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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
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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
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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 .
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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
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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 ),
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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
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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.
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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
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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
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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
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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.
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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.
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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
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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.
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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
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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
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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
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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
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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
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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).
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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
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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)
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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
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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.
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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).
■
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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
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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.)
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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)
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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
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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
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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.
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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
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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
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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?
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Linear Approximations and L’Hôpital’s Rule
555