Reference no: EM131004099

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.)