REMARK Not only can we compare improper integrals with one another but we can compare improper integrals with infinite series.. Probability Density Functions Problems 45 through 48 As me
Trang 1x
y = e –x2
y = e –x
1
Figure 29.13
The first summand is proper We’ll concern ourselves with the second integral and compare with the convergent integral1∞e−xdx
∞ 1
e−x2dx = lim
b→∞
b
1
e−x2dx
We will show that this limit exists and is finite
We know e−x2>0, so for b > 1,1be−x2dxincreases with b Therefore, as b → ∞,
b
1 e−xdxeither grows without bound or is finite
0 <
b
1
e−x2dx <
b
1
e−xdx because e−x2≤ e−xon [1, b]
0 ≤ lim
b→∞
b
1
e−x2dx ≤ lim
b→∞
b
1
e−xdx
lim
b→∞
b
1
e−xdx = lim
b→∞−e−x
b 1
= lim
b→∞−1
eb +1
e=1 e
so 0 ≤1∞e−x2dx ≤1e Therefore1∞e−x2dxis convergent We conclude that0∞e−x2dx converges
y
x
y = e –x2
1
Figure 29.14
REMARK ∞
−∞e−x2dx is an interesting integral We’ve concluded that it is convergent; using more advanced methods it can be shown that its value is √
π If we wanted to approximate−∞∞ e−x2dx, we could proceed as follows
i.−∞∞ e−x2dx = 20∞e−x2dx
ii Cut the tail off of0∞e−x2dxand bound it For instance,6∞e−x2dx <6∞e−xdx < 0.0025 Even better, bound∞e−x2dxby∞xe−x2dx For instance,
Trang 2∞ 5
e−x2dx <
∞ 5
xe−x2dx = 1
2e25<7 × 10−12
The interested reader has many details to fill in here The claim is that xe−x2is a much better bound, a tighter fit, than is e−x.
y
x
y = e –x2
y = xe –x2
1
Figure 29.15
iii Use numerical methods to approximate the proper integral 0ke−x2dxafter the tail,
∞
k e−x2dx, has been amputated
Notice that the graph of e−x2is bell-shaped As stands, e−x2cannot be a probability density function because the area under such a function must be exactly 1 A bit of tinkering takes care of this A standard normal distribution in statistics is described mathematically by the formula
p(x) =√1
2πe
−x22
The Method of Comparison
We’ve used the method of comparison in several instances We state the comparison theorem below We omit the formal proof, but the statements should seem quite reasonable; an informal argument was provided earlier in this section
C o m p a r i s o n T h e o r e m
Let f and g be continuous functions with 0 ≤ g(x) ≤ f (x) for x ≥ a
Ifa∞f (x) dxconverges, thena∞g(x) dxconverges
Ifa∞g(x) dxdiverges, thena∞f (x) dxdiverges
Suppose h(x) is positive and continuous
To showa∞h(x) dx converges, we must produce a larger function whose improper
integral converges
To showa∞h(x) dxdiverges, we must produce a smaller function whose improper
integral diverges
Trang 3Naturally, we can’t produce both, so we begin by taking a guess about whether or not
∞
a h(x)converges
The integral 1∞ x1p dx (and constant multiples of this integral) can be useful for comparison Recall that this integral converges for p > 1 and diverges for p ≤ 1
√
1+sin2x
x dxconvergent?
y
x
y = √1 + sin 2 x
x
Figure 29.16
SOLUTION 0 ≤1x≤
√
1+sin2x x
∞ 1 1
xdxdiverges, so1∞
√
1+sin 2 x
x dxdiverges by comparison
REMARK Not only can we compare improper integrals with one another but we can compare improper integrals with infinite series See Figure 29.17
x y
y =1x
area = 1
area =
area =
1
3
area =14
Figure 29.17
The shaded circumscribed rectangles have areas corresponding to the terms of the harmonic series The rectangles lie abovex1 We can argue that the harmonic series 1 +12+13+14+
· · · diverges becausenk=11kis larger than1n+1x1dxand1∞x1dxis a divergent improper integral
We can argue that the infinite series∞n=2 n12 =12+13+41+ · · · converges by com-paring it to the convergent improper integral1∞ 12 dx
Trang 4x y
y =1
x2
area =
area = 1
9 area =161
Figure 29.18
Similarly, after completing Exercise 29.11 we can argue that∞n=1 n1pconverges for p > 1 and diverges for p ≤ 1
We will formalize the comparison between improper integrals and infinite series by the end of the next chapter
P R O B L E M S F O R S E C T I O N 2 9 4
In Problems 1 through 5, pinpoint all the improprieties in the integral If necessary rewrite the integral as a sum of integrals so that each impropriety occurs at an endpoint and there is only one impropriety per integral.
