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成功大學 109 年度 微積分A

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第 1 題10 分
Evaluate the following limits if they exist.
(a)
lim⁡n→∞3n+22n+1\lim\limits_{n \to \infty} \frac{3n+2}{2n+1}
(b)
lim⁡x→0cos⁡(2x)−1x2\lim\limits_{x \to 0} \frac{\cos(2x)-1}{x^2}
第 2 題10 分
A curve in R2\mathbb{R}^2 is given parametrically by
x=t2+2t+3x = t^2 + 2t + 3
y=t4−3t3y = t^4 - 3t^3
for all t>0t > 0. Find dydx\frac{dy}{dx} at the point (6,−2)(6, -2).
第 3 題10 分
Let x>0x > 0 and △ABC\triangle ABC be a triangle whose side lengths are BC→=5\overrightarrow{BC} = 5, AC→=4\overrightarrow{AC} = 4, AB→=3\overrightarrow{AB} = 3. Choose a point PP on AB→\overrightarrow{AB}, and a point QQ on BC→\overrightarrow{BC}, and a point RR on AC→\overrightarrow{AC} so that
BP→AP→=CQ→BQ→=AR→CR→=x\frac{\overrightarrow{BP}}{\overrightarrow{AP}} = \frac{\overrightarrow{CQ}}{\overrightarrow{BQ}} = \frac{\overrightarrow{AR}}{\overrightarrow{CR}} = x
Let f(x)f(x) be the area of △PQR\triangle PQR. Find the critical point and the minimum of f(x)f(x).
第 4 題10 分
Find the radius of the convergence of the power series ∑n=0∞(2n)nn!xn\sum\limits_{n=0}^{\infty} \frac{(2n)^n}{n!} x^n.
第 5 題10 分
Evaluate the improper integral
∫0π2cos⁡x(ln⁡cos⁡x2+ln⁡sin⁡x2)dx\int_0^{\frac{\pi}{2}} \cos x \left( \ln \cos \frac{x}{2} + \ln \sin \frac{x}{2} \right) dx
第 6 題10 分
Let g:(0,∞)→Rg : (0,\infty) \to \mathbb{R} be a twice differentiable function. Assume that
g(1)=1,g′(1)=3,g′′(1)=−4.g(1) = 1, \quad g'(1) = 3, \quad g''(1) = -4.
Define a real valued function hh on R3∖{(0,0,0)}\mathbb{R}^3 \setminus \{(0,0,0)\} by
h(x,y,z)=g(x2+y2+z2)h(x,y,z) = g\left(\sqrt{x^2 + y^2 + z^2}\right)
Calculate ∂2h∂x2(P)+∂2h∂z2(P)+∂2h∂z2(P)\frac{\partial^2 h}{\partial x^2}(P) + \frac{\partial^2 h}{\partial z^2}(P) + \frac{\partial^2 h}{\partial z^2}(P) where P=(23,23,−13)P = \left(\frac{2}{3}, \frac{2}{3}, -\frac{1}{3}\right).
第 7 題10 分
Let SS be the surface defined by the equation
xcos⁡(xy)+z2y4−7xz=1x \cos(xy) + z^2 y^4 - 7xz = 1
and P(0,1,1)P(0,1,1) be a point on SS. Find an equation that defines the tangent plane to SS at PP and a parametric equation of the normal line to SS at PP.
第 8 題10 分
Evaluate the double integral
∬R(y−x)dA\iint_R (y-x) dA
where R={(x,y)∈R2:1≤x2+y2≤4,x≥0}R = \{(x,y) \in \mathbb{R}^2 : 1 \leq x^2 + y^2 \leq 4, x \geq 0\}.
第 9 題10 分
Let CC be the curve in R3\mathbb{R}^3 defined by the parametric equation
x(t)=cos⁡(t),y(t)=sin⁡(t),z(t)=tx(t) = \cos(t), \quad y(t) = \sin(t), \quad z(t) = t
for 0≤t≤a0 \leq t \leq a. Suppose that the arc length of CC is 2π4\frac{\sqrt{2}\pi}{4}. Evaluate the line integral ∫CF⋅dr\int_C \mathbf{F} \cdot d\mathbf{r} of the vector field F=xzi+yzj+x3k\mathbf{F} = xz\mathbf{i} + yz\mathbf{j} + x^3\mathbf{k} on R3\mathbb{R}^3.
第 10 題10 分
Find the flux of the vector field F\mathbf{F} on R3\mathbb{R}^3 defined by
F=3xi+2yj+5zk\mathbf{F} = 3x\mathbf{i} + 2y\mathbf{j} + 5z\mathbf{k}
through the surface S={(x,y,1−x2−y2)∈R3:x2+y2≤1}S = \left\{(x,y,\sqrt{1-x^2-y^2}) \in \mathbb{R}^3 : x^2 + y^2 \leq 1\right\} oriented with upward pointing normal vector field. 備註：i=(1,0,0)\mathbf{i} = (1,0,0), j=(0,1,0)\mathbf{j} = (0,1,0), k=(0,0,1)\mathbf{k} = (0,0,1).
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