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15 - 15.1 Functions of Several Variables We write a...

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15.1 Functions of Several Variables We write a function of 2 variables as f x, y = z. Ex. Let f x, y = 9 x 2 4 y 2 f 2, 1 = 1 Domain: All x, y R such that x 2 4 y 2 9 (all points on or inside the ellipse x 2 4 y 2 = 9). Operations of functions with 2 variables work the same as operations between functions of 1 variable. (addition/subtraction, multiplication/division, etc.) The graph of the function f(x, y) is the collection of all 3-tuples (ordered triples x, y, z) such that (x, y) D (where D is the domain) and z = f x, y . To graph z = f x, y we use projections onto planes z = c. Note: if the plane z = c intersects the surface z = f x, y , the result is a curve C = f x, y [in other words, a trace]. The set of points (x, y) satisfying C = f x, y are called level curves or contour curves of f at C. ex. Sketch level curves: f x, y = 10 x 2 y 2 is an elliptic paraboloid with vertex z = 10 f x, y, z : let c = f x, y, z to get level surfaces. Suppose a region R 3 is heated so its temp. T @ each pt. x, y, z is given by T x, y, z = 100 x 2 y 2 z 2 degrees Celsius. Draw isothermal surfaces (where temp. is always the same) for T 0. (spheres) Interestingly, linear functions in 3 dimensions form planes. 15.2 proofs : (p105 gives a general grounding) I’m saying that, given an accuracy you want f x or f x, y to, I can find a range of x or (x, y) coordinates that give you that accuracy. It doesn’t matter how small of an accuracy you want, as long as the error is not 0. So your range of x’s or (x, y)’s gets smaller and smaller as it approaches a or (a, b), and your range of f(x)’s or f(x, y)’s gets smaller and smaller as it approaches L.
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