. Differential and integral calculus, an introductory course for colleges and engineering schools. (5) As = V(pA0)2 + (Ap)2, pA0 = sin a, -r1-As As A<? VAp <W P2 + COSa. Taking limits and noting that lim a = , we have (6) Des = VP2+(Dep)2, pDs6 = sin , D8p = cos , [q which are formula? V (a) and (b) of Art. Equations (5) are of course not exact, but they lead to results (6)which are absolutely true. Here, too, we may write differen-tials in place of increments, andthen, by mere inspection of the. 148 DIFFERENTIAL CALCULUS §104 figure, can write down at once the formulae ds = Vdp2
. Differential and integral calculus, an introductory course for colleges and engineering schools. (5) As = V(pA0)2 + (Ap)2, pA0 = sin a, -r1-As As A<? VAp <W P2 + COSa. Taking limits and noting that lim a = , we have (6) Des = VP2+(Dep)2, pDs6 = sin , D8p = cos , [q which are formula? V (a) and (b) of Art. Equations (5) are of course not exact, but they lead to results (6)which are absolutely true. Here, too, we may write differen-tials in place of increments, andthen, by mere inspection of the. 148 DIFFERENTIAL CALCULUS §104 figure, can write down at once the formulae ds = Vdp2 + (pdd)2, -j- = coscf), -p = sin, ^- = tan0. These formulae do not need to be retained in memory, since theyare so readily derived from the figure. Examples. 1. To determine Dxs for the parabola y1 = 2mx. Differentiating the equation, yy = m, y = —; then 1 + V ~ y2 ~ 2x - DxS~ 7—~\~ 2x 2 Til 2. To determine Des for the parabola p = Differentiating, p = 1 — cos 02 m sin0 , , ,„ 8 m2 (1 — cos 0)2 (1 — cos 0)3 m 104. Exercises. 1. Find Dxs for the circle. 2. Find Z^s for the astroid, x* + 2/3 = a$. 3. Find Z)0s for the astroid, x = a cos3 6, y — a sin3 0. 4. Find D^s and Des for the cycloid, x = a(0 — sin 0), ?/ = a{\ — cos 0). 5. Find Des for the lemniscate, p2 = a2 cos 2 0. 6. Find D0s for the cardioid, p = 2 a(l — cos 0). n / - -\ 7. Find Dzs for the catenary, y = ^[ea -\- e °J. 8. Find D#s /or ^e epicycloid, and for the the equation at the foot of page 1
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
