Calculus Help

Discus: SAT/ACT Tests and Test Preparation: October 2003 Archive: Calculus Help
 By Onizuka (Onizuka) on Thursday, October 23, 2003 - 11:14 pm: Edit

Was wondering if someone could solve this?

A flexible tubing system has supports that must be perpendicular to the tubes. If a section of the tube follows the curve y= 4/x^2+1, along which lines must the supports be directed if they are located at x=-1, x=1, x=0?

Any help would be greatly appreciated.
Thanks

 By Miseryxsignals (Miseryxsignals) on Thursday, October 23, 2003 - 11:38 pm: Edit

well, assuming that the curve is y = 4/[(X^2)+1]instead of Y = 1 + [4/(X^2)] because there is a vertical asymptote for the latter at x = 0, you basically need to find the slope at each point (X = -1,0,1), take the inverse of it, and change the (+/-) sign, then use y = mx+b to find the equation of the line.

ex: y = 4/[X^2+1]
slope at given point = y' = -8x/[(X^2+1)^2]

Slope at (x = -1) = f'(-1) = 2
Slope of the support at (x = -1) = perpendicular = -1/2 = m
Point at (x = -1) = f(-1) = 2
y = mx + b, plug in variables to 2 = (-1/2)(-1) + b
2 = 1/2 + b, b = 3/2
so the support for the point (x = -1) would be the line y = (-1/2)x + 3/2

do the same for the last two points. Hope this helps

 By Onizuka (Onizuka) on Friday, October 24, 2003 - 11:25 pm: Edit

Thanks, it did help a lot.