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chapter_10 - 10(sin x/x I call our world Flatland not...

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10 (sin x )/ x I call our world Flatland, not because we call it so, but to make its nature clear to you, my happy readers, who are privileged to live in Space. —Edwin A. Abbott, Flatland (1884) Students of calculus encounter the function sin x /x early in their study, when it is shown that lim x 0 sin x /x = 1; this result is then used to establish the differentiation formulas sin x 0 = cos x and cos x 0 = - sin x . Once this has been done, however, the function is soon forgotten, and the student rarely sees it again. This is unfortunate, for this simple-looking function not only has some remarkable properties, but it also shows up in many applications, sometimes quite unexpectedly. We note, to begin with, that the function is defined for all values of x except 0; but we also know that as x gets smaller and smaller, the ratio sin x /x —provided x is measured in radians— tends to 1. This provides us with a simple example of a removable singularity : we can simply define the value of sin 0 / 0 to be 1, and this definition will assure the continuity of the function near x = 0. Let us denote our function by f x and plot it for various val- ues of x ; the resulting graph is shown in figure 61. Two features make this graph distinct from that of the function g x = sin x : first, it is symmetric about the y -axis; that is, f - x = f x for all values of x (in the language of algebra, f x is an even function , so called because the simplest functions with this property are of the form y = x n for even values of n ). By contrast, the function g x = sin x has the property that g - x = - g x for all x (func- tions with this property are called odd functions , for example y = x n for odd values of n ). To prove that f x = sin x /x is even, we simply note that f - x = sin - x / - x = - sin x / - x = sin x /x = f x . Second, unlike the graph of sin x , whose up-and-down oscil- lations are confined to the range from - 1 to 1 (that is, the sine
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130 CHAPTER TEN x O 1 y 3 π 2 π –2 π –3 π π - π Fig. 61. The graph of sin x /x . wave has a constant amplitude 1), the graph of sin x /x rep- resents damped oscillations whose amplitude steadily decreases as x increases. Indeed, we may think of f x as a sine wave squeezed between the two envelopes y = ± 1 /x . We now wish to locate the extreme points of f x —the points where it assumes its maximum or minimum values. And here a surprise is await- ing us. We know that the extreme points of g x = sin x oc- cur at all odd multiples of π/ 2, that is, at x = 2 n + 1 π/ 2. So we might expect the same to be true for the extreme points of f x = sin x /x . This, however, is not the case. To find the ex- treme point, we differentiate f x using the quotient rule and equate the result to zero: f 0 x = x cos x - sin x x 2 = 0 : (1) Now if a ratio is equal to zero, then the numerator itself must equal to zero, so we have x cos x - sin x = 0, from which we get tan x = x: (2) Equation (2) cannot, unfortunately, be solved by a closed for- mula in the same manner as, say, a quadratic equation can; it is a transcendental equation whose roots can be found graphi- cally as the points of intersection of the graphs of y = x and y = tan x (fig. 62). We see that there is an infinite number of
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