1. What points on the ellipse x2+xy+2y2=1 are furthest from the origin? Hint: Finding then amounts to the same thing as maximizing some expression that has no square roots in it, subject to a condition. Lagrange multipliers works, and it works better if you proceed with the simplest functions f and g that will do the job.
2. Let f(x, y) = e-(x^2 + y^2).
(a) There is a region D1 in which the discriminant of f is positive. There is another region D2 in which ∂2f/∂x2 and ∂2f/∂y2 are both negative. Sketch both regions, and indicate which is which.
(b) Let E be the square -½ ≤ x ≤ ½, -½ ≤ y ≤ ½. Estimate ∫Ef(x,y) dA by cutting E into four smaller square. Take the centers for these for your points for a Riemami sum for ∫Ef(x,y) dA. Should the result be an overestimate, or an underestimate. of the actual integral. And why?(Use a calculator to do the evaluations and arrive a specific number. such as 0.6931 (though 1156 is wrong) 111 1115 Riemann sum.)