PastExamLabPastExamLab

台灣大學 · 數學系 · 轉學考考古題 · 民國111年（2022年）

111 年度 微積分(A)

台灣大學 · 數學系 · 轉學考

PDF
第 1 題20 分
(a)5 分
Assume that f′(a)>0f'(a) > 0. Use the definition of derivative to prove that there is some ϵ>0\epsilon > 0 such that f(x)>f(a)f(x) > f(a) for all x∈(a,a+ϵ)x \in (a, a + \epsilon).
(b)15 分
Suppose that f(x)f(x) is differentiable on an open interval containing [a,b][a, b] with f′(a)≠f′(b)f'(a) \neq f'(b) and mm is a constant between f′(a)f'(a) and f′(b)f'(b). Prove that there is some c∈(a,b)c \in (a, b) such that f′(c)=mf'(c) = m.
第 2 題15 分
Let {fn(x)}\{f_n(x)\} be a sequence of continuous functions defined on [0,1][0, 1]. Assume that fn(x)f_n(x) converges to f(x)f(x) uniformly on [0,1][0, 1].
(a)9 分
Prove that f(x)f(x) is continuous on [0,1][0, 1] and there is some M>0M > 0 such that ∣fn(x)∣<M|f_n(x)| < M for all x∈[0,1]x \in [0, 1] and positive integer nn.
(b)6 分
Prove or disprove lim⁡n→∞∫01−1/nfn(x)dx=∫01f(x)dx\lim\limits_{n\to\infty} \int_0^{1-1/n} f_n(x) dx = \int_0^1 f(x) dx.
第 3 題15 分
(a)9 分
Suppose that 1<b<21 < b < 2 is a fixed constant and an=(1−b2)(1−b3)⋯(1−bn)a_n = (1 - \frac{b}{2})(1 - \frac{b}{3})\cdots(1 - \frac{b}{n}), for n≥2n \geq 2. Show that there are constants 0<m<M0 < m < M such that m<nban<Mm < n^b a_n < M for all n≥2n \geq 2.
(b)6 分
Show that for k>0k > 0, the series ∑n=1∞(kn)\sum_{n=1}^{\infty} \binom{k}{n} converges absolutely.
第 4 題15 分
Suppose that f(x,y)f(x, y), g(x,y)g(x, y) have continuous second derivatives, and on the level curve g(x,y)=0g(x, y) = 0, f(x,y)f(x, y) obtains a local extreme value at (1,2)(1, 2). Assume that at (1,2)(1, 2), fx=3f_x = 3, gx=−1g_x = -1, gy=12g_y = \frac{1}{2}, fxx=0f_{xx} = 0, fxy=1f_{xy} = 1, fyy=−2f_{yy} = -2, gxx=3g_{xx} = 3, gxy=0g_{xy} = 0, and gyy=1g_{yy} = 1.
(a)2 分
Find fy(1,2)f_y(1, 2).
(b)7 分
When restricted to the curve g(x,y)=0g(x, y) = 0, is f(1,2)f(1, 2) a local maximum, local minimum, or saddle point?
(c)6 分
Suppose that on the curve g(x,y)=10−2g(x, y) = 10^{-2}, f(x,y)f(x, y) has a local extreme value at (x1,y1)(x_1, y_1) which is close to (1,2)(1, 2). Estimate f(x1,y1)−f(1,2)f(x_1, y_1) - f(1, 2) by linear approximation.
第 5 題15 分
Suppose that F(x,y,u,v)F(x, y, u, v) and G(x,y,u,v)G(x, y, u, v) have continuous first partial derivatives and equations F(x,y,u,v)=0F(x, y, u, v) = 0 and G(x,y,u,v)=0G(x, y, u, v) = 0 can be solved for x,yx, y as functions of u,vu, v. Express ∂(x,y)∂(u,v)\frac{\partial(x, y)}{\partial(u, v)} in terms of ∂(F,G)∂(u,v)\frac{\partial(F, G)}{\partial(u, v)} and ∂(F,G)∂(x,y)\frac{\partial(F, G)}{\partial(x, y)} where ∂(x,y)∂(u,v)=∣xuxvyuyv∣\frac{\partial(x, y)}{\partial(u, v)} = \begin{vmatrix} x_u & x_v \\ y_u & y_v \end{vmatrix}, ∂(F,G)∂(u,v)=∣FuFvGuGv∣\frac{\partial(F, G)}{\partial(u, v)} = \begin{vmatrix} F_u & F_v \\ G_u & G_v \end{vmatrix}, ∂(F,G)∂(x,y)=∣FxFyGxGy∣\frac{\partial(F, G)}{\partial(x, y)} = \begin{vmatrix} F_x & F_y \\ G_x & G_y \end{vmatrix} are Jacobian determinants.
第 6 題20 分
Let F(x,y,z)=x2+2y2−3z2F(x, y, z) = x^2 + 2y^2 - 3z^2 and SS be the level surface F(x,y,z)=2F(x, y, z) = 2 between z=0z = 0 and y−3x+1=0y - \sqrt{3x + 1} = 0 with downward orientation. Compute ∬S∇F⋅dS\iint_S \nabla F \cdot dS.
廣告區域 (Google AdSense)