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Math 237 Multivariable Calculus questions and solutions.

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Math 237 Multivariable Calculus questions and solutions. Note: ‘Ch. N’ refers to the Chapter N Problem Set in the course notes, for N = 1, 2, 3, ..., 15. You should do these problems as if you w ... ere going to submit them. 1. Consider f(x, y) = p4 + x2 y2. (a) What is the domain and range off? (b) What symmetries do you see? (c) Sketch some level curves and cross-sections off. (d) Sketch the surface z = f(x, y). (e) Then verify your sketch with a maple plot (“plot3d”). (If you do the maple plot before the other parts, the exercise is much less useful. ) Solution: (a) The domain of f is D(f) = {(x, y) 2 R2 ; |y|  p 4 + x2} and the range is R(f) = {z 2 R; z 0}. (b) The function is symmetric in ±x, ±y. (c) The level curves are k = p4 x2 y2; rewriting this is x2 y2 = k2 4, k 0 which are hyperbolae. The cross-sections (constant x) are z = p4 + c2 y2. Rewriting, z2 + y2 =4+ c2 , z 0 which are semi-circles. The cross-sections (constant y) are z = p4 + x2 d2. Rewriting, z2 x2 = 4 d2 , z 0 which are half hyperbolae. For plots, see Figure 1. [Show More]

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