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Math Help - common tangent line?

  1. #1
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    common tangent line?

    find the two points on the curve
    y=x4-2x2​-x that have a common tangent line
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  2. #2
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: common tangent line?

    Quote Originally Posted by pnfuller View Post
    find the two points on the curve
    y=x4-2x2​-x that have a common tangent line
    Try the two points:

    x1=-1, x2=1

    Attached Thumbnails Attached Thumbnails common tangent line?-common-tanget.png  
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  3. #3
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    Re: common tangent line?

    what do i do with those two points?
    Quote Originally Posted by MaxJasper View Post
    Try the two points:

    x1=-1, x2=1

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  4. #4
    MHF Contributor MarkFL's Avatar
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    Re: common tangent line?

    The equation of the line to a function at (x_n,f(x_n) is given in slope intercept form as:

    y=f'(x_n)x+f(x_n)-x_nf(x_n)

    Two lines are the same if they have the same slope and y-intercept. Using the points (x_1,f(x_1)) and (x_2,f(x_2)) will give you two equations and two unknowns, from which you can find two distinct points.
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  5. #5
    Senior Member x3bnm's Avatar
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    Re: common tangent line?

    Quote Originally Posted by pnfuller View Post
    find the two points on the curve
    y=x4-2x2​-x that have a common tangent line
    The slope of the tangent line is:
    m = \frac{dy}{dx} = 4x^3-4x-1

    f'(a) = 4 a^3 -4a -1

    f'(b) = 4 b^3 -4b -1

    f(a) = a^4 -2a^2 -a

    f(b) = b^4 -2 b^2 -b

    Now:

    f'(a) =\frac{f(b) - f(a)}{b -a}

    And:
    f'(b) =\frac{f(b) - f(a)}{b -a}

    So now the equation becomes:

    (4 a^3 -4a -1)(b - a) = (b^4 -2 b^2 -b) - (a^4 -2a^2 -a)..................(1)

    (4 b^3 -4b -1)(b - a) = (b^4 -2 b^2 -b) - (a^4 -2a^2 -a)..................(2)


    Solving these two equations by maple we find a = -1 \text{ and } b = 1 or a = 1 \text{ and } b = -1
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  6. #6
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    Re: common tangent line?

    what do you mean solve it with maple?
    Quote Originally Posted by x3bnm View Post
    The slope of the tangent line is:
    m = \frac{dy}{dx} = 4x^3-4x-1

    f'(a) = 4 a^3 -4a -1

    f'(b) = 4 b^3 -4b -1

    f(a) = a^4 -2a^2 -a

    f(b) = b^4 -2 b^2 -b

    Now:

    f'(a) =\frac{f(b) - f(a)}{b -a}

    And:
    f'(b) =\frac{f(b) - f(a)}{b -a}

    So now the equation becomes:

    (4 a^3 -4a -1)(b - a) = (b^4 -2 b^2 -b) - (a^4 -2a^2 -a)..................(1)

    (4 b^3 -4b -1)(b - a) = (b^4 -2 b^2 -b) - (a^4 -2a^2 -a)..................(2)


    Solving these two equations by maple we find a = -1 \text{ and } b = 1 or a = 1 \text{ and } b = -1
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    Re: common tangent line?

    Maple is a program that does mathematical calculations. Here's a link:

    Maple 16 by Maplesoft

    - Hollywood
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