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題組內容

2.

(a) (10%) For a certain real number a, the following integral
converges. Determine a and evaluate the integral.

其他申論題

8. (10 pts)Find the tangent and normal to the curve x³ + y³ - 9xy = 0 at point (2, 4).

9. (12 pts)Find the horizontal tangents to the graph of the cardioid r=1-cosθ, 0 ≤ θ ≤ 2π. (That is, find the points at which horizontal lines tangent to the curve.)

(a) (5%) ∫ dx

(b) (5%) ∫ dx

(b) (10%) For what values of the constants b and cwill the following limit exist and be equal to 1? dx .

3. (15%) If dis an arbitrary real number, let Sn(d) =. Determine the following limit: .

4. (10%) Let f(x) be a polynomial of degree m. Show that ∫exf(x)dx = ex(f(x) - f'(x) + f"(x) - ... + (-1)mf(m)(x))

5. (15%) Evaluate the following integral along the curve C = {(x,y,z) ∈ R³ | x² + y² = 4, z = -3}, oriented counterclockwise as seen by a person standing at the origin and with respect to the right-handed Cartesian coordinates, where F = yi + xz³j – zy³k, r' = dr/ds is the unit tangent vector, and *sis the arc length of C

6. (15%) Evaluate the following integral I = ∫L(2xyz²) dx + (x²z² + z cos(yz)) dy + (2x²yz + y cos(yz)) dzalong the line segment L from P: (0,0,1) to Q: (2,π/4,2)

7. (15%) Compute the area of the hemisphere of radius t>0 and centered at the origin given by S = {(x,y,z) ∈ R³| x² + y² + z² = t², z≥0}

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