Calculus Made Easy; Being a Very Simplest Introduction to Those Beautiful Methods of Recknoning Which Are Generally Called by the Terrifying Names of

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1914 Excerpt: ... values of x; therefore it must be true for such values of x as will cause x--1 and ac+l to become zero, that is for a? = 1 and for a?=--1 respectively. If we make x = 1, we get 4 = (Ax0) + (jBx2), so that B=2; and if we make Replacing the A and B of the partial fractions by these new values, we find them to become---=-and 2-=.; and the thing is done. x--1 As a farther example, let us take the fraction. ihe denominator becomes zero when If x =--3, we get:-2x-4); whence (7=2. So then the partial fractions are: 3 __ 1_ ( 2 x+l x-l x+S' which is far easier to differentiate with respect to x than the complicated expression from which it is derived. Case II. If some of the factors of the denominator contain terms in x2, and are not conveniently put into factors, then the corresponding numerator may contain a term in x, as well as a simple number; and hence it becomes necessary to represent this unknown numerator not by the symbol A but by Ax+S; the rest of the calculation being made as before.--a?2--3 Try, for instance: J" so that A =, and the partial fractions are: x-1 2--ajs+1 35+1' Take as another example the fraction 352-2 In this case the determination of A, B, C, D is not so easy. It will be simpler to proceed as follows: Since the given fraction and the fraction found by adding the partial fractions are equal, and have identical denominators, the numerators must also be identically the same. In such a case, and for such algebraical expressions as those with which we are dealing here, the coefficients of the same powers of x 'are equal and of same sign. Hence, since This method can always be used; but the method shown first will be found the quickest in the case of factors in x only. Case III. When, among the factors of the denominator there are some ...