y Note that here, we have 1 cos(cos−1 x) = x, for all x ∈ [−1, 1] x q p 1 Figure 6.36 y = cos x on [0, π]. and cos−1 (cos x) = x, for all x ∈ [0, π]. As with the definition of arcsine, it is helpful to think of cos−1 x as that angle θ in [0, π] for which cos θ = x. As with sin−1 x , it is common to use cos−1 x and arccos x interchangeably. smi98485_ch06b.qxd 532 5/17/01 1:36 PM Page 532 Chapter 6 Exponentials, Logarithms and Other Transcendental Functions Example 7.3 Evaluating an Inverse Cosine Evaluate cos−1 (0). Solution You will need to find that angle θ in [0, π] for which cos θ = 0. It’s not hard to see that cos−1 (0) = π2 . If you calculate this on your calculator and get 90, your calculator is in degrees mode. In this event, you should immediately change it to radians mode. ■ Example 7.4 √ Evaluate cos−1 − 22 . y p Solution √ Here, look for the angle θ ∈ [0, π] for which cos θ = − cos 3π = − 22 and 3π ∈ [0, π] . Consequently, 4 4 √ 2 3π cos−1 − = . 2 4 q x 1 Evaluating an Inverse Cosine with a Negative Argument √ 2 . 2 Notice that ■ 1 Figure 6.37 y = cos−1 x. Once again, we obtain the graph of this inverse function by reflecting the graph of y = cos x on the interval [0, π] (seen in Figure 6.36) through the line y = x (see Figure 6.37). We can define inverses for each of the four remaining trig functions in similar ways. For y = tan x, we restrict the domain to the interval − π2 , π2 . Think about why the endpoints of this interval are not included (see Figure 6.38). Having done this, you should readily see that we define the inverse tangent function by y 6 4 y = tan−1 x 2 if and only if tan y = x and − π2 < y < π . 2 (7.4) x q q 2 The graph of y = tan−1 x is then as seen in Figure 6.39, found by reflecting the graph in Figure 6.38 through the line y = x . 4 y 6 q Figure 6.38 y = tan x on − π2 , π2 . 6 4 x 2 2 q Figure 6.39 y = tan−1 x. 4 6 smi98485_ch06b.qxd 5/17/01 1:36 PM Page 533 Section 6.7 y Example 7.5 533 Evaluating an Inverse Tangent Evaluate tan−1 (1). 10 Solution You must look for the angle, θ on the interval − π2 , π2 for which tan θ = 1. This is easy enough. Since tan π4 = 1 and π4 ∈ − π2 , π2 , we have that tan−1 (1) = π4 . 5 p 1 1 x q ■ 5 We now turn to defining an inverse for sec x. First, we must issue a disclaimer. As we have indicated, there are any number of ways to suitably restrict the domains of the trigonometric functions in order to make them one-to-one. With the first three we’ve seen, there has been an obvious choice of how to do this and there is general agreement among mathematicians on the choice of these intervals. In the case of sec x, this is not true. There are several reasonable ways in which to suitably restrict the domain and different authors restrict these differently. We have (arbitrarily) chosen to restrict the domain to be 0, π2 ∪ π2 , π . You might initially think that this looks strange. Why not use all of [0, π]? You need only think about the definition of sec x to see why we needed to exclude the point x = π2 . See Figure 6.40 for a graph of sec x on this domain. (Note the vertical asymptote at x = π2 .) Consequently, we define the inverse secant function by 10 Figure 6.40 y = sec x on [0, π]. y p q y = sec−1 x 10 The Inverse Trigonometric Functions 5 1 1 x 5 Figure 6.41 y = sec−1 x. 10 if and only if sec y = x and y ∈ 0, π2 ∪ π2 , π . (7.5) A graph of sec−1 x is shown in Figure 6.41. Example 7.6 √ Evaluate sec−1 (− 2). Evaluating an Inverse Secant Solution You must look for the angle θ with θ ∈ 0, π2 ∪ π2√, π , for which √ √ sec θ = − 2. Notice that if√sec θ = − 2, then cos θ = − √12 = − 22 . Recall from π = − 22 . Further, the angle 3π , π and so, example√ 7.4 that cos 3π 4 4 is in the interval 2 sec−1 (− 2) = 3π . 4 ■ Calculators do not usually have built-in functions for sec x or sec−1 x . In this case, you must convert the desired secant value to a cosine value and use the inverse cosine function, as we did in example 7.6. Remark 7.2 We can likewise define inverses to cot x and csc x . As these functions are used only infrequently, we will omit them here and examine them in the exercises. Often, as a part of a larger problem (for example, evaluating an integral by some of the methods discussed in Chapter 7) you will need to recognize some relationship between the trigonometric functions and their inverses. We present several clues here. When you are faced with these problems, our best advice is to keep in mind the definitions of the inverse functions and then draw a picture. smi98485_ch06b.qxd 534 5/17/01 1:36 PM Page 534 Chapter 6 Exponentials, Logarithms and Other Transcendental Functions Example 7.7 Simplifying Expressions Involving Inverse Trigonometric Functions Simplify sin(cos−1 x) and tan(cos−1 x). sin u 1 x2 1 u cos1x Solution Do not look for some arcane formula to help you out. Think first: cos−1 x is the angle (call it θ ) for which x = cos θ. First, consider the case where x > 0. Looking at Figure 6.42, we have drawn a right triangle, with hypotenuse 1 and adjacent angle θ. From the definition of the sine and cosine, then, we have that the base of the triangle is cos θ = x and the altitude is sin θ, which by the Pythagorean Theorem is sin(cos−1 x) = sin θ = 1 − x 2 . There are two subtle points that we should illuminate here. First, what if anything in Figure 6.42 changes if x < 0? (Think about this one.) Second, since θ = cos−1 x, we have that θ ∈ [0, π] and hence, sin θ ≥ 0. Finally, you can also read from Figure 6.42 that √ sin θ 1 − x2 −1 tan(cos x) = tan θ = = . cos θ x cos u x Figure 6.42 θ = cos−1 x. Note that this last identity is valid, regardless of whether x = cos θ is positive or negative. ■ Example 7.8 Simplify sin tan−1 43 . 4 5 sin u 5 u tan1 (d) 3 5 cos u Figure 6.43 θ = tan−1 43 . Simplifying an Expression Involving an Inverse Tangent Once again, keep in mind that tan−1 34 is the angle θ, in the interval Solution π π − 2 , 2 for which tan θ = 43 . As an aid, we visualize the triangle shown in Figure 6.43. Note that we have made the side opposite the angle, θ to be of length 4 and the adjacent side of length 3, so that the tangent of the angle is 43 , as desired. Of course, this makes the hypotenuse 5, by the Pythagorean Theorem. Using the picture as a guide, we can read off the desired value. 