. An elementary treatise on the differential and integral calculus. Fig. 31. EXAMPLES. 20? dy _ 4ax dx ~ Sx2 3f cVydx* -8a2 9x* (2a — x)* When x = 0 or 2a, y = 0; .*. the curve cuts the axis ofx at the origin and at x = find the equation of the asymptote, we have y ?(-#—(-£- )• therefore, ?/ = — a? -f fa is the equation of the asymptote,and as the next term of the expression is positive, the curvelies above the asymptote. Evaluating the first derivative for x = 0, y = 0, we have dy 4:ax — dx2 4a — 6x dx ~~ Sy2 ~ dy 6y dx A i£) =£ = cc> wlien *=*=*; fy= /2a _dx \ 3y~ ± °° 9 when y = 0
. An elementary treatise on the differential and integral calculus. Fig. 31. EXAMPLES. 20? dy _ 4ax dx ~ Sx2 3f cVydx* -8a2 9x* (2a — x)* When x = 0 or 2a, y = 0; .*. the curve cuts the axis ofx at the origin and at x = find the equation of the asymptote, we have y ?(-#—(-£- )• therefore, ?/ = — a? -f fa is the equation of the asymptote,and as the next term of the expression is positive, the curvelies above the asymptote. Evaluating the first derivative for x = 0, y = 0, we have dy 4:ax — dx2 4a — 6x dx ~~ Sy2 ~ dy 6y dx A i£) =£ = cc> wlien *=*=*; fy= /2a _dx \ 3y~ ± °° 9 when y = 0. Hence, at the origin there aretwo branches of the curve tangentto the axis of y; and the value of ~ shows that if y be negative as it approaches 0, -j- will be imaginary; and hence the origin is a cusp ofthe first species. When x = i-a, ~- = 0 ; /. there is a maximum ordinate, 3 dx at x = %a. - — en the curve cuts.
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Keywords: ., bookcentury1800, bookdecade1890, bookpublishernewyo, bookyear1892
