Posted by Anonymous on Tuesday, November 24, 2009 at 5:55pm.
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.

calculus  Reiny, Tuesday, November 24, 2009 at 8:23pm
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)
Answer This Question
Related Questions
 Calculus  find the xcoordinate of the point P on the parabola y=1x^2 for 0<...
 math  A tangent line is drawn to the hyerbola xy=c at a point P. 1) show that ...
 math repost!!  please help, i procrastinated and now this is due tomorrow!! A ...
 calculus  Find the point on the curve y =2/3 〖√(18x^2 )〗 (...
 Calculus 151  At what point in the first quadrant on the parabola y=4x^2 does ...
 Calculus  If a tangent line is drawn to the parabola y = 3  x^2 at any point ...
 calculus  Find the area of the region in the first quadrant enclosed by the ...
 Calc.  Find the area of the region bounded by the parabola y=x^2, the tangent ...
 Calculus  4. Find the area of the largest rectangle (with sides parallel to the...
 calculus  If a tangent line is drawn to the parabola y = 3  x^2 at any point ...
More Related Questions