4 −1 4 sin θ = sin tan = . 3 5 While we’re at it, we should also observe that we have found that 3 4 cos θ = cos tan−1 = . 3 5 ■ EXERCISES 6.7 1. Discuss how to compute sec−1 x, csc−1 x and cot−1 x on a calculator that only has built-in functions for sin−1 x, −1 cos x and tan−1 x . 2. Give a different range for sec−1 x than that given in the text. For which x’s would the value of sec−1 x change? Using the calculator discussion in exercise 1, give one reason why we might have chosen the range that we did. 3. Inverse functions are necessary for solving equations. The restricted range we had to use to define inverses of the trigonometric functions also restricts their usefulness in smi98485_ch06b.qxd 5/17/01 1:36 PM Page 535 Section 6.7 equation-solving. Explain how to use sin−1 x to find all solutions of the equation sin u = x . 4. The idea of restricting ranges can be used to define inverses for a variety of functions. Explain how to define an inverse for f (x) = x 2 . In exercises 5–18, evaluate the inverse function by sketching a unit circle and locating the correct angle on the circle. 5. sin−1 0 6. cos−1 0 7. tan−1 1 8. tan−1 0 The Inverse Trigonometric Functions 45. tan(5x) = 1 46. tan(2x) = −1 47. sin(3x) = 48. sin(x/4) = − 12 1 2 49. sec(2x) = 2 535 50. csc(2x) = −2 51. Give precise definitions of csc−1 x and cot−1 x . 52. In baseball, outfielders are able to easily track down and catch fly balls that have very long and high trajectories. Physicists have argued for years about how this is done. A recent explanation involves the following geometry. Ball −1 9. sin (−1) 11. sec−1 1 −1 10. cos (1) 12. csc−1 0 13. tan−1 (−1) 14. cot−1 1 15. sec−1 2 16. csc−1 2 17. sin−1 − 12 18. cos−1 12 20. cos(tan−1 x) 21. sec−1 (cos x) 22. csc−1 (sin x) 23. tan(sec−1 x) 24. cot(cos−1 x) 25. sin−1 (sin x) 26. cos−1 (cos x) 27. sin cos−1 21 28. cos sin−1 21 29. tan cos−1 53 30. csc sin−1 32 31. cos−1 (cos(−π/8)) 32. sin−1 (sin 3π/4) 33. sin(2 sin−1 x/3) 34. cos(2 sin−1 2x) b Outfielder Home plate In exercises 19–34, use a triangle to simplify each expression. Where applicable, state the range of x’s for which the simplification holds. 19. cos(sin−1 x) a c The player can catch the ball by running to keep the angle ψ constant (this makes it appear that the ball is moving in a straight line). Assuming that all triangles shown are right triangles, show that tan ψ = tan α and then solve for ψ. tan β 53. A picture hanging in an art gallery has a frame 20 inches high and the bottom of the frame is 6 feet above the floor. A person who is 6 feet tall stands x feet from the wall. Let A be the angle formed by the ray from the person’s eye to the bottom of the frame and the ray from the person’s eye to the top of the frame. Write A as a function of x and graph y = A(x). 20" A In exercises 35–40, sketch a graph of the function. 35. cos−1 (2x) 36. cos−1 (x 3 ) 37. sin−1 (3x) 38. sin−1 (x/4) 39. tan−1 (5x) 40. tan−1 (x 2 ) 6' x In exercises 41–50, find all solutions. 41. cos(2x) = 0 42. cos(3x) = 1 43. sin(3x) = 0 44. sin(x/4) = 1 54. In golf, the goal is to hit a ball into a hole of diameter 4.5 inches. Suppose a golfer stands x feet from the hole trying to putt the ball into the hole. A first approximation of the margin smi98485_ch06b.qxd 536 5/17/01 1:36 PM Page 536 Chapter 6 Exponentials, Logarithms and Other Transcendental Functions of error in a putt is to measure the angle A formed by the ray from the ball to the right edge of the hole and the ray from the ball to the left edge of the hole. Find A as a function of x. 55. An oil tank with circular cross sections lies on its side. A stick is inserted in a hole at the top and used to measure the depth d of oil in the tank. Based on this measurement, the goal is to compute the percentage of oil left in the tank. 1 1 u d d To simplify calculations, suppose the circle is a unit circle with center at (0, 0). Sketch radii extending from the origin to the top of the oil. The area of oil at the bottom equals the area of the portion of the circle bounded by the radii minus the area of the triangle formed above. 6.8 Start with the triangle, which has area one-half base times height. Explain why the height is 1 − d. Find a right triangle in the figure (there are two of them) with hypotenuse 1 (the radius of the circle) and one vertical side of length 1 − d. The horizontal side has length equal to one-half the base of the larger triangle. Show that this equals 1 − (1 − d)2 . The area of the portion of the circle equals πθ/2π = θ/2, where θ is the angle at the top of the triangle. Find this angle as a function of d. (Hint: Go back to the right triangle used above with upper angle θ/2.) Then find the area filled with oil and divide by π to get the portion of the tank filled with oil. THE CALCULUS OF THE INVERSE TRIGONOMETRIC FUNCTIONS Now that we have defined the inverse trigonometric functions in section 6.7, we will examine the calculus of these functions briefly in the present section. Our first job is to find the derivative of sin−1 x. You might think of the definition of derivative, d sin−1 (x + h) − sin−1 x sin−1 x = lim , h→0 dx h but we don’t know enough about sin−1 x to resolve this limit directly. So, what do you know about this function? Hopefully, you’ll realize that, beyond the definition and a few isolated values, not much is known. Thus, while you might compute this limit approximately, for fixed values of x, you have little hope of resolving this limit symbolically. But remember, this is an inverse function and if you focus on that, you’ll see the following analysis. Recall the definition of sin−1 x given in (7.1): π π y = sin−1 x if and only if sin y = x and y ∈ − , . 