PastExamLabPastExamLab

台灣大學 · 資訊工程學系 · 轉學考考古題 · 民國113年（2024年）

113 年度 微積分(B)

台灣大學 · 資訊工程學系 · 轉學考

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PART 1: Fill in the blanks

Please ensure that each answer is clearly labeled with the corresponding blank number. Please note that only the final answers will be graded, and each blank is worth 5 points.
第 1 題50 分
(a)5 分
lim⁡x→0∣x−tan⁡x∣1∣x∣=\lim\limits_{x \to 0} |x - \tan x|^{\frac{1}{|x|}} = __(1)__.
(b)10 分
The 3rd degree Taylor polynomial of (1+3x)6(1 + 3x)^6 at x=0x = 0 is __(2)__. lim⁡x→∞(x2(1+3x)x−x2−3ex)=\lim\limits_{x \to \infty} \left( x^2 \left( 1 + \frac{3}{x} \right)^x - x^2 - 3ex \right) = __(3)__.
第 2 題15 分
Assume that the equation 2y2=x+y2y^2 = x + y defines yy as an implicit function of xx, denoted by y=f(x)y = f(x), near the point (x,y)=(0,2)(x,y) = (0,2). f′(0)=f'(0) = __(4)__. The tangent plane of the graph of g(x,y)=2y2−x−y+1g(x,y) = 2y^2 - x - y + 1 at (0,2,1)(0,2,1) is __(5)__.
第 3 題15 分
Compute the integrals. ∫14x2+4xdx=\int \frac{1}{\sqrt{4x^2 + 4x}} dx = __(6)__. ∫01x2(x2+1)2dx=\int_0^1 \frac{x^2}{(x^2 + 1)^2} dx = __(7)__. ∬DyxdA=\iint_D \frac{y}{x} dA = __(8)__, where DD is the region in the first quadrant bounded by x2+y2=14x^2 + y^2 = \frac{1}{4}, x2+y2=xx^2 + y^2 = x, y=xy = x, and y=0y = 0.
第 4 題10 分
Let F(x,y,z)=xzi+yzj+(−z+ey2)k\mathbf{F}(x,y,z) = xz\mathbf{i} + yz\mathbf{j} + (-z + ey^2)\mathbf{k} and SS be the part of the cylinder x2+y2=1x^2 + y^2 = 1 between planes z=0z = 0 and z=2+xz = 2 + x with outward orientation. A parametrization of the surface SS is __(9)__. The flux of F\mathbf{F} through SS is __(10)__.

PART 2:

Please solve the following problems and provide computations as well as explanations. Partial credits are allocated according to the level of completeness in your work.
第 1 題30 分
Suppose that f(x)f(x) is a function defined on R\mathbb{R} satisfying the following properties. ∣f(x)−f(0)−3x∣≤x2|f(x) - f(0) - 3x| \leq x^2 for ∣x∣≤1|x| \leq 1. f(x+y)=f(x)+f(y)+xy(x+y)f(x + y) = f(x) + f(y) + xy(x + y) for all x,y∈Rx, y \in \mathbb{R}.
(a)5 分
(5%) Find f(0)f(0) and f′(0)f'(0).
(b)7 分
(7%) Show that f(x)f(x) is differentiable and find f′(x)f'(x).
(c)8 分
(8%) Show that f(x)f(x) is one-to-one and find ddxf−1(x)∣x=103\frac{d}{dx} f^{-1}(x) \Big|_{x=\frac{10}{3}}.
(d)10 分
(10%) Show that for any a<ba < b, ∫f(a)f(b)f−1(x)dx=bf(b)−af(a)−∫abf(x)dx\int_{f(a)}^{f(b)} f^{-1}(x) dx = bf(b) - af(a) - \int_a^b f(x) dx. Find ∫0103f−1(x)dx\int_0^{\frac{10}{3}} f^{-1}(x) dx.
第 2 題20 分
(a)10 分
(10%) Find the maximum value of f(x,y,z)=zf(x,y,z) = z on the curve of the intersection of x+y+z=1x + y + z = 1 and x2+y2+z2=3x^2 + y^2 + z^2 = 3.
(b)5 分
(5%) f(x,y,z),g(x,y,z),h(x,y,z)f(x,y,z), g(x,y,z), h(x,y,z) are differentiable functions. Assume that ff obtains a local maximum value at (x0,y0,z0)(x_0, y_0, z_0) when restricted to g(x,y,z)=cg(x,y,z) = c and h(x,y,z)=kh(x,y,z) = k. It is known that ∇f(x0,y0,z0)=λ∇g(x0,y0,z0)+μ∇h(x0,y0,z0)\nabla f(x_0, y_0, z_0) = \lambda \nabla g(x_0, y_0, z_0) + \mu \nabla h(x_0, y_0, z_0) for some constants λ\lambda and μ\mu. Suppose that f(x,y,z)f(x,y,z) obtains new local maximum at (x1,y1,z1)(x_1, y_1, z_1) when restricted to g(x,y,z)=cg(x,y,z) = c and h(x,y,z)=k+ϵh(x,y,z) = k + \epsilon where ∣ϵ∣|\epsilon| is small and (x1,y1,z1)(x_1, y_1, z_1) is close to (x0,y0,z0)(x_0, y_0, z_0). Show by the linear approximation that we can approximate f(x1,y1,z1)−f(x0,y0,z0)f(x_1, y_1, z_1) - f(x_0, y_0, z_0) by μ⋅ϵ\mu \cdot \epsilon.
(c)5 分
(5%) Estimate, by linear approximation, the maximum value of f(x,y,z)=zf(x,y,z) = z on the curve of the intersection of x+y+z=1x + y + z = 1 and x2+y2+z2=3.02x^2 + y^2 + z^2 = 3.02.
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