3. Let d₂(x, y) = |arctan(x) — arctan(y)| for x, y € R. (a) Show that d₂ is a metric on R. (b) Are de and d₂ equivalent?

Please explain steps in details: Density of water is approximately 1000 kg/m^3 and gravity is 9.81 m/s^2. The function y = x^2 from x = 0 to 2 is rotated about the y-axis. 1) Find the volume of the resulting solid 2) If the resulting solid is filled with water, find the work done in emptying it - assume all units of measurement are in meters. Do we subtract? 3) Find the surface area of the resulting shape (set up the integral, and find a numerical approximation)

Let f: R2 → R. At some point, a = · (~69), v we have that f(a) = 3 and that Vf(a) = (²). What is the equation of the plane tangent to f at the point a? Enter your answer as a function of x and y: 2 =

get, t, n and kappa for the spatial curves of the exercises