2 2 Differentiating the equation sin y = x implicitly, we have d d sin y = x dx dx smi98485_ch06b.qxd 5/17/01 1:36 PM Page 537 Section 6.8 The Calculus of the Inverse Trigonometric Functions 537 and so, cos y Solving this for dy = 1. dx dy , we find (for cos y = 0) that dx dy 1 = . dx cos y This is not entirelysatisfactory, though, since this gives us the derivative in terms of y. Notice that for y ∈ − π2 , π2 , cos y ≥ 0 and hence, √ cos y = 1 − sin2 y = 1 − x 2 . This leaves us with for −1 < x < 1. That is, dy 1 1 = =√ , dx cos y 1 − x2 d 1 sin−1 x = √ , for −1 < x < 1. dx 1 − x2 Alternatively, we can derive this same derivative formula using Theorem 2.3 in section 6.2. Similarly, we can show that −1 d cos−1 x = √ , for −1 < x < 1. dx 1 − x2 We leave the derivation of this as an exercise. Likewise, to find d tan−1 x, we rely on its definition in (7.4). Recall that we have dx π π y = tan−1 x if and only if tan y = x and y ∈ − , . 2 2 Using implicit differentiation, we then have d d tan y = x dx dx and so, (sec2 y) We solve this for dy = 1. dx dy , to obtain dx 1 dy = dx sec2 y 1 = 1 + tan2 y 1 = . 1 + x2 smi98485_ch06b.qxd 538 5/17/01 1:36 PM Page 538 Chapter 6 Exponentials, Logarithms and Other Transcendental Functions That is, d 1 tan−1 x = . dx 1 + x2 You can likewise show that 1 d sec−1 x = √ , for |x| > 1. dx |x| x 2 − 1 This is left as an exercise. The derivatives of the two remaining inverse trigonometric functions are not important and are not discussed here. Example 8.1 Finding the Derivative of an Inverse Trigonometric Function Compute the derivative of (a) cos−1 (3x 2 ) and (b) (sec−1 x)2 . Solution (a) From the chain rule, we have d −1 d cos−1 (3x 2 ) = (3x 2 ) 2 2 dx dx 1 − (3x ) −6x =√ . 1 − 9x 4 (b) Also from the chain rule, we have d d (sec−1 x)2 = 2(sec−1 x) (sec−1 x) dx dx 1 = 2(sec−1 x) √ . |x| x 2 − 1 ■ Example 8.2 Modeling the Rate of Change of a Ballplayer’s Gaze One of the guiding principles of most sports is to “keep your eye on the ball.’’ In baseball, a batter stands 2 feet from home plate as a pitch is thrown with a velocity of 130 ft/s (about 90 mph). At what rate does the batter’s angle of gaze need to change when the ball crosses home plate? Solution First, look at the triangle shown in Figure 6.44. We denote the distance from the ball to home plate by d and the angle of gaze by θ. Since the distance is changing with time, we write d = d(t). The velocity of 130 ft/s means that d (t) = −130. [Why would d (t) be negative?] From Figure 6.44, notice that −1 d(t) θ(t) = tan . 2 The rate of change of the angle is then θ (t) = d (t) 1 d(t) 2 2 1+ 2 = 2d (t) radians/second. 4 + [d(t)]2 smi98485_ch06b.qxd 5/17/01 1:36 PM Page 539 Section 6.8 The Calculus of the Inverse Trigonometric Functions 539 Overhead view d u 2 Figure 6.44 A ballplayer’s gaze. When d(t) = 0 (i.e., when the ball is crossing home plate), the rate of change is then θ (t) = 2(−130) = −65 radians/second. 4 One problem with this is that most humans can accurately track objects only at the rate of about 3 radians/second. Keeping your eye on the ball in this case is thus physically impossible. How do those ballplayers do it? Research indicates that ballplayers must anticipate where the ball is going, instead of continuing to track the ball visually. (See Watts and Bahill, Keep Your Eye on the Ball.) ■ Integrals Involving the Inverse Trigonometric Functions You may have already guessed the next step. Each of our new differentiation formulas gives rise to a new integration formula. First, since 1 d sin−1 x = √ , dx 1 − x2 we also have that 1 dx = sin−1 x + c. √ 1 − x2 (8.1) Likewise, since −1 d cos−1 x = √ , dx 1 − x2 we also have that 1 dx = −cos−1 x + c. √ 2 1−x (8.2) However, since the integral in (8.2) is the same integral as in (8.1), we will ignore (8.2). smi98485_ch06b.qxd 540 5/17/01 1:36 PM Page 540 Chapter 6 Exponentials, Logarithms and Other Transcendental Functions In the same way, we obtain 1 dx = tan−1 x + c 1 + x2 and Example 8.3 1 dx. Evaluate 9 + x2 1 dx = sec−1 x + c. √ |x| x 2 − 1 An Integral Related to tan−1 x Solution Notice that the integrand is nearly the derivative of tan−1 x. However, the constant in the denominator is 9, instead of the 1 we need. With this in mind, we rewrite the integral as 1 1 1 dx = 2 dx. 2 9+x 9 1 + x3 If we let u = x3 , then du = 1 3 dx and hence, 1 1 1 dx = 2 dx 2 9+x 9 1 + x3 1 1 1 = x 2 dx 3 1+ 3 3 du 1 + u2 1 1 du 3 1 + u2 1 = tan−1 u + c 3 x 1 + c. = tan−1 3 3 = ■ We leave it as an exercise to prove the more general formula: 1 1 −1 x dx = tan + c. a2 + x 2 a a Example 8.4 ex dx. Evaluate 1 + e2x An Integral Requiring a Simple Substitution Solution Think about how you might approach this. You probably won’t recognize an antiderivative immediately. Remember that it often helps to look for terms that are derivatives of other terms. You should also recognize that e2x = (e x )2 . With this in smi98485_ch06b.qxd 5/17/01 1:36 PM Page 541 Section 6.8 The Calculus of the Inverse Trigonometric Functions 541 mind, we let u = e x, so that du = e x dx. We then have 1 ex dx = e x dx 1 + e2x 1 + (e x )2 du 1 + u2 1 du 1 + u2 = tan−1 u + c = = tan−1 (e x ) + c. ■ Example 8.5 Another Integral Requiring a Simple Substitution x dx. √ Evaluate 1 − x4 Solution Look carefully at the integrand and recognize that you don’t know an antiderivative. However, you might observe that x x dx = dx. √ 1 − x4 1 − (x 2 )2 Once you’ve recognized this one small step, you can complete the problem with a simple substitution. Letting u = x 2 , we have du = 2x dx and 1 x 2x dx = dx √ 2 1 − x4 1 − (x 2 )2 1 1 = du √ 2 1 − u2 1 −1 sin u + c 2 1 = sin−1 (x 2 ) + c. 2 = ■ You will explore integrals involving the inverse trigonometric functions further in the exercises. Whenever dealing with these functions, it is easiest if you keep the basic definitions in mind (including the domains and ranges). You will only need to know the derivative formulas for sin−1 x, tan−1 x and sec−1 x. With these basic formulas, you can quickly develop anything else that you need, using elementary techniques (such as substitution). EXERCISES 6.8 1. Explain why, as is stated in the text, antiderivatives corresponding to three of the six inverse trigonometric functions “can be ignored.’’ 2. From equations (8.1) and (8.2), explain why it follows that sin−1 x = −cos−1 x + c. From the graphs of y = sin x and y = cos x , explain why this is plausible and identify the constant c for 0 < x < π/2. In exercises 3–16, find the derivative of the function. 