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Chapter 10: Multivariable Calculus Section 10.1: Functions of Two or More Variables Definition: A function of two variables is a rule that assigns a unique real number f (x, y) to each ordered pair (x, y) ∈ D ⊂ R2 . The set D is called the domain of f and {f (x, y)|(x, y) ∈ D} is called the range of f . Example: Evaluate each function at the given point. √ (a) f (x, y) = x + y at (4, 3) (b) f (x, y) = 2x − 3y + ln(xy) at (1, 1) Example: Find the largest possible domain of each function and the corresponding range. p (a) f (x, y) = 9 − x2 − y 2 (b) f (x, y) = e−(x 2 +y 2 ) 1 Example: Find and sketch the domain of the given functions: p (a) f (x, y) = y − x2 (b) f (x, y) = (c) f (x, y) = p x2 + y 2 − 1 + ln(4 − x2 − y 2 ) √ x+ √ y+ p x2 + y 2 − 16 p 9 − x2 − y 2 (d) f (x, y) = 2x + y 2 Definition: If f is a function of two variables with domain D, then the graph of f is the set {(x, y, z) ∈ R3 |z = f (x, y), (x, y) ∈ D}. Note: The graph of a function f of two variables is a surface with equation z = f (x, y). The graph of f can be visualized as lying directly above or below the domain D in the xy-plane. Example: Describe the graph of the following functions: (a) f (x, y) = 2 (b) f (x, y) = 4 − 2x − y (c) f (x, y) = x2 + y 2 3 Definition: The level curves of a function of two variables are the curves defined by f (x, y) = k, where k is a constant in the range of f . The level curves of f (x, y) are the horizontal traces of the graph of f in the plane z = k projected onto the xy-plane. A graph of the level curves is called a contour plot. Note: Contour plots are commonly used in topographic maps. Example: Sketch the level curves for the given functions: (a) f (x, y) = x2 + 4y 2 for k = 0, 4, 16 (b) f (x, y) = y − x2 for k = −1, 0, 1, 2 (c) f (x, y) = p 4 − x2 − y 2 for k = 0, 1, 2 4