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DERIVATIVES
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(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 (one-dimensional) 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 one-dimensional 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 single-valued 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. 4-7.)
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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 x-axis intercept of the tangent line at P0 then 0 f x0 f 0 x0 x x0
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DERIVATIVES
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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
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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
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n X f xk f 0 xk k 0
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f x1 f x f x x0 0 0 0 1 f 0 x1 f x0 f x1
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.