. Applied calculus; principles and applications . a straight line; therefore, taking y = a -- bx, a and h canbe evaluated by measurement on the figure; a = 0.688 = log 4.88, h = 2.473. Hence, log Q = log 4.88 + 2.473 log d = log (4.88 d-^^); whence Q = 4.88 d^-^^s. * (Ziwet and Hopkins.) 7. Increments. — The amount of change in the value ofa variable is called an increment. If the variable is increas-ing, its increment is positive; if it is decreasing, its incrementis negative and is really a decrement. An increment of a variable is denoted by putting the letterA before it; thus Ax, A.y and A
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. Applied calculus; principles and applications . a straight line; therefore, taking y = a -- bx, a and h canbe evaluated by measurement on the figure; a = 0.688 = log 4.88, h = 2.473. Hence, log Q = log 4.88 + 2.473 log d = log (4.88 d-^^); whence Q = 4.88 d^-^^s. * (Ziwet and Hopkins.) 7. Increments. — The amount of change in the value ofa variable is called an increment. If the variable is increas-ing, its increment is positive; if it is decreasing, its incrementis negative and is really a decrement. An increment of a variable is denoted by putting the letterA before it; thus Ax, A.y and A{x^) denote the incrementsof X, y, and x^, respectively. If y = fix), Ax and A?/ denotecorresponding increments of x and y, and ^y = ^f{x)=J{x^-^x)-f{x), ■ ^.^ Aj/_ A/(x)^/(x + Ax)-/(x)^ Ax Ax Ax X denoting any value of x. In the figure, let OPi ... S be the locus oi y = f (x)referred to the rectangular axes OX and OY. If when x = INCREMENTS 13 OMi, Ax = M1M2, then Ay = M2P2 - MiPi = DP2; ifwhen X =OMz, Ax = M3M4, then Ay = MJ^-M^P^ = -EP^.. In the last case Ay is negative and is what algebraicallyadded to M3P3 gives M4P4. When X = OMi = xi, / (x) = MiPi = / (xi); when X = OM2 = xi + Ax, /(x) = M2P2 = /(xi + Ax);hence when X = xi, A/ (x) = M2P2 - MiPi = / (xi + Ax) - / (xi). , EXERCISE I. 1. One side of a rectangle is 10 feet. Express the variable area Aas a function of the other side x. 2. Express the circumference of a circle as a function of its radiusr; of its diameter d. 3. Express the area of a circle as a function of its radius r; of itsdiameter d. 4. Express the diagonal d of a square as a function of a side x. 5. The base of a triangle is 10 feet. Express the variable area Aas a function of the altitude y. 6. If y =f{x), y-^^y =f{x + ^x); :. Ay = A/ (x) =f{x+Ax)-f (x), and hence, Ay ^Af(x) _f(x+Ax) - f (x) Ax Ax Ax AllIf y = mx + 6, find value of Ay and of --^« Ax 7. If ?/ = x2, find value of Aw and of ^ 14 DIFFERENTIAL CALCULUS ^.Ax Aw 8. If w = x^, find value of