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Math Help - derivative tangents

  1. #1
    Joelmalmsteen
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    derivative tangents

    I'd be eternally greatful if someone could help me with these few problems...

    1. The Curve C has the equation

    y=x+ax+b

    Where a and b are constants.

    Given that the minimum point of C has co-ordinates (-2, 5), Find the values of a and b.

    2. Given that

    dy/dx=2x+1,

    and that y=3 when x=0, find the value of y when x=2.

    And finally

    3. f(x)=4x-3x-x

    (a) Fully factorise f(x)=4x-3x-x.

    I hope someone can help! It needs to be in by tomorrow, and one person has already been thrown off the course as their work was not upto standard. I fear mine will be without this help! Thanks in advance
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Joelmalmsteen View Post
    1. The Curve C has the equation

    y=x+ax+b

    Where a and b are constants.

    Given that the minimum point of C has co-ordinates (-2, 5), Find the values of a and b.
    The minimum point of the curve will be the x for which y' = 0. So:
    y'(x) = 2x + a = 0
    has the solution
    x = \frac{-a}{2}

    We know that the minimum point of the curve is at x = -2, so thus
    -2 = \frac{-a}{2}

    or a = 4.

    So y = x^2 + 4x + b
    has a point (-2, 5) on it.

    Thus:
    5 = (-2)^2 + 4(-2) + b

    or
     b = 9

    Thus your curve is y = x^2 + 4x + 9.

    -Dan
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  3. #3
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    Quote Originally Posted by Joelmalmsteen View Post

    3. f(x)=4x-3x-x

    (a) Fully factorise f(x)=4x-3x-x.
    x(4-3x-x^2)
    -x(x^2+3x-4)
    -x(x+4)(x-1)
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Joelmalmsteen View Post
    2. Given that

    dy/dx=2x+1,

    and that y=3 when x=0, find the value of y when x=2.
    \frac{dy}{dx} = 2x^3 + 1

    So
    y = \int dx \, (2x^3 + 1) = \frac{1}{2}x^4 + x + C

    We know that the point (0,3) is on this solution curve, so
    3 = \frac{1}{2}(0)^4 + (0) + C

    Thus C = 3.
    y = \frac{1}{2}x^4 + x + 3

    What is y when x = 2?
    y = \frac{1}{2}(2)^4 + 2 + 3 = 8 + 2 + 3 = 13

    -Dan
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