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Unformatted text preview: (10/7/08) Math 10C. Lecture Examples. Sections 12.1, 12.2, and 12.3. Functions of two variables Example 1 (a) What is the domain of f ( x , y ) = x 2 + y 2 ? (b) What are the values f ( 2 , 3 ) and f (- 2 ,- 3 ) of this function at (2,3) and (- 2 ,- 3 ) ? (c) What is its range? Answer: (a) The domain of f is the entire xy-plane. (b) f (2 , 3) = 13 f (- 2 ,- 3) = 13. (c) The range of f is the closed infinite interval [0 , ). Example 2 Determine the shape of the surface z = x 2 + y 2 in xyz-space by studying its cross sections in the planes x = c perpendicular to the x-axis. Answer: The intersection of the surface z = x 2 + y 2 with the plane x = c is a parabola that opens upward and whose vertex is at the origin if c = 0 and is c 2 units above the xy-plane if c negationslash = 0 Figure A2a The surface has the bowl-like shape in Figure A2b y x z x = c c 2 Figure A2a Figure A2b Lecture notes to accompany Sections 12.1, 12.2, and 12.3 of Calculus by Hughes-Hallett et al. 1 Math 10C. Lecture Examples. (10/7/08) Sections 12.1, 12.2, and 12.3, p. 2 Example 3 Determine the shape of the surface z = x 2 + y 2 of Example 2 by studying its cross sections in the planes y = c perpendicular to the y-axis....
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