To find the arc length function for the curve defined by the equation y = 2x^3, we use the arc length formula:
S = ∫√(1 + (dy/dx)²) dx
Firstly, we need to compute the derivative of y with respect to x:
dy/dx = d(2x^3)/dx = 6x²
Now, we can substitute this expression into the arc length formula:
S = ∫√(1 + (6x²)²) dx
This simplifies to:
S = ∫√(1 + 36x^4) dx
To find the arc length function from the starting point P(0,12), we compute the integral from x = 0 to x. The integral can be complex, but it can be solved using numerical methods or special functions if necessary. However, for practical purposes, we can express the arc length function, in general terms, as:
S(x) = ∫₀ˣ√(1 + 36t^4) dt
Thus, the arc length function for the given curve from point P to any point (x, 2x³) is expressed as:
S(x) = ∫₀ˣ√(1 + 36t^4) dt
This integral needs to be evaluated based on the specific context or requirements of the problem.