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Math Help - help with function

  1. #1
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    help with function

    heyy here is a question i have, i was wondering if anyone could help me:

    The function f is defined by f:x --> -0.5x^2 + 2x + 2.5

    a. Let N be the normal to the curve at the point wehre the graph intercpets the y-axis. Show that the equation of N may be written as y = -0.5x + 2.5

    Let g:x --> -0.5x + 2.5

    b. Find the solutions of f(x) = g(x)
    c. Hence find the coordinates of the other point of intersection of the normal and the curve.

    Thanks!
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  2. #2
    Grand Panjandrum
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    [quote=samantha_malone;60972]heyy here is a question i have, i was wondering if anyone could help me:

    The function f is defined by f:x --> -0.5x^2 + 2x + 2.5

    a. Let N be the normal to the curve at the point wehre the graph intercpets the y-axis. Show that the equation of N may be written as y = -0.5x + 2.5

    Let g:x --> -0.5x + 2.5

    [/tex]

    The curve intercepts the y-axis when x=0, and so y=2.5. The slope of the curve is:

    <br />
\frac{d}{dx}(-0.5x^2 + 2x + 2.5)=-x+2<br />

    so at the intercept the slope is 2, and hence the slope of the normal is -0.5. So the normal has slope -0.5 and passess through (0,2.5), and so its equation is:

    <br />
y=0.5x+2.5<br />

    as required.

    RonL
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  3. #3
    MHF Contributor red_dog's Avatar
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    a) f(0)=2.5, so the point where the graph intercepts the y-axis has the coordinates (0,2.5).
    The normal to the graph in this point is perpendicular to the tangent in the same point.
    The slope of the tangent is f'(0).
    We have f'(x)=x+2\Rightarrow f'(0)=2.
    The slope of the normal is m=-\frac{1}{2}=-0.5.
    The equation of the normal is y-y_0=m(x-x_0), where x_0=0,y_0=2.5,m=-0.5.
    Then the equation is y-2.5=-0.5x\Leftrightarrow y=-0.5x+2.5

    b) f(x)=g(x)\Leftrightarrow 0.5x^2+2x+2.5=-0.5x+2.5\Leftrightarrow
    \Leftrightarrow 0.5x^2+2.5x=0\Leftrightarrow x(0.5x+2.5)=0\Rightarrow
    \Rightarrow x_1=0,x_2=-5

    c) f(-5)=5, so the second point where the normal intercepts the graph has the coordinates (-5,5).
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  4. #4
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    thanks!
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