. Differential and integral calculus, an introductory course for colleges and engineering schools. 000 f 1 -\ J Now by giving suitable values to n we find what the amount in one year will be if the interest is com-pounded weekly, daily, hourly, etc., and by conceiving n to becomeinfinite we reach the idea of interest being compounded continu-ously. In that case the amount in one year is given by theformula A = lim$100o(l + — VNow by formula (C) of Art. 68, lim(l+**)- Hence* A = $1000 e008 = $ i y the same method of reasoning we arrive at the general formulaor the amount in one year of
. Differential and integral calculus, an introductory course for colleges and engineering schools. 000 f 1 -\ J Now by giving suitable values to n we find what the amount in one year will be if the interest is com-pounded weekly, daily, hourly, etc., and by conceiving n to becomeinfinite we reach the idea of interest being compounded continu-ously. In that case the amount in one year is given by theformula A = lim$100o(l + — VNow by formula (C) of Art. 68, lim(l+**)- Hence* A = $1000 e008 = $ i y the same method of reasoning we arrive at the general formulaor the amount in one year of a principal P compounded contin-uously at r per cent per annum, A = limP(l+~)n=Pe™.„=oo \ 100 n) The amount in t years is given by the formula At = Pem. This equation is what is termed the compound-interest law. At kis function and t is argument and the equation is of the formy = Ceax,where C and a are constants. A characteristic property of such a * Logio e = 96 DIFFERENTIAL CALCULUS §72 function is that its derivative is proportional to the function itself. For Dxy = Cae° There are many phenomena which proceed according to the com-pound-interest law. Following is an example. Let R be a reservoir fitted with two pipes Px and P2, throughwhich liquid can be admitted to and drawn from R. Let Pi andP2 be of such relative sizes that the amount of liquid that flowsin through Pi in a time t is exactly equal to the amount that flows out through P2, so that when both pipesare open the level of the liquid in Rremains unchanged. Suppose R at thestart to be filled with water, and sup-pose alcohol to be admitted through Piwhile P2 is left open. Suppose, further,that a stirrmg mechanism keeps thewater and alcohol thoroughly mixed,so that the liquid that flows out through P2 is completely homo-geneous. The level of the liquid in R will remain unchanged,*and the problem is, given the rate of flow through the pipes todetermine the proportions of water and alcohol in R afte
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