MediumPartial derivatives

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §13.3 Tangent Planes and Linear Approximations, #24 · Multivariable Calculus · Partial derivatives

The _Fundamental Lemma_ behind equation (13) writes

The _Fundamental Lemma_ behind equation (13) writes $Δ f = a Δ x + b Δ y$ . The Lemma says that $a \to f_{x} (x_{0}, y_{0})$ and $b \to f_{y} (x_{0}, y_{0})$ when $Δ x \to 0$ and $Δ y \to 0$ . The proof takes $Δ x$ first and then $Δ y$ :
(1) $f (x_{0} + Δ x, y_{0}) - f (x_{0}, y_{0}) = Δ x f_{x} (c, y_{0})$ where $c$ is between ______ and ______ (by which theorem?)
(2) $f (x_{0} + Δ x, y_{0} + Δ y) - f (x_{0} + Δ x, y_{0}) = Δ y f_{y} (x_{0} + Δ x, C)$ where $C$ is between ______ and ______.
(3) $a = f_{x} (c, y_{0}) \to f_{x} (x_{0}, y_{0})$ provided $f_{x}$ is ______.
(4) $b = f_{y} (x_{0} + Δ x, C) \to f_{y} (x_{0}, y_{0})$ provided $f_{y}$ is ______.

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §13.3 Tangent Planes and Linear Approximations, #24

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