A treatise on generating functions by Hari Srivastava, H. L. Manocha

By Hari Srivastava, H. L. Manocha

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B) If the point ͑5, 3͒ is on the graph of an odd function, what other point must also be on the graph? 72. A function f has domain ͓Ϫ5, 5͔ and a portion of its graph is 20 x 71. (a) If the point ͑5, 3͒ is on the graph of an even function, what x x x 12 x shown. (a) Complete the graph of f if it is known that f is even. (b) Complete the graph of f if it is known that f is odd. x x y x 64. A cell phone plan has a basic charge of $35 a month. The plan includes 400 free minutes and charges 10 cents for each additional minute of usage.

D1 d2 d3 dn ϩ 2 ϩ 3 ϩ иии ϩ n ϩ иии 10 10 10 10 Therefore some infinite sums, or infinite series as they are called, have a meaning. But we must define carefully what the sum of an infinite series is. Returning to the series in Equation 3, we denote by sn the sum of the first n terms of the series. 99998474 4 2 Observe that as we add more and more terms, the partial sums become closer and closer to 1. In fact, it can be shown that by taking n large enough (that is, by adding sufficiently many terms of the series), we can make the partial sum sn as close as we please to the number 1.

The velocity v when t ෇ 2 is the limiting value of this average velocity as t approaches 2; that is, 20 10 0 change in position f ͑t͒ Ϫ f ͑2͒ ෇ time elapsed tϪ2 P { 2, f(2)} 1 FIGURE 8 2 3 4 v ෇ lim 5 t tl2 f ͑t͒ Ϫ f ͑2͒ tϪ2 and we recognize from Equation 2 that this is the same as the slope of the tangent line to the curve at P. A PREVIEW OF CALCULUS 5 Thus, when we solve the tangent problem in differential calculus, we are also solving problems concerning velocities. The same techniques also enable us to solve problems involving rates of change in all of the natural and social sciences.

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