Point Slope Equation f(x) for Curved Line

[morrobay]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

y-3 = 1.7 ( x - 1 )
y = 1.7 x +1.3

 

[HallsofIvy]

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 $y= ax^3+ bx^2+ cx+ d$ them $dy/dx= 4ax^2+ 2bx+ c$ 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.