5.7 Applications of Partial Derivatives

Having established how to differentiate a function of several variables, we now put the partial derivatives to work.

Three applications follow, and they share one idea: near a point, a smooth surface is well approximated by a plane, and the partial derivatives are exactly what specify that plane. Maxima and minima come first, located where the tangent plane is horizontal, with the new possibility of a saddle point to be distinguished. The tangent plane itself is then constructed explicitly. Finally the linear approximation is used quantitatively, to estimate how much a calculated quantity changes when its inputs are altered slightly — the basis of error estimation in measurement.

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