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題組內容

1.

(a) (6 分) Determine whether the series converges absolutely or converges conditionally or diverges and give reasons for your answer.

其他申論題

5. Find the derivative of f(x,y,x) = xyz in the direction of the velocity vector of the helix r(t) = (cos3t)i + (sin 3t)j + 3tk at t = π/3.

6. Evaluate the integraldydx.

7. Evaluate the integral dA, where R is the region inside the upper semicircle of radius 2 centered at the origin, but outside the circle x2 + (y - 1)2 = 1.

8. Eraluate the integal

(b) (6 分) Find all values of x for whichconverges and give reasons for your answer.

(a) (4 分) Show that f is not continuous at (0, 0).

(b) (4 分) Find the partial derivative at (2,y) if x3 +y3≠0.

(c) (4分) Use the definition of the partial derivative to find af at (x, y) = (0,0).

3. Find the absolute maximum and minimum values of f(x,y) = x2 + 3y2 + 2y on the unit disk x2 + y2 ≤1.

1. Find the slope of the tangent line to the graph of f(x) =xlnx at the point (e,e).

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