The approximation, Landsburg says, is good "assuming that your contributions are small relative to the initial endowments". Here's the thing: why? Suppose Δx/X, Δy/Y and Δz/Z are indeed very small - what then? Why does it follow that the linear approximation works? There's no explanation, and if you think this is because it's immediately obvious - well, it isn't. It may sound plausible, but the math isn't there. We need to go deeper.

What shapes for U(X,Y,Z) could make the linear approximation not work? It would have to be a curve that had sudden local changes. It would be kinked or fractal. That would be surprising. If U(X,Y,Z) is continuous, smooth, monotonic, and its first and second derivatives are monotonic, I can't imagine how the linear approximation could fail.