Need help with these. Detailed development please and do not copy answers from elsewhere, please.

(1) Consider the function f : R3 -> R given by zk (ex2+y? - 1) (ac2 + 92 + 22) k if (x, y, z) # (0, 0, 0), f(x, y, z ) = 0 if ( x, y, z) = (0, 0, 0), where k is a positive constant. (a) Find all values of k for which f is continuous at the origin. In other words, find all positive real numbers k for which lim zk (ex2+y? - 1) (x,y,z) -0 (x2 + 32 + 22) k . = 0. Hint: if you want, you can use the fact that et - 1 ~ t, as t - 0. (b) Find all values of k for which the partials fx (0, 0, 0), fy (0, 0, 0) and fz (0, 0, 0) exist. For each such value, compute fx (0, 0, 0), fy (0, 0, 0) and fz (0, 0, 0).(2) Consider the function f : R4 > R2 given by at/,2, 10) = (1 + 22 - zyw + ism'1, (1 + .11)Sin(w2 - 2%))- (3) Find the quadratic approximation of f at the point P = (0, 0, 0, 0), and use this approximation to obtain an estimate for f(0.1, 0.1, 0.1, 0.1). (b) Now consider the function g : R2 4 R3 given by g(:1:,y) = (mew, tan(2 + yz ID), ycos(:1:y)). We can compose the maps f and gto obtain a smooth function g o f : R4 ) R3. Use the chain rule to compute DP (9 o f), where P = (0, 0,0,0)