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Point Slope Equation f(x) for Curved Line

  1. Mar 26, 2012 #1

    morrobay

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    Gold Member

    1. The problem statement, all variables and given/known data
    On a displacement versus time graph with y axis in feet and x axis in seconds
    and variable speed. Is it possible to find the equation of this curve , f(x)
    With two points: x =1sec and y = 3' and dy/dx = 1.7 ft/sec , at that point
    And x = 2.5 sec and y= 5' and dy/dx = 0 at that point

    2. Relevant equations
    point slope equation : y - y1 = m (x - x1)
    Not sure if this applies to a curved line


    3. The attempt at a solution
    y-3 = 1.7 ( x - 1 )
    y = 1.7 x +1.3
     
    Last edited: Mar 26, 2012
  2. jcsd
  3. Mar 26, 2012 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    No, that function is "linear" and so represents a straight line. Exactly what the equation is depends on exactly what the speed function is. Here, you are given 4 values (position and speed at x= 1 and = 2.5) so you can "fit" a cubic function to that data.

    If [itex]y= ax^3+ bx^2+ cx+ d[/itex] them [itex]dy/dx= 4ax^2+ 2bx+ c[/itex] so that y(1)= a+ b+ c+ d= 3, y(2.5)= 15.625a+ 6.25b+ 2.4c+ d= 5, y'(1)= 4a+ 2b+ c= 1.7, y'(2.5)= 25a+ 5b+ c= 0. You can solve those four linear equations for a, b, c, and d to find the simplest polynomial that will fit the given data. There are, however, an infinite number of functions that will fit the given data.
     
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