Let H be the set of all points (x, y) in ℝ2 such that x2 + xy 3y2 = 3. Show that H is a closed subset of ℝ2(using Euclidean metric). Is H bounded?

For each of the sets find the set of boundary points and the set of interior points. Also state if the set is open (explain), closed (explain), bounded, and compact (explain). (a) [0, 1] U {4} (b) {(x, y) = R² : ² + y² = 1}

Consider the 3D domain W = {(x, y, 2)| 2 2 > 0}. ;, y, z)| 2 2z > 0}. 2. Calculate x2 + y? + z? d(x, y, z) using / || dpdø d0 set up. W

The concept of the linear approximation builds from the fact that differentiable functions are locally a)smooth b)concave  c)convex d)linear

Suppose that the set A={X₁, X2..., Xn}, X₁ ER" spans the entire space R", i.e. for any arbitrary n -vector y E R",3 a set of coefficients {t₁, t₂,... tn} with at least one t; #0 such that 1tixi = y. Prove that the coefficients t₁, t₂,.., t are unique.

Let X=ℝ2 and define d2,:ℝ2×ℝ2→ℝ by d2((x1 ,y1),(x2,y2)) = max{|x1 - x2|,|y1 - y2|}. a) Verify that d2 is a metric on ℝ2. b.) Draw the neighborhood N((0; 1) for d2, where 0 is the origin in ℝ2.

Let d: R2 × R2 →R given by xỉ + yỉ + x3 + y½ , if (x1,y1) # (x2, Y2) 0, if (x1, Y1) = (x2, Y2) d((x1.y1).(x2.y2))= { (a) Show that d is a metric. (b) Show that (R2, d) is a complete metric space. (c) Show that (R2, d no a connected metric space