Reference no: EM131072926

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) lim_x→0,y→0 e^xy-(1+xy)/x²+y². (b) lim_x→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.