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Solutions to various mathematical problems from a university-level final exam, including finding the center, foci, and vertices of an ellipse, evaluating derivatives, converting between polar and rectangular coordinates, analyzing critical numbers and extrema of a function, and finding the equation of a tangent line.
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MAT321Prof. Thistleton1. Consider the ellipse described by the equation Final 5 May, 1999 4 x^2 − 16 x + 16y^2 − 32 y = 32. (b) Where are the foci of this ellipse?^ (a) Where is the center of this ellipse? (d) Make a quick sketch of the ellipse on the axes below.(c) Where are the vertices of this ellipse?
(b) Where is f (x) increasing and where is it decreasing? (c) Where is f (x) concave up and where is concave down? (d) Find all extrema on the interval − 2 ≤ x ≤ 3.
Using differentials, approximate √80.