3. sin−1 (3x 2 ) 4. sin−1 (x 2 + 1) smi98485_ch06b.qxd 542 5/17/01 1:36 PM Page 542 Chapter 6 Exponentials, Logarithms and Other Transcendental Functions 5. cos−1 (x 3 ) 6. cos−1 (x 2 + 1) 7. tan−1 (x 2 ) √ 8. cot−1 ( x) 9. sec−1 (x 2 ) √ 10. csc−1 ( x) 11. 41. 13. cos−1 (sin x) 14. tan−1 (cos x) −1 In this exercise, we give a different derivation from the text for the derivative of sin−1 x . Use Theorem 2.3 to show that d 1 sin−1 x = . Then, evaluate cos(sin−1 x). dx cos(sin−1 x) 43. In example 8.2, it was shown that by the time the baseball reached home plate, the rate of rotation of the player’s gaze (θ ) was too fast for humans to track. Given a maximum rotational rate of θ = −3 radians per second, find d such that θ = −3. That is, find how close to the plate a player can track the ball. In a major league setting, the player must start swinging by the time the pitch is halfway (30 ) to home plate. How does this correspond to the distance at which the player loses track of the ball? 44. Suppose the pitching speed x in example 8.2 is different. Then θ will be different and the value of x for which θ = −3 will change. Find x as a function of x for x ranging from 30 ft/s (slowpitch softball) to 140 ft/s (major league fastball) and sketch the graph. 45. Suppose a painting hangs on a wall. The frame extends from 6 feet to 8 feet above the floor. A 5-foot person stands x feet from the wall and views the painting, with a viewing angle A formed by the ray from the person’s eye to the top of the frame and the ray from the person’s eye to the bottom of the frame. Find the value of x that maximizes the viewing angle A. 46. What changes in exercise 45 if the person is 6 feet tall? 47. When the astronauts blasted off from the moon, the lunar excursion module (LEM) burned its rockets briefly. Suppose the acceleration of the LEM was a(t) = 40 if 0 ≤ t ≤ 2 ft/s2 . A camera was set up on the moon to 0 if 2 < t enable viewers at home to watch the takeoff. Obviously, there was no camera operator, but the motion of the camera had to be preprogrammed. Suppose that the camera was 200 feet from the LEM. Find the camera angle between the ground and LEM as a function of the height h of the LEM. Integrate the equation h (t) = a(t) to find the height and hence the camera angle as a function of time. To rotate the camera to the correct angle, a torque would have to be applied at the pivot point of the camera. Torque is proportional to the second derivative of the angle. Find this derivative. 48. You have already seen that one of the important features of calculus is that one technique (e.g., differentiation) can be used to solve a wide variety of problems. Mathematicians take this laziness principle (don’t rework a problem that has already been solved) to an extreme. The idea of a homeomorphism helps to identify when superficially different sets 16. cot (sin x) In exercises 17–38, evaluate the given integral. 2 6 dx dx 18. 17. 2 1+x 9 + x2 19. 21. 23. 25. 27. 29. 31. 20. 2x dx 1 + x4 22. 4x dx √ 1 − x4 24. 2x dx √ 2 x x4 − 1 26. 2 dx 4 + x2 28. ex dx √ 1 − e2x 30. cos x dx 4 + sin2 x 32. 2 33. 0 35. 37. 0 2 x 1/4 √ −1 2 dx √ |x| x 2 − 1 3x 2 dx 1 + x6 2x 2 dx √ 1 − x6 3 dx √ |x| x 6 − 1 2x dx 4 + x2 cos x dx 1 − sin2 x 2x dx 9 + 9x 4 √ 34. 1 4 x2 6 dx 4 + x2 2 √ 4 dx √ 1 − x2 dx 3 dx √ 1 − 4x 2 4 36. 0 1 dx 2 + 2x 2 √ 2 x x4 − 1 2 38. 3 1 dx 2 dx √ 4 − x2 39. Derive the formula −1 d cos−1 x = √ . dx 1 − x2 40. Derive the formula 1 d sec−1 x = √ . dx |x| x 2 − 1 x + c for any a 42. −1 15. tan (sec x) 1 1 dx = tan−1 2 2 a +x a positive constant a. 12. sin x sin−1 (2x) x cos−1 (2x) Derive the formula smi98485_ch06b.qxd 5/17/01 1:36 PM Page 543 Section 6.9 are essentially the same. For example, the intervals (0, 1), (0, π) and (−π/2, π/2) are all finite open intervals. They are homeomorphic. By definition, intervals I1 and I2 are homeomorphic if there is a function f from I1 onto I2 that has an inverse with f and f −1 both being continuous. A homeomorphism from (0, 1) to (0, π) is f (x) = π x . Show that this is a homeomorphism by finding its inverse and verifying that both are continuous. Find a homeomorphism from (0, π) to (−π/2, π/2). (Hint: Sketch a picture, and decide how you 6.9 The Hyperbolic Functions 543 could move the interval (0, π) to produce the interval (−π/2, π/2).) This will take some thinking, but try to find a homeomorphism for any two finite open intervals (a, b) and (c, d). It remains to decide whether the interval (−∞, ∞) is different because it is infinite or the same because it is open. In fact, (−∞, ∞) is homeomorphic to (−π/2, π/2) and hence to all other open intervals. Show that tan−1 x is a homeomorphism from (−∞, ∞) to (−π/2, π/2). THE HYPERBOLIC FUNCTIONS The Gateway Arch in Saint Louis is one of the most distinctive and recognizable architectural structures in the United States. There are several surprising features of its shape. For instance, is the arch taller than it is wide? Most people think that it is taller, but this is the result of a common optical illusion. In fact, the arch has the same width as height. A slightly less mysterious illusion of the arch’s shape is that it is not a parabola. Its shape corresponds to the graph of the hyperbolic cosine function (called a catenary). This function and the other five hyperbolic functions are introduced in this section. You may be wondering why we need more functions. Well, these functions are not entirely new. They are simply common combinations of exponentials. We study them because of their usefulness in applications (e.g., the Gateway Arch) and their convenience in solving equations (in particular, differential equations). The hyperbolic sine function is defined by The Gateway Arch, St. Louis, MO sinh x = e x − e−x , 2 for all x ∈ (−∞, ∞). The hyperbolic cosine function is defined by cosh x = e x + e−x , 2 again for all x ∈ (−∞, ∞) . You can easily use the preceding definitions to verify the important identity cosh2 u − sinh2 u = 1, (9.1) for any value of u. (We leave this as an exercise.) In light of this identity, notice that if x = cosh u and y = sinh u, then x 2 − y 2 = cosh2 u − sinh2 u = 1, which you should recognize as the equation of a hyperbola. This identity is the source of the name “hyperbolic” for these functions. You