calculus
posted by Anonymous .
Find the xcoordinate of the point P on the parabola y=1x^2 (0<x=<1) where the triangle enclosed by the tangent line at p and the coordinate axes has the smallest area.

let the point of contact be P(a,1a^2)
dy/dx = 2x, so at our point P the slope of the tangent is 2a
equation of tangent:
y  (1a^2) = 2a(xa)
y  1 + a^2 = 2ax +2a^2
2ax + y = a^2 + 1
the base of the triangle is the xintercept of this line,
the height of the triangle is the yintercept of this line.
xintercept: x = (a^2 + 1)/(2a)
yintercept: y = a^2 + 1
Area of triangle
= (1/2)(a^2+1)^2/(2a)
= (a^4 + 2a^2 + 1)/a
= a^3 + 2a + 1/a
d(Area)/da = 3a^2 + 2  1/a^2
= 0 for a max/min of Area
the only real solution for the above is
a = ± 1/√3
so P is (1/√3 , 2/3)
Respond to this Question
Similar Questions

Calc.
Find the area of the region bounded by the parabola y=x^2, the tangent line to this parabola at (1,1) and the xaxis. I don't really get what this question is asking. It looks like the area of right triangle to me...try the graph, … 
math
A tangent line is drawn to the hyerbola xy=c at a point P. 1) show that the midpoint of the line segment cut from the tangent line by the coordinate axes is P. 2) show that the triangle formed by the tangent line and the coordinate … 
math repost!!
please help, i procrastinated and now this is due tomorrow!! A tangent line is drawn to the hyerbola xy=c at a point P. 1) show that the midpoint of the line segment cut from the tangent line by the coordinate axes is P. 2) show that … 
calculus
Find the area of the region in the first quadrant enclosed by the coordinate axes and graph of x^3+y^3=1. 
Calculus
find the xcoordinate of the point P on the parabola y=1x^2 for 0<x<1 where the triangle that is enclosed by the tangent line at P and the coordinate axes has the smallest area. 
Calculus 151
At what point in the first quadrant on the parabola y=4x^2 does the tangent line, together with the coordinate axis, determine a triangle of minimum area? 
calculus
If a tangent line is drawn to the parabola y = 3  x^2 at any point on the curve in the first quadrant, a triangle is formed with the axes. At what point on the curve should the tangent be drawn to form a triangle of least are? 
Calculus
If a tangent line is drawn to the parabola y = 3  x^2 at any point on the curve in the first quadrant, a triangle is formed with the axes. At what point on the curve should the tangent be drawn to form a triangle of least area? 
Calculus
4. Find the area of the largest rectangle (with sides parallel to the coordinate axes) that can be inscribed in the region enclosed by the graphs of f(x) = 18 – x^2 and g(x) = 2x^2 – 9. 
calculus
Find the point on the curve y =2/3 〖√(18x^2 )〗 (first quadrant) where a tangent line may be drawn so that the area of the triangle formed by the tangent line and the coordinate axes is a minimum.