Machinery's Handbook, 31st Edition
132 Calculus Integral Rules.—Integration rules are somewhat similar to differentiation rules, since they undo what the derivative does. The analogous processes for the chain, product, and quotient rules of derivatives are covered by either u -substitution or integration by parts for integrals. Table of Derivatives and Integrals contains integration formulas. Newton’s Method for Solving Equations.—Algebraic (polynomial, rational, and root) and transcendental (trigonometric, exponential and logarithmic) equations often can be quickly solved directly, using the processes described in ALGEBRA and TRIGONOM- ETRY (for example, cos x = 1 or log x = 4). But, long before there were calculators that could solve less straightforward equations to a high degree of accuracy, approxima- tion methods were developed. One such method is Newton’s (or the Newton-Raphson ) method, which produces excellent approximations of the solution of more difficult equations, with the help of differentiation. Some equations that can be solved by New- ton’s method are: x 2 = 101 x 3 – 2 x 2 = 5 cos x = x Rewriting any of these equations as a function f ( x ) = 0 converts the problem into one of finding the roots of f ( x )—that is, those values where the function crosses the x -axis. For example, x 2 = 101 is rewritten as f ( x ) = x 2 – 101 = 0. This function has two real roots (see ALGEBRA ). A good first estimate of the positive root is 10. Rewriting the other two equations, x 3 – 2 x 2 = 5 and cos x = x in the f ( x ) = 0 form does not give as obvious an esti- mate of the root(s), but by inspection (trial and error) or a rough graph, an estimate can be made. In each case, this first estimate is called r 1 . From these, successive estimates r 2 , r 3 , . . . are made, each progressively closer to the exact value of the root. After estimating r 1 , the first derivative of the function, f ′ ( x ), is found. This is the equation for the function’s instantaneous rate of change at any value of x . f ′ ( x ) is the equation that gives the slope of the line tangent to the function’s curve at a given x . In the above examples, f ′ ( x ) is, respectively, 2 x , 3 x 2 - 4 x , and –sin x + 1. These were found by the methods described in Table of Derivatives and Integrals on page 133. Starting with the first estimate, the steps of Newton’s method are as follows: r 1 is the first estimate of the value of the root of f ( x ) = 0. Find f ( r 1 ), the value of f ( x ) at x = r 1. Find f ′ ( x ), the first derivative of f ( x ). Find f ′ ( r 1 ), the value of f ′ ( x ) at x = r 1 . Get the second approximation of the root of f ( x ) = 0, r 2 , by calculating r 2 r 1 f r 1 ( ) f ′ r 1 ⁄ ( ) = – and, further approximations, r n r n – 1 f r n – 1 ( ) f ′ r n – 1 ( ) ⁄ – = Example: Find the square root of 101 using the Newton-Raphson method. Solution: The problem is restated as the algebraic equation x 2 = 101, rewritten as x 2 – 101 = 0, and solved for the positive root. r 1 = 10 is a good first estimate. Then, apply the steps of the algorithm: f r 1 ( ) f 10 ( ) 10 2 – 101 –1 = = =
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