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Fig. 4-11 4.83. A variable is called an in nitesimal if it has zero as a limit. Given two in nitesimals and , we say that is an in nitesimal of higher order (or the same order) if lim = 0 (or lim = l 6 0). Prove that as x ! 0, (a) sin2 2x and 1 cos 3x are in nitesimals of the same order, (b) x3 sin3 x is an in nitesimal of higher order than fx ln 1 x 1 cos xg. Why can we not use L Hospital s rule to prove that lim x2 sin 1=x 0 (see Problem 3.91, Chap. 3) x!0 sin x
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Can we use L Hospital s rule to evaluate the limit of the sequence un n3 e n , n 1; 2; 3; . . . Explain. (1) Determine decimal approximations with at least three places of accuracy for each of the following p p irrational numbers. (a) 2; b 5; c 71=3 3 2 (2) The cubic equation x 3x x 4 0 has a root between 3 and 4. Use Newton s Method to determine it to at least three places of accuracy. Using successive applications of Newton s method obtain the positive root of (a) x3 2x2 2x 7 0, (b) 5 sin x 4x to 3 decimal places. Ans. (a) 3.268, (b) 1.131 If D denotes the operator d=dx so that Dy  dy=dx while Dk y  d k y=dxk , prove Leibnitz s formula Dn uv Dn u v n C1 Dn 1 u Dv n C2 Dn 2 u D2 v n Cr Dn r u Dr v uDn v where n Cr n are the binomial coe cients (see Problem 1.95, 1). r
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dn 2 x sin x fx2 n n 1 g sin x n=2 2nx cos x n=2 . dxn
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If f 0 x0 f 00 x0 f 2n x0 0 but f 2n 1 x0 6 0, discuss the behavior of f x in the neighborhood of x x0 . The point x0 in such case is often called a point of in ection. This is a generalization of the previously discussed case corresponding to n 1. Let f x be twice di erentiable in a; b and suppose that f 0 a f 0 b 0. Prove that there exists at least 4 one point  in a; b such that j f 00  j A f f b f a g. Give a physical interpretation involving b a 2 velocity and acceration of a particle.
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INTRODUCTION OF THE DEFINITE INTEGRAL The geometric problems that motivated the development of the integral calculus (determination of lengths, areas, and volumes) arose in the ancient civilizations of Northern Africa. Where solutions were found, they related to concrete problems such as the measurement of a quantity of grain. Greek philosophers took a more abstract approach. In fact, Eudoxus (around 400 B.C.) and Archimedes (250 B.C.) formulated ideas of integration as we know it today. Integral calculus developed independently, and without an obvious connection to di erential calculus. The calculus became a whole in the last part of the seventeenth century when Isaac Barrow, Isaac Newton, and Gottfried Wilhelm Leibniz (with help from others) discovered that the integral of a function could be found by asking what was di erentiated to obtain that function. The following introduction of integration is the usual one. It displays the concept geometrically and then de nes the integral in the nineteenth-century language of limits. This form of de nition establishes the basis for a wide variety of applications. Consider the area of the region bound by y f x , the x-axis, and the joining vertical segments (ordinates) x a and x b. (See Fig. 5-1.)
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