PastExamLabPastExamLab

成功大學 112 年度 微積分A

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第 1 題10 分
Evaluate the limit.
lim⁡x→44−x2−x\lim\limits_{x \to 4} \frac{4-x}{2-\sqrt{x}}
第 2 題10 分
Find the point on the curve y=xy = \sqrt{x} that is closest to the point (1,0)(1,0).
第 3 題10 分
If f(x)=∫0x1+t2dtf(x) = \int_0^x \sqrt{1+t^2}dt, find (f−1)′(0)(f^{-1})'(0).
第 4 題10 分
Find the length of the polar curve r=sin⁡3(θ/3)r = \sin^3(\theta/3), 0≤θ≤π0 \leq \theta \leq \pi.
第 5 題10 分
Find the radius of convergence and interval of convergence of the power series
∑n=1∞(−1)n2nnxn\sum\limits_{n=1}^{\infty} \frac{(-1)^n 2^n}{\sqrt{n}} x^n
第 6 題10 分
Find the unit tangent vector and the unit normal vector for the curve r(t)=cos⁡3ti+sin⁡3tj+4tk\mathbf{r}(t) = \cos 3t\mathbf{i} + \sin 3t\mathbf{j} + 4t\mathbf{k}.
第 7 題10 分
Find equations of the tangent plane and the normal line to the surface x+2y+3z=sin⁡(xyz)x+2y+3z = \sin(xyz) at the point (2,−1,0)(2,-1,0).
第 8 題10 分
Evaluate the integral ∫01/2∫3y4−y2x2ydxdy\int_0^{1/2} \int_{\sqrt{3y}}^{\sqrt{4-y^2}} x^2 y dx dy.
第 9 題10 分
Evaluate ∫CF⋅dr\int_C \mathbf{F} \cdot d\mathbf{r}, where F=⟨yexy,xexy⟩\mathbf{F} = \langle ye^{xy}, xe^{xy} \rangle and CC is the unit circle x2+y2=1x^2 + y^2 = 1.
第 10 題10 分
Evaluate ∬SF⋅dS\iint_S \mathbf{F} \cdot d\mathbf{S}, where F(x,y,z)=(z2x+ey)i+(x2y+cosh⁡z)j+(y2z+x)k\mathbf{F}(x,y,z) = (z^2x + e^y)\mathbf{i} + (x^2y + \cosh z)\mathbf{j} + (y^2z + x)\mathbf{k} and SS is the top half of sphere x2+y2+z2=1x^2 + y^2 + z^2 = 1.
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