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Unformatted text preview: MAT 1332: Calculus for the Life Sciences II University of Ottawa, Winter 2008 Prof. F. Lutscher Practice Problems, Several variables Question 1: Find the general expression of the level sets L c of the function f ( x, y ) = 16 − 4 x 2 − y 2 , i.e., find a formula for all the points ( x, y ) such that f ( x, y ) = c. Draw the level sets in the x- y-plane for c = 4 and c = 0 . Solution: We write f ( x, y ) = c and solve for y to get y = ± 16 − c 2 − 4 x 2 . Hence, for c = 4 we get y = ± √ − 4 x 2 , which gives only one solution, namely the point (0 , 0) . For c = 0 we get y = ± √ 16 − 4 x 2 which is an ellipse in the x- y-plane. Question 2: Consider the function of two variables f ( x, y ) = 4 x 4 − 5 x 2 y 2 + y 4 . (a) Find the zero level set of the function, i.e., all the points ( x, y ) where f ( x, y ) = 0 . Graph the level sets in the x- y-plane where x, y range from -2 to 2. [Hint: use z = y 2 , then solve the quadratic equation for z and finally replace y = ±...
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