. Applied calculus; principles and applications . ATE FORMS. 204. Law of the Mean. — The mean value of / (x)between the values / (a) and / (b) is, by Art. 133, I f(x)dx b — a where c is some value between a and h. If the function of x is (x) = f (x) and Xi is some constantvalue between a and x, then X — a X — a or f(x)=f(a)+f{x{)(x-a), (1) the Law of the Mean, or Theorem of Mean Value. If the curve in the figure be the graph oi y = f{x); thenthe ordinate at Pi will be f{xi), the mean value of f{x)between/(a) at Pa and/(a:) at any value x, the integral beingrepresented by the area under the cur
. Applied calculus; principles and applications . ATE FORMS. 204. Law of the Mean. — The mean value of / (x)between the values / (a) and / (b) is, by Art. 133, I f(x)dx b — a where c is some value between a and h. If the function of x is (x) = f (x) and Xi is some constantvalue between a and x, then X — a X — a or f(x)=f(a)+f{x{)(x-a), (1) the Law of the Mean, or Theorem of Mean Value. If the curve in the figure be the graph oi y = f{x); thenthe ordinate at Pi will be f{xi), the mean value of f{x)between/(a) at Pa and/(a:) at any value x, the integral beingrepresented by the area under the curve from x = aiox = x. If the curve \sy = f (x), it may be seen that there must beat least one point Pi between the points (a, / (a)) and (x,f (x)) at which the slope of the tangent is equal to the slopeof the secant through those points; that is ^ ^^^^ ~ x-a ~ Axand hence (1). 401: 402 INTEGRAL CALCULUS This may be put in the form, which may be used to determine increments approximately,and is the Theorem of Finite The theorem may be extended so as to express in terms ofthe second derivative the difference made in using the firstderivative at x = a in place of its value at a; = Xi. Thus,if the function of x is 0 (x) = f (x), and x^ is some constantvalue between a and x, then, /(%) = Jj (x) dx _ f (^) ~ / (^) /by mean value, VX — a X — a \ Art. 133, / or f (x)=/(a)+r(x2)(x-a). Integrating this equation between the limits x = a andX = X, f (a) and / (X2) being constants, gives fix) = /(a) +/ (a) (X - a) +/ (x,) (£^, (2)a second Theorem of Mean Value, or the Law of the Mean. OTHER FORMS OF THE LAW OF THE MEAN 403 If the tangent at Pa meet the ordinate MP produced at R,then, MR=f (a) + f (a) {x-a); MP = f (x),and, therefore, both in sign and in magnitude, RP = MP - MR = } (X2) ^^Z_^. Here the deviation of the curve at P is below the tangent atPa, J {X2) being negative, and, measured along the line of the ordinate MP, is equal to f (X2) -—q^ ^^^^ ^^® cu
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