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DERIVATIVES in .NET
DERIVATIVES QR Code Decoder In Visual Studio .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in VS .NET applications. Creating QR Code In .NET Framework Using Barcode creator for .NET Control to generate, create Quick Response Code image in Visual Studio .NET applications. [CHAP. 4
QR Code 2d Barcode Scanner In .NET Using Barcode reader for .NET Control to read, scan read, scan image in .NET framework applications. Bar Code Generation In .NET Using Barcode printer for .NET Control to generate, create barcode image in .NET framework applications. (Notice that the order of di erentiation in each succeeding case is two greater. The nature of the intermediate possibilities is suggested in the next paragraph.) It is possible that the slope of the tangent line to the graph of f is positive to the left of P1 , zero at the point, and again positive to the right. Then P1 is called a point of in ection. In the simplest case this point of in ection is characterized by f 0 x1 0, f 00 x1 0, and f 000 x1 6 0. 2. Particle motion The fundamental theories of modern physics are relativity, electromagnetism, and quantum mechanics. Yet Newtonian physics must be studied because it is basic to many of the concepts in these other theories, and because it is most easily applied to many of the circumstances found in everyday life. The simplest aspect of Newtonian mechanics is called kinematics, or the geometry of motion. In this model of reality, objects are idealized as points and their paths are represented by curves. In the simplest (onedimensional) case, the curve is a straight line, and it is the speeding up and slowing down of the object that is of importance. The calculus applies to the study in the following way. If x represents the distance of a particle from the origin and t signi es time, then x f t designates the position of a particle at time t. Instantaneous velocity (or speed in the onedimensional case) is dx f t t change in distance represented by lim (the limiting case of the formula for speed when dt t!0 t change in time the motion is constant). Furthermore, the instantaneous change in velocity is called acceleration and d 2x represented by 2 . dt Path, velocity, and acceleration of a particle will be represented in three dimensions in 7 on vectors. 3. Newton s method It is di cult or impossible to solve algebraic equations of higher degree than two. In fact, it has been proved that there are no general formulas representing the roots of algebraic equations of degree ve and higher in terms of radicals. However, the graph y f x of an algebraic equation f x 0 crosses the xaxis at each singlevalued real root. Thus, by trial and error, consecutive integers can be found between which a root lies. Newton s method is a systematic way of using tangents to obtain a better approximation of a speci c real root. The procedure is as follows. (See Fig. 47.) Recognize Barcode In VS .NET Using Barcode scanner for .NET Control to read, scan read, scan image in VS .NET applications. QR Generation In C#.NET Using Barcode printer for VS .NET Control to generate, create QR Code JIS X 0510 image in .NET applications. Fig. 47 QR Code JIS X 0510 Generator In .NET Framework Using Barcode creation for ASP.NET Control to generate, create Denso QR Bar Code image in ASP.NET applications. Create QR Code ISO/IEC18004 In VB.NET Using Barcode drawer for .NET Control to generate, create QR Code 2d barcode image in .NET framework applications. Suppose that f has as many derivatives as required. Let r be a real root of f x 0, i.e., f r 0. Let x0 be a value of x near r. For example, the integer preceding or following r. Let f 0 x0 be the slope of the graph of y f x at P0 x0 ; f x0 . Let Q1 x1 ; 0 be the xaxis intercept of the tangent line at P0 then 0 f x0 f 0 x0 x x0 Making UCC128 In Visual Studio .NET Using Barcode creation for .NET Control to generate, create GS1128 image in Visual Studio .NET applications. Make Matrix 2D Barcode In .NET Framework Using Barcode maker for .NET Control to generate, create Matrix 2D Barcode image in VS .NET applications. CHAP. 4] Creating Bar Code In .NET Framework Using Barcode creation for VS .NET Control to generate, create barcode image in Visual Studio .NET applications. Intelligent Mail Maker In Visual Studio .NET Using Barcode creation for Visual Studio .NET Control to generate, create USPS Intelligent Mail image in .NET applications. DERIVATIVES
Reading Bar Code In Visual C#.NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in .NET applications. Generate Barcode In ObjectiveC Using Barcode generation for iPhone Control to generate, create barcode image in iPhone applications. where the two representations of the slope of the tangent line have been equated. The solution of this relation for x1 is x1 x0 f x0 f 0 x0 Code 128 Code Set B Encoder In None Using Barcode maker for Microsoft Word Control to generate, create Code 128 image in Office Word applications. Drawing ECC200 In ObjectiveC Using Barcode maker for iPhone Control to generate, create DataMatrix image in iPhone applications. Starting with the tangent line to the graph at P1 x1 ; f x1 and repeating the process, we get x2 x1 and in general xn x0 Decode Code 39 Extended In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. Bar Code Maker In None Using Barcode generator for Word Control to generate, create bar code image in Word applications. n X f xk f 0 xk k 0
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Under appropriate circumstances, the approximation xn to the root r can be made as good as desired. Note: Success with Newton s method depends on the shape of the function s graph in the neighborhood of the root. There are various cases which have not been explored here.