1 (a) 0∞x21
−4dx
2 (a) 0∞x12 dx (b) −∞∞ x12 dx
3 (a) −∞∞ x21+4dx (b) −∞∞ x21−4dx
4 (a)
π 2
−π 2
tan x dx (b) 0 tan x dx
5 (a) 0∞tan−1x dx (b) −∞∞ tan−1x dx
6 Show that1∞ x1pdxconverges for p > 1 and diverges for p ≤ 1
7 Show that01x1pdxconverges for p < 1 and diverges for p ≥ 1
8 Show that−1∞ x14 dxdiverges
9 (a) Evaluate0∞xe−x2dx (b) Evaluate−∞∞ xe−x2dx
10 Show4∞e−x2dx <0.0000001 Hint: Compare it to4∞xe−x2dx
In Problems 11 through 36, determine whether the integral is convergent or divergent Evaluate all convergent integrals Be efficient If limx→∞ a∞f (x) dx is
divergent.
Trang 512.0∞x dx
13.0∞cos x + 1 dx
14.0∞cos x dx
15.0∞xe−xdx
16.−11 5x12 dx
17.−∞∞ x13dx
18.01ln x dx
19.1∞ln x dx
20.1∞ x(x+1)1 dx
21.0∞ x(x+1)1 dx
22.e∞ xln x1 dx
23.e∞2 1
x(ln x) 2 dx
24.0∞ x
2+x 2 dx
25.0∞ √1
x+1dx
26.−11 √1
x+1dx
27.15 x−31 dx
28.1∞ln x dx
29.1∞ √x
3+x 2 dx
30.1∞ arctan x
1+x 2 dx
31.12 xln x1 dx
32.1∞arctan x dx
33. tan x dx
Trang 634.0∞ xx+12+3 dx
35.1∞ 1
(x+1) 3 dx
36.1e2 dx
x √
ln x
Use the comparison theorem to determine whether the integral is convergent or diver-gent.
37.1∞ (sin x)x2 2 dx
38.2∞ x(x+1)1 dx
39.2∞ x22ln x dx
40.1∞ √1
x 7 +1dx
41.0∞sin xe−xdx
42.1∞ cos xx2 dx
43 (a) Show that1∞ 1
1+x 4 dxconverges
(b) Approximate1∞ 1
1+x 4 dxwith error < 0.01 This involves making some choices, but the gist should be as follows
i Snip off the tail,c∞ 1
1+x 4 dx, for some constant c Bound it usingc∞ x14 dx
ii Approximate1c 1
1+x 4 dxusing numerical methods
iii Be sure the sum of the bound in part (i) and the error in part (ii) is less than 0.01
44 The surface formed by revolving the graph of y =x1 on [1, ∞) about the x-axis is
known as Gabriel’s horn Find the volume of the horn Curiously, you will find that the
volume is finite even though the area under y =x1on [1, ∞) is infinite.
Probability Density Functions (Problems 45 through 48)
As mentioned at the beginning of this section, statisticians use probability density
functions to determine the probability of a random variable falling in a certain interval.
If p(x) is a probability density function, then p(x) ≥ 0 for all x and−∞∞ p(x) dx = 1.
45 A probability density function of the form
p(x) =
λe−λx for x ≥ 0,
0 for x < 0 where λ is a positive constant
describes what is known as an exponential distribution Verify that
∞
−∞p(x) dx = 1
Trang 746 A cumulative density function, C(x), gives the probability of a random variable taking
on a value less than or equal to x It is given by
C(x) =
x
−∞
p(y) dy
Show that for an exponential distribution (refer to Problem 45), the cumulative density function is given by
C(x) =
1 − e−λx for x ≥ 0,
0 for x < 0
Find limx→∞C(x)
47 The mean of a probability distribution is given by
µ =
∞
−∞
xp(x) dx, where p(x) is the probability density function Think of this as p(x) giving a fractional weight to each value of x Show that the mean of an exponential distribution (see Problem 45) isλ1 (Note: You will need to use integration by parts to find µ.)
