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台灣聯合大學系統 108 年度 微積分 A2

原始 PDF
壓縮檔內：轉學考/微積分A2.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。
第 1 題8 分
Find the average value of g(x)=∣x∣−1g(x) = |x| - 1 on [−1,3][-1, 3]. Answer: __________
第 2 題8 分
Find ∫xsec⁡2x dx\int x \sec^2 x \, dx. Answer: __________
第 3 題8 分
Evaluate ∫02∫y/21yex3 dxdy\int_0^2 \int_{y/2}^1 ye^{x^3} \, dx dy. Answer: __________
第 4 題8 分
Find the values of aa and bb that makes the function f(x)={2sin⁡2xx,if x>0ax+bcos⁡x,if x≤0f(x) = \begin{cases} \frac{2\sin^2 x}{x}, & \text{if } x > 0 \\ ax + b \cos x, & \text{if } x \leq 0 \end{cases} differentiable at x=0x = 0. Answer: __________
第 5 題8 分
Find the tangent line to the curve x2cos⁡2y−sin⁡y=0x^2 \cos^2 y - \sin y = 0 at (0,π)(0, \pi). Answer: __________
第 6 題8 分
Find the volume of the solid obtained by revolving the region bounded by the curves y=−x2+4xy = -x^2 + 4x and y=x2y = x^2 about the xx-axis. Answer: __________
第 7 題8 分
Evaluate lim⁡x→0+(sin⁡x)x\lim\limits_{x \to 0^+} (\sin x)^x. Answer: __________
第 8 題8 分
Find the sum of the series ∑n=3∞ln⁡(1+1n)(ln⁡n)ln⁡(n+1)\sum\limits_{n=3}^{\infty} \frac{\ln(1 + \frac{1}{n})}{(\ln n) \ln(n + 1)}. Answer: __________

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
An open rectangular box is to be constructed from material that costs $3/ft2\$3/\text{ft}^2 for the bottom and $1/ft2\$1/\text{ft}^2 for its sides. Find the dimensions of the box of greatest volume that can be constructed for $36\$36.
第 2 題12 分
Use the limit definition to show that g′(0)g'(0) exists but g′(0)≠lim⁡x→0g′(x)g'(0) \neq \lim\limits_{x \to 0} g'(x), where g(x)={x2sin⁡1x,if x≠00,if x=0.g(x) = \begin{cases} x^2 \sin \frac{1}{x}, & \text{if } x \neq 0 \\ 0, & \text{if } x = 0. \end{cases}
第 3 題12 分
Determine if the series converges or diverges.
(a. (6 分))6 分
∑n=0∞e−n2\sum\limits_{n=0}^{\infty} e^{-n^2}
(b. (6 分))6 分
∑n=1∞sin⁡1n\sum\limits_{n=1}^{\infty} \sin \frac{1}{n}