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...Math calculus iii final exam practice problems spring let p g x y z a describe the shapes of level surfaces b in three di erent graphs sketch cross sections to surface for which i ii each section label axes and any intercepts c find equation plane tangent at point suppose f is erentiable function such that fx fy fxx fxy fyy gradf vector perpendicular contour line e graph d what rate change direction j use quadratic approximation estimate classify critical points maximum minimum values subject constraint approximate value explain your answer terms lagrange multipliers integral some over region r given polar coordinates as dr integration xy convert this cartesian evaluate you may either or w be solid where other words bounded by yz xz density material express total mass triple rectangular cylindrical yourexpressions should complete enough principle they could evaluated but not integrals we say space normal intersection if s de ned parametric equations found will intersect two one them co...