1. Determine whether the series converges or diverges.
(a) n=2Σ∞ 1/√(n ln n)
(b) n=2Σ∞ 1/(ln n)2
(c) n=2Σ∞ cosn/n2ln(n+cosn)
2. Calculate the limits or show that they don't exist: (a) limx→0,y→0 exy-(1+xy)/x2+y2. (b) limx→0,y→0 x2/x2+2y2.
3. A particle's path is described by the position vector r→(t) = (t2 + 1, (1 + t)2, t).
(a) Determine the velocity vector at t = 1.
(b) Determine an equation for the line going through the point (2, 4, 1) in the direction of the velocity vector at t = 1.
4. Let f(x, y) be a function depending on x, y. Let x0, y0, a1 ≠ 0, a2 ≠ 0 be constants. Suppose x(t) = x0 + a1t, y(t) = y0 + a2t. Let w(t) = f(x(t), y(t)).
(a) Determine w'(0) in terms of f and its derivatives.
(b) Determine w''(0) in terms of f and its derivatives.
(c) Write down the first three terms in the Maclaurin series for w(t) in terms of f and its derivatives and x, y (no t is allowed, note that t = (x - x0)/a1 = (y-y0)/a2).
5. Consider the function G(x, y, z) = x2y2 + yz2 + zx2.
(a) Compute the gradient vector ∇G.
(b) Determine the tangent plane at (1, 2, -2) for the level surface G(x, y, z) = 10.
(c) Find the distance from (1, 0, 1) to the tangent plane determined in (b).
6. Find the critical points for f(x, y) = (x2/2) + xy - (y3/3). For each critical points, determine whether it is a local max, local min, or a saddle point.
7. Find the absolute maximum and minimum values of f(x, y) = x2 - y2 - 2x + 4y + 1 in the rectangular region in the first quadrant bounded by the coordinate axes and the lines x = 4 and y = 2.