Final answer:
The linear approximation of a function at a point is given by the function's value at that point plus the dot product of the gradient and the displacement from that point.
Step-by-step explanation:
The linear approximation of a function f(x,y) at the point (x₀,y₀,z₀) is typically represented as f(x₀,y₀) + fx(x₀,y₀)⋅(x - x₀) + fy(x₀,y₀)⋅(y - y₀), where fx and fy represent the partial derivatives of f with respect to x and y, respectively, evaluated at the point (x₀,y₀). This formula stems from the first-order Taylor expansion and is used to approximate the function near the point of interest assuming it is differentiable at that point. In the context of the problem domain presented, if the function includes dynamics in three dimensions or considerations of time, additional terms will be required to create a linear approximation that accounts for those variables.