1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Calculus exercise

  1. Aug 17, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the equation of a tangent line to the curve y = x³ - 1, that is perpendicular to y = -x (I mean the tangent line should be perpendicular to y=-x, sorry for my bad english).

    2. Relevant equations
    3. The attempt at a solution

    [tex]y = x^3 - 1; \;\; y' = 3x^2[/tex]

    If the tangent line must be perpendicular to y = -x then its slope must be +1, right? So we need to know at what value of x the slope is +1:

    [tex]1 = 3x^2; \;\; x = \pm \frac{\sqrt{3}}{3}[/tex]

    [tex]y - \left ( \frac{\sqrt{3}^3}{3^3} -1 \right ) = \left ( x - \frac{\sqrt{3}}{3} \right )[/tex]


    [tex]27y - 3\sqrt{3} + 27 = x - 9\sqrt{3}[/tex]

    [tex]x - 27y - 9\sqrt{3} + 3\sqrt{3} + 27 = 0[/tex]

    [tex]x - 27y +6\sqrt{3} + 27 = 0[/tex]

    The correct answer is
    [tex]3\sqrt{3} x - 3\sqrt{3} y - 3\sqrt{3} -2 = 0; \;\; 3\sqrt{3} x - 3\sqrt{3} y - 3\sqrt{3} +2 = 0[/tex]

    I hope this was not an error in arithmetics...

    Thank you for the help...
     
  2. jcsd
  3. Aug 17, 2010 #2

    Mark44

    Staff: Mentor

    Right.
    What you have above looks fine, but there are two points at which the slope of the curve y = x3 - 1 is 1. You need to find the normal line at each of these points.
     
  4. Aug 17, 2010 #3

    Mark44

    Staff: Mentor

    Starting from here:
    [tex]y - \left ( \frac{\sqrt{3}^3}{3^3} -1 \right ) = \left ( x - \frac{\sqrt{3}}{3} \right )[/tex]

    and rewriting as:
    [tex]y - \left ( \frac{1 }{3\sqrt{3}} -1 \right ) = \left ( x - \frac{1}{\sqrt{3}} \right )[/tex]
    Just multiply both sides by 3 sqrt(3).

    There is a mistake in your work. In this equation -
    [tex]27y - 3\sqrt{3} + 27 = x - 9\sqrt{3}[/tex]
    you forgot to multiply the x on the right side by 27.
     
  5. Aug 17, 2010 #4
    Ah, yes, arithmetic error again ;( haha

    Thank you for the help...
     
  6. Aug 18, 2010 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I always say "I am only in Advanced Calculus", I haven't taken arithmetic yet!":rofl:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook