Consider the curve `y=x^n` where `n > 1` in the `1^st` quadrant. Ifthe areabounded by the curve, the x-axis and the tangent line to the graph of `y=x^n` at the point `(1, 1)` is maximum then find the value of n.

Consider the curve x^(2/3)+y^(2/3)=2 Find the slope of the tangent to the curve at the point (1,1).

Find the inclination to the positive X-axis of the tangent to the curve y= -3x- x^4 at the point where x=-1

Let A_n be the area bounded by the curve y = x^n(n>=1) and the line x=0, y = 0 and x =1/2 .If sum_(n=1)^n (2^n A_n)/n=1/3 then find the value of n.

The area bounded by the curves y=sqrt(x),2y+3=x , and x-axis in the 1st quadrant is

For 0 lt r lt 1, let n_(r) dennotes the line that is normal to the curve y = x ^® at the point (1,1) Let S _(r) denotes the region in the first quadrant bounded by the curve y = x ^(r ) ,the x-axis and the line n _(r )' Then the value of r the minimizes the area of S_(r ) is :

If the area of the triangle included between the axes and any tangent to the curve x^(n)y=a^(n) is constant,then find the value of n.

The length of the subtangent to the curve x^m y^n = a^(m + n) at any point (x_1,y_1) on it is