Math E-21a – Things to know for the 2nd midterm exam Higher order derivatives, interpretation of 2nd derivatives, equality of mixed partials (Clairaut’s Theorem), quadratic approximation. Extrema of functions and the 2nd Derivative test; local maxima, local minima, and saddle points; unconstrained optimization. Constrained optimization and the Method of Lagrange Multipliers; extrema of a function in a bounded region. Integration of a function f(x, y) over a region in R2. Calculation of integrals over regions R2via iterated single integrals; and the Fubini Theorem. Interchanging the order of integration; calculation of double integrals using polar coordinates. Integration of a function f(x, y, z) over a region in R3. Calculation of integrals over regions R3via iterated single integrals using Cartesian, cylindrical, and spherical coordinates. Applications of multiple integrals: area of a region, volume of a solid, mass, average value of a function, centroid (geometric center).
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