. Applied calculus; principles and applications . epresented graphically by a curve, ■dsthen the slope of the curve is m = -r. = S2 t = v, the veloc-ity; and the flexion of the curve is 6 = -rr = -7: = 32 = a«, the acceleration. Example 3. — Let the function to be differentiated be: y = mx + h. (1) y -\- Ay = m {x -\- Ax) + 6 = mx -\-m* Ax -{-h, (2) Ay = m- Ax, (3) !=£-. (^) .*. dy = mdx. (6) Here again the ratio of the increments, being shown by (4) to be a constant m, does not approach a hmit; hence, as dv Allshown by (5) the derivative j~ = T~ = ^j ^tie constant slope of the line y = mx + h
. Applied calculus; principles and applications . epresented graphically by a curve, ■dsthen the slope of the curve is m = -r. = S2 t = v, the veloc-ity; and the flexion of the curve is 6 = -rr = -7: = 32 = a«, the acceleration. Example 3. — Let the function to be differentiated be: y = mx + h. (1) y -\- Ay = m {x -\- Ax) + 6 = mx -\-m* Ax -{-h, (2) Ay = m- Ax, (3) !=£-. (^) .*. dy = mdx. (6) Here again the ratio of the increments, being shown by (4) to be a constant m, does not approach a hmit; hence, as dv Allshown by (5) the derivative j~ = T~ = ^j ^tie constant slope of the line y = mx + h. That the ordinate is changing m times as fast as theabscissa is shown by (6). It is evident that for a linear function not only is the ratioof the increments the derivative, but the increments are thedifferentials as defined. 34 DIFFERENTIAL CALCULUS Example 4. — Let the function to be differentiated be 1 , y = - or xy = 1. X (1) (x + Ax) (y-{-Ay) = xy-{-x\y-\-yAx + Ax Ay = 1, . (2)xAy + yAx-\-AxAy=0, or (x + Ax) Ay = —y Axj (3) ^y y. , average slope over Ax, (4) y = tan (j), Ax X + Ax Ax=o V x-\- AxJslope at any point (x, y); (5) /. dy=-ldx, (6) showing that the . oi y is - times the increase increase of X, at any point {x, y). decrease Example 5. — In compressing air, if thetemperature of the air is kept con-stant, the pressure and the volumeare connected by the relation pV =constant. To find the rate of changeof the pressure with respect to the dijvolume, that is, the derivative 7^- dV Let pV = K, (1) ovt (p + Ap) (V + AV) = pV+pAV + VAp + ApAV = K, (2) pAV-{-VAp + ApAV = 0, or (V + AV)Ap = -pAF, (3) Ap _ p AV~~V+AVaverage rate of change from F to F + A T, ^ ^ =]!??„[If] =ii?o(- F+af) = -f rate of change for any corresponding values of p and Vi (4) (5) ILLUSTRATIVE EXAMPLES 35 .*. dv = — —dV, showing that the . of pressure is ^ V increase p ,. ,, increase r- i ^ ^^ ^ ^ times the , oi volume at any corresponding values of pressure and volume. Example 6.
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