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台灣聯合大學系統 113 年度 微積分 A3/A4/A6

原始 PDF
壓縮檔內：轉學考/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。請將答案卷上列出題號依序作答。
第 1 題8 分
Evaluate the limit lim⁡x→21x−2∫4x2sin⁡ttdt\displaystyle\lim\limits_{x \to 2} \frac{1}{x-2} \int_4^{x^2} \frac{\sin t}{t} dt.
第 2 題8 分
Evaluate the integral ∫0∞e−2xcos⁡xdx\displaystyle\int_0^\infty e^{-2x} \cos x dx.
第 3 題8 分
Evaluate the integral ∫0ln⁡17exex−1ex+15dx\displaystyle\int_0^{\ln 17} \frac{e^x \sqrt{e^x - 1}}{e^x + 15} dx.
第 4 題8 分
Evaluate the double integral ∬D(x2+y2)3/2dA\displaystyle\iint_D (x^2 + y^2)^{3/2} dA, where DD is the region in the first quadrant bounded by the lines y=0y = 0 and y=3xy = \sqrt{3}x and the circle x2+y2=9x^2 + y^2 = 9.
第 5 題8 分
Evaluate the double integral ∬R(x+y)ex2−y2dA\displaystyle\iint_R (x + y)e^{x^2-y^2} dA, where RR is the rectangle enclosed by the line x−y=0x - y = 0, x−y=2x - y = 2, x+y=0x + y = 0, and x+y=3x + y = 3.
第 6 題8 分
Evaluate the triple integral ∭Ex2dV\displaystyle\iiint_E x^2 dV, where EE is the solid hemisphere x2+y2+z2≤4x^2 + y^2 + z^2 \leq 4, y≥0y \geq 0.
第 7 題8 分
Find the length of the curve r(t)=cos⁡ti+sin⁡tj+ln⁡(cos⁡t)k\mathbf{r}(t) = \cos t \mathbf{i} + \sin t \mathbf{j} + \ln(\cos t) \mathbf{k}, 0≤t≤π/40 \leq t \leq \pi/4.
第 8 題8 分
Find the work done by the force field F(x,y)=xi+(y+2)j\mathbf{F}(x,y) = x \mathbf{i} + (y + 2) \mathbf{j} in moving an object along an arch of the cycloid r(t)=(t−sin⁡t)i+(1−cos⁡t)j\mathbf{r}(t) = (t - \sin t) \mathbf{i} + (1 - \cos t) \mathbf{j}, 0≤t≤2π0 \leq t \leq 2\pi.

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
(a)6 分
Find the radius of convergence of the power series ∑n=0∞(−1)nxn32n(2n)!\displaystyle\sum\limits_{n=0}^\infty \frac{(-1)^n x^n}{3^{2n}(2n)!}. Then find the sum of the series ∑n=0∞(−1)nπn32n(2n)!\displaystyle\sum\limits_{n=0}^\infty \frac{(-1)^n \pi^n}{3^{2n}(2n)!}.
(b)6 分
Determine whether the series ∑n=1∞(−1)nn(1+ln⁡2n)\displaystyle\sum\limits_{n=1}^\infty \frac{(-1)^n}{n(1 + \ln^2 n)} is absolutely convergent, conditionally convergent, or divergent.
第 2 題12 分
Find the limit or show that the limit does not exist.
(a)6 分
lim⁡x→(π/2)−(tan⁡x)cos⁡x\displaystyle\lim\limits_{x \to (\pi/2)^-} (\tan x)^{\cos x}.
(b)6 分
lim⁡(x,y)→(0,0)y5sin⁡2xx4+y4\displaystyle\lim\limits_{(x,y) \to (0,0)} \frac{y^5 \sin^2 x}{x^4 + y^4}.
第 3 題12 分
Find the local maximum and minimum values and saddle point of the function
f(x,y)=(x2+y2)e−x.f(x,y) = (x^2 + y^2) e^{-x}.