PastExamLabPastExamLab

成功大學 112 年度 微積分C

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第 1 題10 分
Evaluate the limit
lim⁡x→1x−1x−1.\lim\limits_{x \to 1} \frac{\sqrt{x} - 1}{x - 1}.
第 2 題10 分
Evaluate
∫0π2sin⁡2xcos⁡x dx.\int_0^{\frac{\pi}{2}} \sin^2 x \cos x \, dx.
第 3 題10 分
Evaluate
∫011x2+2 dx.\int_0^1 \frac{1}{x^2 + 2} \, dx.
第 4 題10 分
Find the arc length of the cycloid
C:{x=r(θ−sin⁡θ)y=r(1−cos⁡θ),0≤θ≤π2.C: \begin{cases} x = r(\theta - \sin \theta) \\ y = r(1 - \cos \theta) \end{cases}, \quad 0 \leq \theta \leq \frac{\pi}{2}.
第 5 題10 分
Find the 5th derivative f(5)(0)f^{(5)}(0) of the function f(x)=ln⁡(1+x)⋅tan⁡−1xf(x) = \ln(1 + x) \cdot \tan^{-1} x.
第 6 題10 分
Find the maximum value of the function f(x,y,z)=x+2y+3zf(x, y, z) = x + 2y + 3z on the curve of intersection of the plane x−y+z=0x - y + z = 0 and the cylinder x2+y2=29x^2 + y^2 = 29.
第 7 題10 分
Find the surface area of the part of the paraboloid z=x2+y2z = x^2 + y^2 that lies between the planes z=1z = 1 and z=9z = 9.
第 8 題10 分
Evaluate
∭Be(x2+y2+z2)32 dV,\iiint_B e^{(x^2+y^2+z^2)^{\frac{3}{2}}} \, dV,
where B={(x,y,z)∣x2+y2+z2≤1}B = \{(x, y, z) | x^2 + y^2 + z^2 \leq 1\} is the unit ball.
第 9 題10 分
Evaluate
∮C(y2+sin⁡x) dx+(3xy−ey) dy\oint_C (y^2 + \sin x) \, dx + (3xy - e^y) \, dy
where CC is the boundary of the semiannular region DD in the upper half-plane between the circles x2+y2=4x^2 + y^2 = 4 and x2+y2=9x^2 + y^2 = 9.
第 10 題10 分
Evaluate ∬SF⋅dS\iint_S \mathbf{F} \cdot d\mathbf{S}, where
F(x,y,z)=xyi+(y2+exz)j+cos⁡(xy)k\mathbf{F}(x, y, z) = xy\mathbf{i} + (y^2 + e^{xz})\mathbf{j} + \cos(xy)\mathbf{k}
and SS is the surface of the region EE bounded by the parabolic cylinder z=1−x2z = 1 - x^2 and the planes z=0,y=0,y+z=2z = 0, y = 0, y + z = 2.
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