should also notice some parallel with the trigonometric functions cos x and sin x. This will become even more apparent with what follows. smi98485_ch06b.qxd 544 5/17/01 1:36 PM Page 544 Chapter 6 Exponentials, Logarithms and Other Transcendental Functions The remaining four hyperbolic functions are defined in terms of the hyperbolic sine and hyperbolic cosine functions, in a manner analogous to their trigonometric counterparts. That is, we define the hyperbolic tangent function tanh x, the hyperbolic cotangent function coth x, the hyperbolic secant function sech x and the hyperbolic cosecant function csch x as follows: sinh x , cosh x 1 sech x = , cosh x tanh x = cosh x sinh x 1 csch x = . sinh x coth x = These functions are remarkably easy to deal with, and we can readily determine their behavior, using what we have already learned about exponentials. First, note that d d sinh x = dx dx e x − e−x 2 = e x + e−x = cosh x. 2 Similarly, we can establish the remaining derivative formulas: and d d cosh x = sinh x, tanh x = sech2 x dx dx d d coth x = −csch2 x, sech x = −sech x tanh x dx dx d csch x = −csch x coth x. dx These are all elementary applications of earlier derivative rules and are left as exercises. As it turns out, only the first three of these are of much significance. Example 9.1 Computing the Derivative of a Hyperbolic Function Compute the derivative of f (x) = sinh2 (3x). Solution From the chain rule, we have d d sinh2 (3x) = [sinh(3x)]2 dx dx d = 2 sinh(3x) [sinh(3x)] dx d = 2 sinh(3x) cosh(3x) (3x) dx = 2 sinh(3x) cosh(3x)(3) f (x) = = 6 sinh(3x) cosh(3x). ■ smi98485_ch06b.qxd 5/17/01 1:36 PM Page 545 Section 6.9 The Hyperbolic Functions 545 Example 9.2 An Integral Involving a Hyperbolic Function Evaluate x cosh(x 2 ) dx. Solution Notice that you can evaluate this integral using a substitution. If we let u = x 2 , we get du = 2x dx and so, x cosh(x 2 ) dx = 1 2 cosh(x 2 ) (2x) dx cosh u du = 1 2 = 1 sinh(x 2 ) + c. 2 cosh u du = 1 sinh u + c 2 ■ For f (x) = sinh x, note that y f (x) = sinh x = 10 5 4 x 2 2 4 lim sinh x = ∞ and x→∞ Figure 6.45 y = sinh x. > 0 if x > 0 . < 0 if x < 0 This is left as an exercise. Further, since f (x) = cosh x > 0, sinh x is increasing for all x. Next, note that f (x) = sinh x. Thus, the graph is concave down for x < 0 and concave up for x > 0. Finally, you can easily verify that 5 10 e x − e−x 2 lim sinh x = −∞. x→−∞ It is now a simple matter to produce the graph seen in Figure 6.45. Similarly, you should be able to produce the graphs of cosh x and tanh x seen in Figures 6.46a and 6.46b, respectively. We leave the graphs of the remaining three hyperbolic functions to the exercises. If a flexible cable or wire (such as a power line or telephone line) hangs between two towers, it will assume the shape of a catenary curve (derived from the Latin word catena meaning “chain”). As we will show at the end of this section, this naturallyoccurring curve corresponds to the graph of the hyperbolic cosine function f (x) = a cosh ax . y 10 y 8 1 6 4 4 x 2 2 2 4 2 x 2 Figure 6.46a y = cosh x. 4 1 Figure 6.46b y = tanh x. 4 smi98485_ch06b.qxd 546 5/17/01 1:36 PM Page 546 Chapter 6 Exponentials, Logarithms and Other Transcendental Functions Example 9.3 Finding the Amount of Sag in a Hanging Cable x For the catenary f (x) = 20 cosh 20 , for −20 ≤ x ≤ 20, find the amount of sag in the cable and the arc length. y Solution From the graph of the function in Figure 6.47, it appears that the minimum value of the function is at the midpoint x = 0, with the maximum at x = −20 and x = 20. To verify this observation, note that x f (x) = sinh 20 30 10 20 x 10 10 20 Figure 6.47 y = 20 cosh 20x . and hence, f (0) = 0, while f (x) < 0 for x < 0 and f (x) > 0, for x > 0. Thus, f decreases to a minimum at x = 0. Further, f (−20) = f (20) ≈ 30.86 is the maximum for −20 ≤ x ≤ 20 and f (0) = 20, so that the cable sags approximately 10.86 feet. From the usual formula for arc length, developed in section 5.4, the length of the cable is 20 20 x 2 L= dx. 1 + [ f (x)] dx = 1 + sinh2 20 −20 −20 Notice that from (9.1), we have 1 + sinh2 x = cosh2 x. Using this identity, the arc length integral simplifies to 20 20 x x L= dx = dx 1 + sinh2 cosh 20 20 −20 −20 20 x = 20 sinh = 20[sinh(1) − sinh(−1)] 20 −20 = 40 sinh(1) ≈ 47 feet. ■ The Inverse Hyperbolic Functions You should note from the graphs of sinh x and tanh x that these functions are one-to-one. Also, cosh x is one-to-one for x ≥ 0. Thus, we can define inverses for these functions, as follows. For any x ∈ (−∞, ∞), we define the inverse hyperbolic sine by y = sinh−1 x For any x ≥ 1, we define the inverse hyperbolic cosine by y y = cosh−1 x 2 4 if and only if cosh y = x, and y ≥ 0. Finally, for any x ∈ (−1, 1), we define the inverse hyperbolic tangent by x 2 if and only if sinh y = x. 2 2 Figure 6.48a y = sinh−1 x. 4 y = tanh−1 x if and only if tanh y = x. Inverses for the remaining three hyperbolic functions can be defined similarly and are left to the exercises. We show the graphs of y = sinh−1 x, y = cosh−1 x and y = tanh−1 x in Figures 6.48a, 6.48b and 6.48c, respectively. (As usual, you can obtain these by reflecting the graph of the original function through the line y = x.) smi98485_ch06b.qxd 5/17/01 1:36 PM Page 547 Section 6.9 y 547 The Hyperbolic Functions We can find derivatives for the inverse hyperbolic functions using implicit differentiation, just as we have for previous inverse functions. We have that 4 y = sinh−1 x 2 if and only if sinh y = x. (9.2) Differentiating both sides of this last equation with respect to x yields x 2 6 d d sinh y = x dx dx 10 Figure 6.48b y = cosh−1 x. or cosh y y dy = 1. dx Solving for the derivative, we find 4 1 dy 1 1 =√ = = , dx cosh y 1 + x2 1 + sinh2 y 2 since we know that x 1 cosh2 y − sinh2 y = 1, 1 2 from (9.1). That is, we have shown that d 1 sinh−1 x = √ . dx 1 + x2 4 Figure 6.48c y = tanh−1 x. Note the similarity with the derivative formula for sin−1 x. We can likewise establish derivative formulas for the other five inverse hyperbolic functions. We list these below for the sake of completeness. 