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1. Use traces to sketch the graph of f(x, y) = p 16 − x 2 − 16y 2 . 2. For each of the following, identify the surface and sketch its graph. a) x = 4y 2 + z 2 − 4z + 4. b) x 2 + 4y 2 + z 2 − 2x = 0 3. a) What does the equation x 2 + y 2 = 1 represent as a curve in R 2 ? b) What does it represent as a surface in R 3 ? 4. Find an equation of the tangent plane and normal to the surface at the specified point. a) z = 4x 2 − y 2 + 2y; (−1, 2, 4) b) z = y ln x; (1, 4, 0) 5. Find an equation for the tangent plane and find symmetric equations for the normal line to the surface z = arctan( y x ) at the point (1, 1, π 4 ). 6. Sketch the region bounded above by the cone z = p x 2 + y 2 and below by the sphere x 2 + y 2 + z 2 = z 7. Find the angle of inclination θ of the tangent plane to the surface 3x 2 + 2y 2 − z = 15 at the point (2, 2, 5).
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