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台灣聯合大學系統 112 年度 微積分 A3/A4/A6

原始 PDF
壓縮檔內：微積分A3A4A6.pdf
中央大學圖書館考古題頁

甲、填充題

共 8 題，每題 8 分，共 64 分。請在答案卷上列出題號依序作答。請注意：本（甲）部分，共 8 題，命題型態為填充題，請依題號順序獨立列出，勿同時條列出計算過程。倘若答案被包含在演算過程，將被視為試算流程，不予行計算計分。
第 1 題8 分
Find the limit lim⁡x→0+(cos⁡(2x))1x\lim\limits_{x \to 0^+} \left(\cos(2\sqrt{x})\right)^{\frac{1}{x}}.
第 2 題8 分
Find f′(−1)f'(-1) if f(x)=eg(x)f(x) = e^{g(x)} and g(x)=tan⁡−1(x2)+∫1x2sec⁡(t−1)dtg(x) = \tan^{-1}(x^2) + \int_1^{x^2} \sec(t-1) dt.
第 3 題8 分
Find the length of the curve x=ln⁡(sec⁡t+tan⁡t)−sin⁡tx = \ln(\sec t + \tan t) - \sin t, y=cos⁡ty = \cos t, 0≤t≤π/30 \leq t \leq \pi/3.
第 4 題8 分
Consider the region bounded by the graphs of y=ln⁡xy = \ln x, y=0y = 0, and x=ex = e. Find the volume of the solid formed by revolving the region about the xx-axis.
第 5 題8 分
Find the derivative of f(x,y,z)=xyzf(x, y, z) = xyz in the direction of the velocity vector of the helix r(t)=(cos⁡3t)i+(sin⁡3t)j+3tk\mathbf{r}(t) = (\cos 3t)\mathbf{i} + (\sin 3t)\mathbf{j} + 3t\mathbf{k} at t=π/3t = \pi/3.
第 6 題8 分
Evaluate the integral ∫08∫x21y4+1dydx\int_0^8 \int_{\sqrt{x}}^2 \frac{1}{y^4 + 1} dy dx.
第 7 題8 分
Evaluate the integral ∬Rx2+y2dA\iint_R \sqrt{x^2 + y^2} dA, where RR is the region inside the upper semicircle of radius 2 centered at the origin, but outside the circle x2+(y−1)2=1x^2 + (y-1)^2 = 1.
第 8 題8 分
Evaluate the integral ∫−14dx∣x∣\int_{-1}^4 \frac{dx}{\sqrt{|x|}}.

乙、計算、證明題

共 3 題，每題 12 分，共 36 分。須詳細寫出計算及證明過程，否則不予計分。
第 1 題12 分
Determine whether the series ∑n=2∞(−1)nln⁡nn−ln⁡n\sum\limits_{n=2}^\infty (-1)^n \frac{\ln n}{n - \ln n} converges absolutely or converges conditionally or diverges and give reasons for your answer.
(a)6 分
Determine whether the series ∑n=2∞(−1)nln⁡nn−ln⁡n\sum\limits_{n=2}^\infty (-1)^n \frac{\ln n}{n - \ln n} converges absolutely or converges conditionally or diverges and give reasons for your answer.
(b)6 分
Find all values of xx for which ∑n=1∞(x+4)nn3n\sum\limits_{n=1}^\infty \frac{(x+4)^n}{n3^n} converges and give reasons for your answer.
第 2 題12 分
Let f(x,y)={x2yx3+y3,x3+y3≠00,x3+y3=0f(x,y) = \begin{cases} \frac{x^2y}{x^3+y^3}, & x^3 + y^3 \neq 0 \\ 0, & x^3 + y^3 = 0 \end{cases}
(a)4 分
Show that ff is not continuous at (0,0)(0,0).
(b)4 分
Find the partial derivative ∂f∂x\frac{\partial f}{\partial x} at (x,y)(x,y) if x3+y3≠0x^3 + y^3 \neq 0.
(c)4 分
Use the definition of the partial derivative to find ∂f∂x\frac{\partial f}{\partial x} at (x,y)=(0,0)(x,y) = (0,0).
第 3 題12 分
Find the absolute maximum and minimum values of f(x,y)=x2+3y2+2yf(x,y) = x^2 + 3y^2 + 2y on the unit disk x2+y2≤1x^2 + y^2 \leq 1.