PastExamLabPastExamLab

成功大學 109 年度 微積分C

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第 1 題10 分
Find the following limits:
(a)5 分
lim⁡x→1ex−1x\lim\limits_{x \to 1} \frac{e^x - 1}{x}
(b)5 分
lim⁡x→+∞(x2+x+1+x)\lim\limits_{x \to +\infty} \left(\sqrt{x^2 + x + 1} + x\right)
第 2 題10 分
Find the tangent line to the curve defined by the equation 4x/y=x−y4^{x/y} = x - y when x=0x = 0.
第 3 題10 分
Compute the definite integral
∫042∫x2/82eyydydx.\int_0^{4\sqrt{2}} \int_{x^2/8}^2 \frac{e^y}{\sqrt{y}} dy dx.
第 4 題10 分
Find the arc length of the polar curve r=θr = \theta with 0≤θ≤π0 \leq \theta \leq \pi.
第 5 題10 分
Find the volume of the solid of revolution about the xx-axis bounded by the curve f(x)=1xx2+1f(x) = \frac{1}{x\sqrt{x^2 + 1}} with x∈[1,∞)x \in [1, \infty).
第 6 題10 分
Consider the power series ∑n=1∞(−3)n(x+2)nn+1\sum\limits_{n=1}^{\infty} \frac{(-3)^n(x+2)^n}{\sqrt{n+1}}. Find its maximal domain of convergence.
第 7 題10 分
Recall that the hyperbolic tangent function
tanh⁡x=ex−e−xex+e−x,x∈R\tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}}, x \in \mathbb{R}
is one to one and has the inverse function f(x)=tanh⁡−1xf(x) = \tanh^{-1} x. Find the Maclaurin series of f(x)=tanh⁡−1xf(x) = \tanh^{-1} x.
第 8 題10 分
Find the absolute maximum of f(x,y)=e−xyf(x, y) = e^{-xy} on the region x2+4y2≤1x^2 + 4y^2 \leq 1.
第 9 題10 分
Evaluate the line integral ∫Cy2dx−x3dy\int_C y^2 dx - x^3 dy, where CC is the circle x2+y2=8x^2 + y^2 = 8 with positive orientation.
第 10 題10 分
Let F(x,y,z)=xi+yj−2zk\mathbf{F}(x, y, z) = x\mathbf{i} + y\mathbf{j} - 2z\mathbf{k}. Suppose that the closed surface SS is defined by the paraboloid y=x2+z2y = x^2 + z^2 with 0≤y≤10 \leq y \leq 1. Find the flux of F\mathbf{F} outward across SS.
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