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研究所、轉學考(插大)-微積分
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99年 - 99 國立政治大學轉學生招生考試_應用數學系：微積分(二)#122370
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題組內容
1. Evaluate the following integrals:
(a) (5%) ∫
dx
其他申論題
6. (12 pts)The usual way to evaluate the improper integral I = dx is first to calculate its square: I² = dxdy. Evaluate the integral and solve the resulting equation for I.
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7. (10 pts)Find the values of ∂z/∂x and ∂z/∂y at point (x,y,z)=(1, 1, 1) and z³ - xy + yz + y³ - 2 = 0.
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8. (10 pts)Find the tangent and normal to the curve x³ + y³ - 9xy = 0 at point (2, 4).
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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.)
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(b) (5%) ∫ dx
#521191
(a) (10%) For a certain real number a, the following integral converges. Determine a and evaluate the integral.
#521192
(b) (10%) For what values of the constants b and cwill the following limit exist and be equal to 1? dx .
#521193
3. (15%) If dis an arbitrary real number, let Sn(d) =. Determine the following limit: .
#521194
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))
#521195
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
#521196
dx 
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dx