Results 1 to 4 of 4

Math Help - tangent equations parrallel or perpendicular to ...

  1. #1
    Newbie
    Joined
    Mar 2009
    Posts
    15

    tangent equations parrallel or perpendicular to ...

    ok i have two question's i can't answer;

    #1) find the equations of the tangents to the curve y = x^3 -3x^2 -7x +5 that are parrallel to the line with equation y = 2x - 3

    i realise that it being parrallel the gradient would be the same. i also know you can solve these algabraicly just lacking the know how

    #2) find the equation of the tangent line to the curve with equation y = e^2x-6 that is perpendicular to the line with equation y = 5 - \frac {x}{2}

    same sort of deal, i realise it being perpendicular it has the gradient of a normal. don't know how to figure these out algabraicly.

    thankin you kind people of math help forum.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
    1,823
    We know that the slope of the line y=2x+3is 2. So therefore, all we have to do is find the point at which a line tangent to the curve y=x^3-3x^2-7x+5, and then determine the equation of this line, knowing that it's slope is 2.

    So, how do we do this? we know that the slope of a line that goes throgh any 2 points on the curve is given by \frac{y_2-y_1}{x_2-x_1}, or in functional notation \frac{f(x+\Delta{x})-f(x)}{\Delta{x}}. But this is only 2 points that goes through the curve, it is not the tangent line. To find the tangent line to a curve at any point on the curve, we'd have to take the limit as \Delta{x} tends to 0. Or

    \lim_{\Delta{x}\to0}\frac{x^3-3x^2-7x+5-f(x)}{\Delta{x}}.

    So, Let's do it.

    \lim_{\Delta{x}\to0}\frac{f(x+\Delta{x})-f(x)}{\Delta{x}}=\lim_{\Delta{x}\to0}\frac{(x+\Del  ta{x})^3-3(x+\Delta{x})^2-7(x+\Delta{x})+5-f(x)}{\Delta{x}}= \lim_{\Delta{x}\to0}\frac{(x^3+3x^2\Delta{x}+3x\De  lta{x^2}+\Delta{x^3})-3(x^2+2x\Delta{x}+x\Delta{x})-7(x+\Delta{x})+5-f(x)}{\Delta{x}}

    after multiplying and adding we have

    \lim_{\Delta{x}\to0}\frac{3x^2\Delta{x}+3x\Delta{x  ^2}+\Delta{x^3}-6x\Delta{x}-3\Delta{x^2}-7\Delta{x}}{\Delta{x}}

    Factoring out \Delta{x}

    \lim_{\Delta{x}\to0}3x^2+3x\Delta{x}+\Delta{x^2}-6x-3\Delta{x}-7

    Taking the limit by substituting 0 for \Delta{x} we get

    \lim_{\Delta{x}\to0}3x^2+3x\Delta{x}+\Delta{x^2}-6x-3\Delta{x}-7=3x^2-6x-7

    and we have a formula for finding the slope at any point on the curve of f(x).

    setting our formula equal to 2 allows us to solve for x

    3x^2-6x-7=2

    x^2-2x=2+\frac{7}{3}

    x^2-2x+1=3+\frac{7}{3}

    (x-1)^2=3+\frac{7}{3}

    x=\pm{\sqrt{3+\frac{7}{3}}}+1

    Now all you have to do is sub \pm{\sqrt{3+\frac{7}{3}}}+1
    into f(x) to obtain the corresponding values of y, and knowing that the slope of the line tangent to the curve at that point is 2, you can write an equation to that line.
    Last edited by VonNemo19; May 17th 2009 at 07:51 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2009
    Posts
    15
    alotta latex there... lol

    i understand everything so far, so what next??
    Follow Math Help Forum on Facebook and Google+

  4. #4
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
    1,823
    You got it?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: May 13th 2011, 02:41 AM
  2. Perpendicular linear equations.
    Posted in the Geometry Forum
    Replies: 4
    Last Post: May 29th 2010, 01:07 PM
  3. Replies: 1
    Last Post: February 21st 2010, 04:47 PM
  4. Perpendicular Tangent Lines
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 10th 2010, 11:13 PM
  5. Tangent line and Perpendicular
    Posted in the Calculus Forum
    Replies: 4
    Last Post: March 11th 2009, 04:47 PM

Search Tags


/mathhelpforum @mathhelpforum