1 d sinh−1 x = √ dx 1 + x2 d 1 cosh−1 x = √ 2 dx x −1 d 1 tanh−1 x = dx 1 − x2 d 1 coth−1 x = dx 1 − x2 d −1 sech−1 x = √ dx x 1 − x2 d −1 csch−1 x = √ dx |x| 1 + x 2 Before closing this section, we wish to point out that the inverse hyperbolic functions have a significant advantage over earlier inverse functions we have discussed. It turns out that we can solve for the inverse functions explicitly in terms of more elementary functions. Example 9.4 Finding a Formula for an Inverse Hyperbolic Function Find an explicit formula for sinh−1 x. Solution Recall from (9.2) that y = sinh−1 x if and only if sinh y = x. Using this definition, we have x = sinh y = e y − e−y . 2 (9.3) smi98485_ch06b.qxd 548 5/17/01 1:36 PM Page 548 Chapter 6 Exponentials, Logarithms and Other Transcendental Functions We can solve this equation for y, as follows. First, recall also that cosh y = e y + e−y . 2 Now, notice that adding these last two equations and using the identity (9.1), we have e y = sinh y + cosh y = sinh y + cosh2 y = sinh y + sinh2 y + 1 = x + x 2 + 1, from (9.3). Finally, taking the natural logarithm of both sides, we get y = ln(e y ) = ln x + x 2 + 1 . That is, we have found a formula for the inverse hyperbolic sine function: sinh−1 x = ln x + x 2 + 1 . ■ Similarly, we can show that for x ≥ 1, cosh−1 x = ln x + x 2 − 1 and for −1 < x < 1, tanh−1 x = 1 1+x ln . 2 1−x We leave it to the exercises to derive these formulas and corresponding formulas for the remaining inverse hyperbolic functions. There is little point in memorizing any of these formulas. You need only realize that these are always available by performing some elementary algebra. Derivation of the Catenary y y f(x) T (x, y) H T sin u u T cos u x Figure 6.49 Forces acting on a section of hanging cable. We close this section by deriving a formula for the catenary. As you follow the steps, pay special attention to the variety of calculus results that we use. In Figure 6.49, we assume that the lowest point of the catenary curve is located at the origin. We further assume that the cable has constant linear density ρ (measured in units of weight per unit length) and that the function y = f (x) is twice continuously differentiable. We focus on the portion of the cable from the origin to the general point (x, y) indicated in the figure. Since this section is not moving, the horizontal and vertical forces must be balanced. Horizontally, this section of cable is pulled to the left by the tension H at the origin and is pulled to the right by the horizontal component T cos θ of the tension T at the point (x, y). Notice that these forces are balanced if H = T cos θ. (9.4) Vertically, the section of cable is pulled up by the vertical component T sin θ of the tension. The section of cable is pulled down by the weight of the section. Notice that the weight of the section is given by the product of the density ρ (weight per unit length) and the length of the section. Recall from your x study of arc length in Chapter 5 that the arc length of this section of cable is given by 0 1 + [ f (t)]2 dt . So, the vertical forces will balance if x T sin θ = ρ 1 + [ f (t)]2 dt. (9.5) 0 smi98485_ch06b.qxd 5/17/01 1:36 PM Page 549 Section 6.9 The Hyperbolic Functions 549 We can combine equations (9.4) and (9.5) by multiplying (9.4) by tan θ to get H tan θ = T sin θ , then using (9.5) to conclude that x H tan θ = ρ 1 + [ f (t)]2 dt. 0 Notice from Figure 6.49 that tan θ = f (x), so that we have x H f (x) = ρ 1 + [ f (t)]2 dt. 0 Differentiating both sides of this equation, the Fundamental Theorem of Calculus gives us H f (x) = ρ 1 + [ f (x)]2 . (9.6) Now, divide both sides of the equation by H and name b = f (x). Equation (9.6) then becomes u (x) = b 1 + [u(x)]2 , ρ H . Further, substitute u(x) = which you should recognize as a separable differential equation. Putting together all of the u terms and integrating with respect to x gives us 1 u (x) dx = b dx. 1 + [u(x)]2 You should recognize the integral on the left-hand side as sinh−1 (u(x)), so that we now have sinh−1 (u(x)) = bx + c. Notice that since f (x) has a minimum at x = 0, we must have that u(0) = f (0) = 0. So, taking x = 0, we get c = sinh−1 (0) = 0. From sinh−1 (u(x)) = bx, we obtain u(x) = sinh(bx). Now, recall that u(x) = f (x), so that f (x) = sinh(bx). Integrating this gives us f (x) = sinh(bx) dx = 1 b cosh(bx) + c. . f (x) = Finally, recall that f (0) = 0 and so, we must have c = −1 b This leaves us with 1 −1 1 a = cosh(bx) . Finally, writing , we obtain the catenary equation b b b f (x) = a cosh ax − a. EXERCISES 6.9 1. 2. Compare the derivatives and integrals of the trigonometric functions to the derivatives and integrals of the hyperbolic functions. Also note that the trigonometric identity cos2 x + sin2 x = 1 differs only by a minus sign from the corresponding hyperbolic identity cosh2 x − sinh2 x = 1. As noted in the text, the hyperbolic functions are not really new functions. They provide names for useful combinations of exponential functions. Explain why it is advantageous to assign special names to these functions instead of leaving them as exponentials. 3. Briefly describe the graphs of sinh x, cosh x and tanh x . Which simple polynomials do the graphs of sinh x and cosh x resemble? 4. The catenary (hyperbolic cosine) is the shape assumed by a hanging cable because this distributes the weight of the smi98485_ch06b.qxd 550 5/17/01 1:36 PM Page 550 Chapter 6 Exponentials, Logarithms and Other Transcendental Functions cable most evenly throughout the cable. Knowing this, why was it smart to build the Gateway Arch in this shape? Why would you suspect that the profile of an egg has this same shape? 41. Use the first and second derivatives to explain the properties of the graph of cosh x . 42. Prove that cosh2 x − sinh2 x = 1. 43. Find an explicit formula, as in example 9.4, for cosh−1 x . 44. Find an explicit formula, as in example 9.4, for tanh−1 x . 45. Suppose that a hanging cable has the shape 10 cosh(x/10) for −20 ≤ x ≤ 20. Find the amount of sag in the cable. 46. Find the length of the cable in exercise 45. 47. Suppose that a hanging cable has the shape 15 cosh(x/15) for −25 ≤ x ≤ 25. Find the amount of sag in the cable. 48. Find the length of the cable in exercise 47. 