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Math Help - show that a tangent line to a curve is tangent at another point

  1. #1
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    show that a tangent line to a curve is tangent at another point

    Show that the tangent line to the curve y = (x^2 + x - 2)^3 + 3 at (1, 3) is also tangent to the curve at another point.

    Like my previous question I got the derivative

    being 3(2x+1)(x^2 +x -2)^2

    I am not sure where to go from here.

    Thank you!
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  2. #2
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    Re: show that a tangent line to a curve is tangent at another point

    What is the gradient at that point?
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    Re: show that a tangent line to a curve is tangent at another point

    solve dy/dx for x=1 I get a slope of 18
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    Re: show that a tangent line to a curve is tangent at another point

    No, if you let x = 1 in dy/dx you should get 0.
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    Re: show that a tangent line to a curve is tangent at another point

    Quote Originally Posted by Jonroberts74 View Post
    solve dy/dx for x=1 I get a slope of 18
    Nope. Your derivative is correct, but you computed the slope incorrectly. At x = 1:
    y' = 3(2(1) + 1)((1)^2 + (1) - 2)^2 = ?

    Now set your derivative equal to 0. What values of x are there where y' = 0?

    -Dan

    Gah! ProveIt beat me to it.
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    Re: show that a tangent line to a curve is tangent at another point

    x= -1/2, x = 1, x = -2

    need more work on my factoring skills.

    thank you!
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    Re: show that a tangent line to a curve is tangent at another point

    To find the other point that the tangent line touches, note that
     \frac{dy}{dx}=0 \text{ and also, your tangent line is given by } y=3, \text{ so set } y=3 \implies 3=(x^2+x-2)^3 + 3 \implies 0=(x^2+x-2)^3 \implies 0=(x^2+x-2) \implies \cdots \text{ (you can practice your factoring here! =) )}

    Just a note, in this question the tangent line touches 2 local extrema at the same level (i.e. y=3), but not necessarily.
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    Re: show that a tangent line to a curve is tangent at another point

    Quote Originally Posted by chen09 View Post
    Just a note, in this question the tangent line touches 2 local extrema at the same level (i.e. y=3), but not necessarily.
    Actually the points (-2, 3) and (1, 3) are inflection points.

    -Dan
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    Re: show that a tangent line to a curve is tangent at another point

    Thanks, I didn't check that. I was rushing, and got 2 problem sets due Monday PDE + topology =(

    I apologize for being emo in this thread.
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    Re: show that a tangent line to a curve is tangent at another point

    dang, I hope someday to understand PDEs and topology
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    Re: show that a tangent line to a curve is tangent at another point

    Quote Originally Posted by Jonroberts74 View Post
    dang, I hope someday to understand PDEs and topology
    Me too!

    -Dan
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