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6. (5分) Use the Ratio or Root Test to determine the convergence or divergence of the following series.

其他申論題

(a) (5分) ∫ x²ex3 dx

4. (10分) Let g(x) =f(t)dt, where f is a function whose graph is shown as Fig. 1. Calculate g(6) and g'(6).

5. (5分) Use the Integral test to determine the convergence or divergence of the following series.

7. (10分) Find the tangent line to the curve of intersection of the surfaces z = x² + y² and x + y + z = 33 at a point (1, 2, 5).

8. (10分) Find the maximum value of f(x, y) = 2x + 2xy + ysubject to the constraint 2x + y = 100 by Lagrange multiplier.

9. (20分) Using Green's Theorem, evaluateclockwise around the boundary curve C of the region R = {(x, y) ∈ R | 1 ≤ y ≤ 2 - x²}, where F = (x² + y²)i + (x² - y²)j. (10分) (請說明 Green's theorem 之公式) (5分) (畫出 R 之圖形)

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