49. Suppose that a hanging cable has the shape a cosh(x/a) for −b ≤ x ≤ b. Show that the amount of sag is given by a cosh(b/a) − a and the length of the cable is 2a sinh(b/a). 50. Show that cosh(−x) = cosh x (i.e., cosh x is an even function) and sinh(−x) = −sinh x (i.e., sinh x is an odd function). 51. Show that e x = cosh x + sinh x. In fact, we will show that this is the only way to write e x as the sum of even and odd functions. To see this, assume that e x = f (x) + g(x), where f is even and g is odd. Show that e−x = f (x) − g(x). Adding equations and dividing by two, conclude that f (x) = cosh x. Then conclude that g(x) = sinh x. 52. Show that both cosh x and sinh x are solutions of the differential equation y − y = 0. By comparison, show that both cos x and sin x are solutions of the differential equation y + y = 0. 53. Show that lim tanh x = 1 and lim tanh x = −1. In exercises 5–12, sketch the graph of each function. 5. cosh 2x 6. sinh 3x 7. tanh 4x 8. tanh x 2 9. cosh 2x sinh 2x 11. x 2 sinh 2x 10. e−x sinh x 12. x 3 sinh x In exercises 13 –24, find the derivative of each function. 13. cosh 4x 14. cosh x 2 15. sinh 2x 16. sinh 17. tanh 4x 18. tanh x 2 19. cosh−1 2x 20. sinh−1 3x 21. x 2 sinh 2x 23. tanh−1 4x 22. √ x x 3 sinh x 24. sinh−1 x 2 In exercises 25–36, evaluate each integral. cosh 6x dx sinh 2x dx 25. 26. tanh 3x dx 27. 1 29. 0 31. 33. 2 0 1 30. 0 dx √ 1 + x2 32. cos x sinh(sin x) dx 34. e2x − e−2x dx e2x + e−2x 2x dx √ 1 + x4 x cosh(x 2 ) dx 1 cosh x esinh x dx 35. 37. e4x − e−4x dx 2 sech2 x dx 28. 36. 0 1 cosh 2x dx 3 + sinh 2x d d cosh x = sinh x and tanh x = Derive the formulas dx dx 2 sech x. 38. Derive the formulas for the derivatives of coth x, sech x and csch x. 39. Using the properties of exponential functions, prove that sinh x > 0 if x > 0 and sinh x < 0 if x < 0. 40. Use the first and second derivatives to explain the properties of the graph of tanh x . x →∞ x →−∞ e −1 . e2x + 1 2x 54. Show that tanh x = 55. In this exercise, we solve the initial value problem for the vertical velocity v(t) of a falling object subject to gravity and air drag. Assume that mv (t) = −mg + kv 2 for some positive constant k. 1 k v (t) = . v 2 − mg/k m 1 1 1 1 = − (b) Use the identity 2 with v − a2 2a v − a v+a mg a= to solve the equation in (a). k √ mg c e2 kg/m t − 1 √ . (c) Show that v(t) = − k c e2 kg/m t + 1 (a) Rewrite the equation as (d) Use the initial condition v(0) = 0 to show that c = 1. smi98485_ch06b.qxd 5/17/01 1:36 PM Page 551 Chapter Review Exercises (e) Use the result of exercise 54 to conclude that mg kg tanh t . v(t) = − k m matches the arch’s measurements of 630 feet wide and 630 feet tall. What would the 127.7 in the model have to be to match the measurements exactly? Now, consider a parabolic model. To have x-intercepts x = −315 and x = 315, explain why the model must have the form y = −c(x + 315)(x − 315) for some positive constant c. Then find c to match the desired y-intercept of 630. Graph the parabola and hyperbolic cosine on the same axes for −315 ≤ x ≤ 315. How much difference is there between the graphs? Find the maximum distance between the curves. The authors have seen mathematics books where the arch is modeled by a parabola. How wrong is it to do this? (f ) Find the terminal velocity by computing lim v(t). t →∞ 56. Integrate the velocity function in exercise 55 part (e) to find the distance fallen in t seconds. 57. Two skydivers of weight 800 N drop from a height of 1000 m. The first skydiver dives head-first with a drag coefficient of k = 18 . The second skydiver is in a spread-eagle position with k = 12 . Compare the terminal velocities and the distances fallen in 2 seconds; 4 seconds. 58. A skydiver with an open parachute has terminal velocity 5 m/s. If the weight is 800 N, determine the value of k. 59. Long and Weiss derive the following equation for the horizontal velocity of the space shuttle during re-entry (see section 4.1): v(t) = 7901 tanh(−0.00124t + tanh−1 (v0 /7901)) m/s, where v0 is the velocity at time t = 0. Find the maximum acceleration experienced by the shuttle from this horizontal motion (i.e., maximize |v (t)|). 60. Graph the velocity function in exercise 59 with v0 = 2000. Estimate the time t at which v(t) = 0. 61. The Saint Louis Gateway Arch is both 630 feet wide and 630 feet tall. Its shape looks very much like a parabola, but is actually a hyperbolic cosine. You will explore the difference between the two functions in this exercise. First, consider the model y = 757.7 − 127.7 cosh(x/127.7) for y ≥ 0. Find the x- and y-intercepts and show that this model (approximately) 62. Suppose a person jumps out of an airplane from a great height. There are two primary forces acting on the skydiver: gravity and air resistance. In this situation, the air resistance would be proportional to the square of the velocity. Then an equation for the (downward) velocity would be v = g − cv 2 , where g is the gravitational constant and c is a constant determined by the orientation of the jumper’s body. Replace c with g/vT2 and explain why the initial condition v(0) = 0 is reasonable. Then show that the solution of the IVP can be written in the form v = vT tanh(gt/vT ). Show that v, the downward velocity, is an increasing function and find the limiting velocity, usually called the terminal velocity, as t → ∞. As mentioned above, the constant c depends on the position of the jumper’s body. If spread-eagle represents a c-value four times as large as a head-first dive, compare the corresponding terminal velocities. You may have seen video of skydivers jumping out of a plane at different times but catching up to each other and forming a circle. Explain how one diver could catch up to someone who jumped out of the plane earlier. Now, integrate the velocity function to obtain the distance function. Finally, answer the following two-part question. How much time and height does it take for a skydiver to reach 90% of terminal velocity? 11. tan−1 (cos 2x) CHAPTER REVIEW EXERCISES In exercises 1–16, find the derivative of the function. 1. ln(x 3 + 5) 3. ln 5. e √ x4 + x 2. ln(1 − sin x) 4. ln x2 − 1 3 x + 2x + 1 −x 2 6. etan x 3 8. 31−2x 7. 