MAT-2130 Midterm: MATH 2130 App State Fall2011 Test2
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If f (x, y) = c, then dy dx. Fz: (9 points) consider the surface z = px2 + y2. (a) write down an equation for the level curves of this surface. Setting z = c (a constant), we get c = px2 + y2. Therefore, the level curves are: x2 + y2 = c2. So intersecting z = px2 + y2 and x = 0 gives us z = py2. This is z = |y| (whose graph looks like ). (c) make a rough sketch of this surface. Horizontal cross sections (level curves) are circles and vertical cross sections are -shaped. The graph is a cone: (10 points) show that the limit lim (x,y) (0,0) Let"s approach the origin along the y-axis (i. e. let x = 0 and y 0): Now let"s approach the origin along the curve y = x2, we get:
