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1 Homework week 3 Consider the function f x y x2 x y2 1. Find: x x2 y2 a) Domain: we do b) level curves: 1 c x x2 y2 1 0 so D x y x2 y2 0 so y cx x2 2. Plot the surface which it represents: a) 3. Find if the function is continuous at 0 0 by using limits: we want to calculate x ∞ x ylim 0 0 x2 y2 so the function is not continuous at the point 0 0 , see Domain and figure. 4. Calculate the partial derivatives of this same function: Take f x y x x 2 y2 ∂f ∂x 2x2 x2 y2 and ∂f ∂y 2 1 x2 y2 2yx y2 x2 1 2 5. Find the slope of the tangent lines to the surface at the points 0 0 2 and 0 2 0 . Draw them in your Plot. Solution: We evaluate the previous formulas ∂f ∂x ∂f ∂y ∂f ∂x ∂f ∂y 2 02 02 0 2 0 0 2 0 0 2 0 2 0 0 2 0 2 2 2 0 2 0 02 0 2 2 2 0 2 2 0 2 2 02 2 0 0 2 0 2 2 02 1 02 2 2 2 1 0 2 2 25 0 0 0 2 1 2 02 25 1 Homework week 3 y= + 3 c<0 cx−x2 c>0 y= + y cx−x2 c=1/6 c=1/5 c=1/4 c=1/3 c=1/2 c=1 2 1 1 2 3 4 5 x 6 −1 −2 c<0 y= − cx−x2 c>0 −3 y= − Figure 1.1: Level curves 2 cx−x2 1 Homework week 3 z z=1 z=1/2 z=1/4 y x Figure 1.2: Surface 3 1 Homework week 3 m=−25 z m=0 (0.2,0) y x Figure 1.3: Tangent lines at (0.2,0) the slopes corresponding to the derivatives of the function with respect to x and y are -25 and 0 respectively. 4