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Thread: Tangent lines at origin---Need help!!

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    Tangent lines at origin---Need help!!

    Find the equations for all of the lines through the origin that are tangent to the curve y=1/5(4-x)(x+1)^3
    I figured y'= (-(x+1)^2(4x-11)) /5 = 11/5 (when you plug in 0 to y') and y=11/5x + 4/5 as my solution, however when graphing my original problem my tangent line doesn't go through the origin....Where did I go wrong??
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    Re: Tangent lines at origin---Need help!!

    $y=\dfrac 1 5 (4-x)(x+1)^3$

    $y^\prime = \dfrac{3}{5} (4-x) (x+1)^2-\frac{1}{5} (x+1)^3$

    we have two points on the line

    $(x,~y(x))$ and $(0,0)$

    this gives us a slope of

    $m = \dfrac{y(x)}{x}$

    and this needs to equal $y^\prime(x)$ so

    $\dfrac{\frac 1 5 (4-x)(x+1)^3}{x} = \dfrac{3}{5} (4-x) (x+1)^2-\frac{1}{5} (x+1)^3$

    a nasty but solvable polynomial equation.

    A hint. If you expand the above equation out so that the right side is $0$,

    the polynomial on the left side has a double root of $x=-1$

    That should let you easily find the other 2 roots.
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    Re: Tangent lines at origin---Need help!!

    Thank You!
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    Re: Tangent lines at origin---Need help!!

    Quote Originally Posted by tomas View Post
    Find the equations for all of the lines through the origin that are tangent to the curve y=1/5(4-x)(x+1)^3
    I figured y'= (-(x+1)^2(4x-11)) /5 = 11/5 (when you plug in 0 to y')
    The derivative of y at x= 0 is irrelevant- especially since the graph does not go through the origin. You are misunderstanding the question. The tangent line goes through the origin. It is not tangent to the graph at the origin (again, the graph itself does not go through the origin).

    Any line through the origin must be of the form y= mx for some number m. In order to be tangent to the curve, it must meet the curve at some point and have slope equal to the derivative at that point.

    So we must have, for some x, y= mx= (1/5)(4- x)(x+ 1)^3. And at that x we must have m= -(1/5)(x+ 1)^3+ (1/5)(4- x)(x+ 1)^2. That gives two equations to solve for m and x.

    and y=11/5x + 4/5 as my solution, however when graphing my original problem my tangent line doesn't go through the origin....Where did I go wrong??
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