48 Suppose the number of minutes a caller spends on hold when calling a health clinic can be modeled using the probability density function
p(x) =
10e−10x for x ≥ 0,
0 for x < 0
The probability that a random caller will wait at least 5 minutes on hold is given by
∞
5 p(x) dx Find this probability Note: it is not necessary to compute an improper integral in order to answer this question
49 Essay Question
Two of your classmates are having some trouble with improper integrals Todd believes that improper integrals ought to diverge He reasons that if f is positive, then the accumulated area keeps increasing, even if only by a little bit, so how can we get anything other than infinity?
Dylan, on the other hand, is convinced that if limx→∞f (x) = 0, then0∞f (x) dx ought to converge After all, he reasons, the rate at which area is accumulating is going
to zero Why isn’t that enough to assure convergence?
Write an essay responding to Todd and Dylan’s misconceptions Your essay should
be designed to help your classmates see the errors in their reasoning
Trang 9X Series
30
Series
30.1 APPROXIMATING A FUNCTION BY A POLYNOMIAL
Preview
Addition and multiplication—these are our fundamental computational tools A high-powered computer, for all its computational sophistication, ultimately relies on these basic operations How then can a computer numerically approximate values of transcendental functions? How are values of exponential, logarithmic, and trigonometric functions com-puted?
Consider the sine function, for example A calculator can approximate sin 0.1 with a high degree of accuracy, accuracy not readily accessible from unit circle or right triangle definitions of sin x How can such a good approximation be obtained?
If we know the value of a differentiable function f at the point x = b, then we can use the tangent line to f at x = b to approximate the function’s values near x = b The tangent line is the best linear approximation of f near x = b; higher degree polynomials offer the possibility of staying even closer to the values of f near x = b and following the shape
of f over a larger interval around b In this section we will improve upon the tangent line approximation, obtaining quadratic, cubic, and higher degree polynomial approximations of
faround x = b We will generally find the “fit” improving with the degree of the polynomial
919
Trang 10Such polynomial approximations are convenient because they involve only the operations
of addition and multiplication; they are easily evaluated, easily differentiated, and easily integrated
The process of approximating an elusive quantity, successively refining the approxi-mation, and using a limiting process to nail it down is at the heart of theoretical calculus In this chapter we obtain successively better polynomial approximations of a function about a point by computing increasingly higher degree polynomial approximations By computing the limit as the degree of the polynomial increases without bound, we will discover that, under certain conditions, we can represent a function as an infinite “polynomial” known
as a power series The fact that sin x, cos x, and exhave representations as power series is remarkable in its own right In addition, this alternative representation turns out to be com-putationally very useful Power series representation of functions was known to Newton who used it as a computational aid, particularly for integrating functions lacking elemen-tary antiderivatives It was the subject of work published by the English mathematician Brook Taylor in 1712 and was popularized by the Scottish mathematician Colin Maclaurin
in a textbook published in 1742 Although mathematicians had been using the ideas as early
as the 1660s, the names of Taylor and Maclaurin have been associated with power series representations of functions
Polynomial Approximations of sin x around x = 0
In this section we will use polynomials to numerically estimate values of some transcen-dental functions
EXAMPLE 30.1 A calculator or computer gives sin 0.1 to ten decimal places, displaying 0.0998334166
Obtain this result by using a polynomial to approximate sin x near x = 0 and evaluating this polynomial at x = 0.1
SOLUTION We will approach this problem via a sequence of polynomial approximations to f (x) = sin x
for x near zero until we arrive at the desired result We denote by Pk(x)the kth degree polynomial approximation Pk(x)is of the form a0+ a1x + a2x2+ · · · + akxk, where
a0, a1, , akare constants We must determine the values of these constants so the Pk(x)
“fits” the graph of f well around x = 0
Constant Approximation
Because sin x is continuous and 0.1 is near 0, we know sin 0.1 ≈ sin 0 = 0
P0(x) = 0; sin 0.1 ≈ P0(0.1) = 0
Tangent Line Approximation
The tangent line passes through (0, 0) and has a slope of f(0) = cos 0 = 1
P1(x) = x; sin 0.1 ≈ P1(0.1) = 0.1