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題組內容

4.

(a) (6 分) Find the limit of the sequence {aₙ}, where aₙ = .

其他申論題

2. (10 分) Find the length of the parameterized curve x = , 0 ≤ t ≤ 3.

(a) (10 分) Finddx.

(b) (10 分) Solve the differential equation .

(c) (10 分) Solve the differential equation 2y' = eˣ/² + y.

(b) (8 分) Find the radius and interval of convergence of the series .

(c) (6 分) Evaluate.

(a) (8 分) Find the value of at the point (1, 1, 1) if the equationxy + z³x - 2yz = 0defines z as a function of the two independent variables x and y and the partial derivative exists.

(b) (10 分) Find the derivative of f(x,y,z) = x³ - xy² - z at point P₀ (1,1,0) in the direction of the vector v = 2i - 3j + 6k.

7. (12 分) The usual way to evaluate the improper integraldx (from 0 to ∞) is first to calculate its square: dxdy (double integral from 0 to ∞ for both x and y). Evaluate the integral and solve the resulting equation for I.

一、王老師發現小明沒有和同學一起去上體育課，反而在教室裡偷取同學的財物。王老師該如何輔導小明？試舉出三項說明之。

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