. Applied calculus; principles and applications . ct area of figures bounded by curves,mathematics up to that time having furnished no methodapplicable to all curves whose equations were known. It is true too that historically the method of Integrationwas discovered before the method of Differentiation wasdeveloped. The Differential Calculus arose through theproblem of determining the direction of the tangent at anypoint of a curve. (See Note, Art. 75.) (d) Let OPn be the locus oi y - f {x) and s the lengthfrom 0 along the curve. Suppose the point (x, y) to movealong the curve to P and thencea
. Applied calculus; principles and applications . ct area of figures bounded by curves,mathematics up to that time having furnished no methodapplicable to all curves whose equations were known. It is true too that historically the method of Integrationwas discovered before the method of Differentiation wasdeveloped. The Differential Calculus arose through theproblem of determining the direction of the tangent at anypoint of a curve. (See Note, Art. 75.) (d) Let OPn be the locus oi y - f {x) and s the lengthfrom 0 along the curve. Suppose the point (x, y) to movealong the curve to P and thencealong the tangent at that at the value x = OM, thechange of x and y would becomeuniform with respect to eachother, as the point (x, y) wouldbe moving along a straight change of s would becomeuniform also with respect to bothX and y. As x is the independentvariable it may be taken to varyuniformly, making PD or dx =Ax or MM I, the actual change in x as the point moves alongthe curve from P to Pi. Then dy is DT, the correspondmg. 18 DIFFERENTIAL CALCULUS uniform change of y, and ds is PT, the corresponding uniformchange of s. It is evident that while dx = Ax, dy is notequal to Ai/ and ds is not equal to As. When, and only when,the locus is a straight line will dy = Ay and ds = As, after dxhas been taken equal to Ax. It should be noted that it is not essential that dx should bemade equal to Aa:, for dx may be taken as any value otherthan zero, and then dy will be the perpendicular distancefrom the end of dx to the tangent and ds will be the distancefrom the point (x, y) along the tangent to end of dy. Fromfigure, {dsY = {dxY + {dyY. 11. Rate, Slope, and Velocity. — The differential triangle PDT in figure for {d) Art. 10, gives -^^ = tan 0 = slope of the dvcurve y= f(x) at point {x, y), and -^ is the ratio of the change of y to the change of x at the point {x, y), or for any diicorresponding values of x and y, and -r- is called the rate of y with respect to x.
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