. An elementary course of infinitesimal calculus . data. Ex. 1. To calculate the difference for one minute in a tableof log sines. If ^ = logio sin X, we have dyjdx = ju, cot x, and 8y = /* cot aSa;, approximately, provided Saj be expressed in circular Sx = circular measure of 1 = = -0002909, iOoOU we find 8y = -0001263 x cot x. Tlie numerical factor agrees with the difference for 1, in theneighbourhood of 45°, given in the tables. Ex. 2. Two sides a, 6 of a triangle and the included angle Gare measured; to find the error in the computed length of thetliird side c due to a smal
. An elementary course of infinitesimal calculus . data. Ex. 1. To calculate the difference for one minute in a tableof log sines. If ^ = logio sin X, we have dyjdx = ju, cot x, and 8y = /* cot aSa;, approximately, provided Saj be expressed in circular Sx = circular measure of 1 = = -0002909, iOoOU we find 8y = -0001263 x cot x. Tlie numerical factor agrees with the difference for 1, in theneighbourhood of 45°, given in the tables. Ex. 2. Two sides a, 6 of a triangle and the included angle Gare measured; to find the error in the computed length of thetliird side c due to a small error in the angle. Wehave c = a^ + 6^ - 2as6 cos C (2), and therefore, supposing (7 and c alone to vary, cSc = ah sin (78(7, whence 8c = —sin (78(7 = a sin 58(7 (3). 134 INFINITESIMAL CALCULUS. [CH. Ill result may also be obtained geometrically; thus, if in3 L liCB = SC, and 5iV be drawn perpendicular to AB, This r€the figurewe have, ultimately, Sc = BN=BB cos BBN= aW. sin CBA = aSO. sin B, neglecting small quantities of the second Again, to find the error in c due to a small error in themeasured length of a, we have, on the hypotliesis that a and calone vary, c8c ={a—b cos C) Sa = ccosBSa, or Sc = cosBoa (4), a result which, like the former, admits of easy geometricalproof. The above method is defective in one respect, in thatthere is no indication of the magnitude of the error involvedin the approximation. This is supplied, however, by thetheorem of Art. 56. It was there shewn that Sy = (}) (x + 6Bw) Sx (5), where 0 is some quantity between 0 and 1. Hence if A andB be the greatest and least vabies which the derived functionassumes in the interval from a; to a; + Sx, the error committedin (1) cannot be greater than \{A—B) Sx \. 58-59] APPLICATIONS OF THE DERIVED FUNCTION. 135 59. Maxima and Minima of Functions of severalVariables. We close this chapter with a few indications concerningthe extension of some of the preceding results to functionsof two or more
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