25-26-2-高等数学A（下）-期末（国际学院）

Q1 (40 marks, 4 marks for each)

  1. The series ∑n=1∞(−1)nn\sum_{n=1}^{\infty} \frac{(-1)^n}{n} is 【暂无答案】. (Fill in: absolutely convergent / conditionally convergent / divergent)

  2. The radius of convergence for the power series ∑n=3∞ln⁡nnxn\sum_{n=3}^{\infty} \frac{\ln n}{n} x^n is 【暂无答案】.

  3. If f(x,y)=x3y2f(x,y) = x^3y^2, then fxy(x,y)=f_{xy}(x,y) = 【暂无答案】.

  4. The equation of the tangent plane to the surface xyz=1xyz = 1 at the point (1,1,1)(1, 1, 1) is 【暂无答案】.

  5. ∬(R)y dσ=\iint_{(R)} y \,\mathrm{d}\sigma = 【暂无答案】, where (R)={(x,y)∣−1≤x≤1,−2≤y≤2}(R) = \{(x,y) | -1 \le x \le 1, -2 \le y \le 2\}.

  6. ∬(D)(x2+y2) dσ=\iint_{(D)} (x^2 + y^2) \,\mathrm{d}\sigma = 【暂无答案】, where (D)={(x,y)∣x2+y2≤4}(D) = \{(x,y) | x^2 + y^2 \le 4\}.

  7. ∭(V)2z dV=\iiint_{(V)} 2z \,\mathrm{d}V = 【暂无答案】, where (V)={(x,y,z)∣x2+y2≤1,0≤z≤1}(V) = \{(x,y,z) | x^2 + y^2 \le 1, 0 \le z \le 1\}.

  8. ∫(C) ds=\int_{(C)} \,\mathrm{d}s = 【暂无答案】, where (C)(C) is the line segment from (0,0,0)(0,0,0) to (1,1,1)(1,1,1).

  9. ∬(S)x dS=\iint_{(S)} x \,\mathrm{d}S = 【暂无答案】, where (S)={(x,y,z)∣x2+y2+z2=1}(S) = \{(x,y,z) | x^2 + y^2 + z^2 = 1\}.

  10. Let F(x,y,z)=(arctan⁡(y3z)+z2,sin⁡(x5+z),xyz2)F(x,y,z) = \left(\arctan(y^3z) + z^2, \sin(x^5 + z), xyz^2\right). Then the divergence ∇⋅F=\nabla \cdot F = 【暂无答案】.

Q2 (12 marks, 6 marks for each)

a) Find the Fourier series of the periodic function with period 2π2\pi defined by f(x)={0,−π≤x<0−1,0≤x<πf(x) = \begin{cases} 0, & -\pi \le x < 0 \\ -1, & 0 \le x < \pi \end{cases} on the interval [−π,π)[-\pi, \pi), and write down its sum function.

b) Suppose that the function y=y(x)y = y(x) and z=z(x)z = z(x) are determined by the equations x2+2y2+3z2=10x^2 + 2y^2 + 3z^2 = 10 and x2+y2−z=0x^2 + y^2 - z = 0. Find dydx\frac{\mathrm{d}y}{\mathrm{d}x} and dzdx\frac{\mathrm{d}z}{\mathrm{d}x}.

Q3 (10 marks, 5 marks for each)

a) Find the extremum values of the function f(x,y)=3xy−x3−y3f(x,y) = 3xy - x^3 - y^3.

b) Find the volume of the region (VV) enclosed by the surfaces z=x2+3y2z = x^2 + 3y^2 and z=8−x2−y2z = 8 - x^2 - y^2.

Q4 (12 marks, 6 marks for each)

a) Evaluate the integral ∬(D)xy dσ\iint_{(D)} x\sqrt{y} \,\mathrm{d}\sigma, where (D)(D) is enclosed by the curves y=xy = \sqrt{x} and y=x2y = x^2.

b) Evaluate the integral ∭(V)zx2+y2 dV\iiint_{(V)} z \sqrt{x^2 + y^2} \,\mathrm{d}V, where (V)(V) is the region in the first octant bounded by the cylinder x2+y2−2x=0x^2 + y^2 -2x = 0 and the planes y=0y = 0, z=0z = 0, z=1z = 1.

Q5 (10 marks, 5 marks for each)

a) Evaluate ∫(C)y ds\int_{(C)} y \,\mathrm{d}s, where (C)(C) is the cycloid given by x=t−sin⁡(t)x = t - \sin(t), y=1−cos⁡(t)y = 1 - \cos(t), 0≤t≤2π0 \le t \le 2\pi.

b) Evaluate the integral ∫(C)y dx+x dy\int_{(C)} y \,\mathrm{d}x + x \,\mathrm{d}y, where (C)(C) is the curve given by y=x2+x(x−1)sin⁡(x2)y = x^2 + x(x - 1)\sin(x^2), from (0,0)(0,0) to (1,1)(1,1).

Q6 (10 marks, 5 marks for each)

a) Compute the surface integral ∬(Σ)(x+y+z) dS\iint_{(\Sigma)} (x+y+z) \,\mathrm{d}S, where (Σ)(\Sigma) is the part of the plane x+y+z=1x + y + z = 1 cut by the cylinder x2+y2=1x^2 + y^2 = 1.

b) Compute the surface integral ∬(Σ)(x3+y) dy∧dz+(y3+z) dz∧dx+(z3+x) dx∧dy\iint_{(\Sigma)} (x^3 + y) \,\mathrm{d}y\wedge\mathrm{d}z + (y^3 + z) \,\mathrm{d}z\wedge\mathrm{d}x + (z^3 + x) \,\mathrm{d}x\wedge\mathrm{d}y, where (Σ)(\Sigma) is the upper hemisphere {(x,y,z)∣x2+y2+z2=1,z≥0}\{(x,y,z) \mid x^2 + y^2 + z^2 = 1, z \ge 0\}, oriented outward. (i.e., the normal vector points away from the origin.)

Q7 (6 marks)

Let (D)={(x,y)∣x2+y2≤1}(D) = \{(x,y) | x^2 + y^2 \le 1\}, and let ∂D={(x,y)∣x2+y2=1}\partial D = \{(x,y) | x^2 + y^2 = 1\}. Suppose that f∈C1(D‾)f \in C^1(\overline{D}) is a function satisfying the boundary condition f∣∂D=0f\mid_{\partial D} = 0. Prove the following inequality:

∬(D)∣f(x,y)∣ dx dy≤π3max⁡(x,y)∈(D)(∂f∂x)2+(∂f∂y)2.\iint_{(D)} |f(x,y)| \,\mathrm{d}x\,\mathrm{d}y \le \frac{\pi}{3} \max_{(x,y) \in (D)} \sqrt{\left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2}.