4x −1 9. sin 2x −1 10. cos 13. cosh x √ x 15. sinh−1 3x 12. sec−1 (3x 2 ) 14. sinh(e2x ) 16. tanh−1 (3x + 1) In exercises 17–40, evaluate the integral. x2 e2x dx dx 17. 18. x3 + 4 e2x + 4 2 551 19. 0 1 x dx x2 + 1 π /4 20. π /12 cos 2x dx sin 2x smi98485_ch06b.qxd 552 5/17/01 1:36 PM Page 552 Chapter 6 Exponentials, Logarithms and Other Transcendental Functions 21. 23. sin(ln x) dx x 22. e−4x dx 24. 25. 33. 35. 37. The half-life of nicotine in the human bloodstream is 2 hours. If there is initially 2 mg of nicotine present, find an equation for the amount at any time t and determine when the nicotine level reaches 0.1 mg. 2−5x dx 58. If the half-life of a radioactive material is 3 hours, what percentage of the material will be left after 9 hours? 11 hours? 6 dx √ 4 − x2 59. If you invest $2000 at 8% compounded continuously, how long will it take the investment to double? e−x dx 1 + e−2x 60. If you invest $4000 at 6% compounded continuously, how much will the investment be worth in 10 years? 61. A cup of coffee is served at 180◦ F in a room with temperature 68◦ F. After 1 minute, the temperature has dropped to 176◦ F. Find an equation for the temperature at any time and determine when the temperature will reach 120◦ F. 62. A cold drink is served at 46◦ F in a room with temperature 70◦ F. After 4 minutes, the temperature has increased to 48◦ F. Find an equation for the temperature at any time and determine when the temperature will reach 58◦ F. 0 57. 2 e−3x dx −2 31. An organism has population 100 at time t = 0 and population 140 at time t = 2. Find an equation for the population at any time and determine the population at time t = 6. 28. 0 29. 56. xe−x dx e3x dx A bacterial culture has an initial population 104 and doubles every 2 hours. Find an equation for the population at any time t and determine when the population reaches 106 . e2x cos(e2x ) dx 26. 2 27. 55. √ e x √ dx x ln x + 1 dx x 34x dx 30. 3 dx x2 + 4 32. x2 dx √ 1 − x6 34. 9x dx √ 2 x x4 − 1 36. 4 dx √ 1 + x2 38. √ 9x 3 x4 −1 dx 4 dx x2 − 1 cosh 4x dx 39. tanh 3x dx 40. In exercises 41–44, determine if the function is one-to-one. If so, find its inverse. 41. 42. e−4x x3 − 1 43. e 2x 2 44. x − 2x + 1 47. x 5 + 2x 3 − 1, a = 2 √ x 3 + 4x, a = 4 46. x 3 + 5x + 2, a = 2 48. e x 3 +2x ,a =1 In exercises 49–54, solve the IVP. 49. y = 2y, y(0) = 3 51. 2x , y(0) = 2 y = y 53. y = 50. 63. y = 2x 3 y 65. y = 3 In exercises 45–48, do both parts without solving for the inverse: (a) find the derivative of the inverse at x = a and (b) graph the inverse. 45. In exercises 63 –66, solve each separable equation, explicitly if possible. 4 (y 2 + y)(1 + x 2 ) 64. y y = √ 1 − x2 66. y = ex + y In exercises 67–70, find all equilibrium solutions and determine which are stable and which are unstable. 67. y = 3y(2 − y) 68. y = y(1 − y 2 ) 69. y = −y 1 + y 2 70. y = y + 2y 1−y y = −3y, y(0) = 2 In exercises 71–74, sketch the direction field. √ x y, y(1) = 4 52. y = −3x y , y(0) = 4 71. y = −x(4 − y) 72. y = 4x − y 2 54. y = x + y 2 x, y(0) = 1 73. y = 2x y − y 2 74. y = 4x − y 2 smi98485_ch06b.qxd 5/17/01 1:36 PM Page 553 Chapter Review Exercises In exercises 75 and 76, use Euler’s method with (a) h = 0.1 and (b) h = 0.05 to approximate y(1) and y(2). Show the first two steps by hand. 75. y = 2x − y, y(0) = 3 76. cise, you will explore the margin of error for successfully serving. First, consider a straight serve (this essentially means a serve hit infinitely hard) struck 9 feet above the ground. Call the starting point (0, 9). The back of the service box is 60 feet away, at (60, 0). The top of the net is 3 feet above the ground and 39 feet from the server, at (39, 3). Find the service angle θ (i.e., the angle as measured from the horizontal) for the triangle formed by the points (0, 9), (0, 0), and (60, 0). Of course, most serves curve down due to gravity. Ignoring air resistance, the path of the ball struck at angle θ and initial speed v ft/s is y = 16 − x 2 − (tan θ)x + 9. To hit the back of the service (v cos θ)2 line, you need y = 0 when x = 60. Substitute in these values along with v = 120. Multiply by cos2 θ and replace sin θ with √ 1 − cos2 θ . Replacing cos θ with z gives you an algebraic equation in z. Numerically estimate z. Similarly, substitute x = 39 and y = 3 and find an equation for w = cos θ . Numerically estimate w. The margin of error for the serve is given by cos−1 z < θ < cos−1 w. y = 3x y, y(0) = 1 In exercises 77–80, evaluate the quantity using the unit circle. 77. sin−1 1 78. cos−1 − 12 79. tan−1 (−1) 80. csc−1 (−2) In exercises 81–84, simplify the expression using a triangle. 81. sin(sec−1 2) 82. tan(cos−1 (4/5)) 83. sin−1 (sin(3π/4)) 84. cos−1 (sin(−π/4)) In exercises 85 and 86, find all solutions of the equation. 85. sin 2x = 1 86. cos 3x = 553 u 9 1 2 3 60 In exercises 87–92, sketch a graph. 96. 87. y = cosh 2x 88. y = tanh−1 3x 89. y = sin−1 2x 90. y = tan−1 3x 91. y = e−x 92. y = ln 2x 93. x for A hanging cable assumes the shape of y = 20 cosh 20 −25 ≤ x ≤ 25. Find the amount of sag in the cable. 94. Find the arc length of the cable in exercise 93. 95. 2 In tennis, a serve must clear the net and then land inside of a box drawn on the other side of the net. In this exer- Baseball players often say that an unusually fast pitch rises or even hops up as it reaches the plate. One explanation of this illusion involves the players’ inability to track the ball all the way to the plate. The player must compensate by predicting where the ball will be when it reaches the plate. Suppose the height of a pitch when it reaches home plate is h = −(240/v)2 + 6 feet for a pitch with velocity v ft/s (this equation takes into consideration gravity but not air resistance). Halfway to the plate, the height would be h = −(120/v)2 + 6 feet. Compare the halfway heights for pitches with v = 132 and v = 139 (about 90 and 95 mph, respectively). Would a batter be able to tell much difference between them? Now compare the heights at the plate. Why might the batter think that the faster pitch hopped up right at the plate? How many inches did the faster pitch hop? smi98485_ch06b.qxd 5/17/01 1:36 PM Page 554 smi98485_ch06b.qxd 5/17/01 1:36 PM Page 555 Section 3.1 Linear Approximations and L’Hôpital’s Rule 555