PastExamLabPastExamLab

成功大學 105 年度 微積分C

PDF

一、填充題(不需計算過程)請於答案卷上作答，否則不予計分

第 1 題12 分
Find the following limits:
(a)6 分
For a,b,c>0a, b, c > 0, lim⁡x→0(ax+bx+cx3)1x=\lim\limits_{x \to 0} \left(\frac{a^x + b^x + c^x}{3}\right)^{\frac{1}{x}} =
(b)6 分
Denote ∏k=1nAk=A1A2⋯An\prod_{k=1}^n A_k = A_1 A_2 \cdots A_n. Evaluate lim⁡n→∞∏k=1n(1+kn2)=\lim\limits_{n \to \infty} \prod_{k=1}^n \left(1 + \frac{k}{n^2}\right) =
第 2 題5 分
Suppose that the function f(x)={ax+b3x+1−x+3,x≠14,x=1f(x) = \begin{cases} \frac{ax + b}{\sqrt{3x + 1} - \sqrt{x + 3}} & , x \neq 1 \\ 4 & , x = 1 \end{cases} is continuous at x=1x = 1. Then (a,b)=(a, b) =
第 3 題5 分
Define f(x)=tan⁡3xf(x) = \tan^3 x on (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) and let f−1f^{-1} be the inverse function of ff. Find (f−1)′(33)=\left(f^{-1}\right)'(3\sqrt{3}) =
第 4 題12 分
Compute the integrals
(a)6 分
∫ln⁡x(1−x)2dx=\int \frac{\ln x}{(1 - x)^2} dx =
(b)6 分
∫1∞1ex+1+e3−xdx=\int_1^{\infty} \frac{1}{e^{x+1} + e^{3-x}} dx =
第 5 題6 分
Let hh be a differentiable function of xx and yy and let f(r,s)=h(rs,r+s)f(r, s) = h(rs, r + s). Assume that ∂h∂x(6,5)=1\frac{\partial h}{\partial x}(6, 5) = 1 and ∂h∂y(6,5)=2\frac{\partial h}{\partial y}(6, 5) = 2. Find ∂f∂s(2,3)=\frac{\partial f}{\partial s}(2, 3) =

二、計算題(無計算過程不給分)

第 6 題10 分
Find the general solution of the differential equation (x2+1)y′+2xy=4x2(x^2 + 1)y' + 2xy = 4x^2
第 7 題10 分
Consider the series ∑k=10∞1kln⁡k[ln⁡(ln⁡k)]p\sum\limits_{k=10}^{\infty} \frac{1}{k \ln k [\ln(\ln k)]^p}. Determine all values of pp such that the series converges.
第 8 題20 分
Let f(x)=11+x4f(x) = \frac{1}{\sqrt{1 + x^4}}.
(a)10 分
Find the Taylor series of f(x)f(x) at x=0x = 0. (Need to write down the general form.)
(b)10 分
Find the interval of convergence of the Taylor series in Problem(a).
第 9 題10 分
Let f(x,y)=x2+y2−12x+16yf(x, y) = x^2 + y^2 - 12x + 16y. Find the maximum of ff on the set {(x,y)∣x2+y2≤25}\{(x, y) | x^2 + y^2 \leq 25\}.
第 10 題10 分
Let EE be the solid cone bounded below by z=x2+y2z = \sqrt{x^2 + y^2} and above by z=2z = 2. Let F(x,y,z)=xi+yj+zk\mathbf{F}(x, y, z) = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}. Evaluate ∭EdivF dV\iiint_E \text{div}\mathbf{F} \, dV
廣告區域 (Google AdSense)