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台灣聯合大學系統 107 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。請注意：本(甲)部分，共 8 題，每題型態為填充題，必須以填充題形式將答案寫在答案卷第一頁，倘若答案該包含在演算過程中，將被視為試算草稿，無法採計分。
第 1 題8 分
Determine the limits of integration where a≤ba \leq b such that ∫ab(x2−16)dx\int_a^b (x^2 - 16) dx has minimal value.
第 2 題8 分
Evaluate ∫0π/21−sin⁡xdx\int_0^{\pi/2} \sqrt{1 - \sin x} dx.
第 3 題8 分
Evaluate the integral ∬R3−x2−y2dA\iint_R \sqrt{3 - x^2 - y^2} dA, where R={(x,y)∣x2+y2≤3}R = \{(x,y)|x^2 + y^2 \leq 3\}.
第 4 題8 分
Find the interval of convergence of the power series ∑n=0∞2n(x−3)n(n+1)!\sum\limits_{n=0}^{\infty} \frac{2n(x-3)^n}{(n+1)!}.
第 5 題8 分
Find the volume of the solid bounded above by the surface z=f(x,y)z = f(x,y) and below by the plane region RR, where f(x,y)=ln⁡xf(x,y) = \ln x and RR is bounded by the graphs y=2xy = 2x and y=0y = 0 from x=1x = 1 to x=3x = 3.
第 6 題8 分
Let z=f(x,y)=ln⁡(xy)1/2z = f(x,y) = \ln(xy)^{1/2}. Find the approximate change in zz when the point changes from (5,10)(5,10) to (5.03,9.96)(5.03,9.96).
第 7 題8 分
Consider a differential equation dydt=kv(10−y)\frac{dy}{dt} = \frac{k}{v}(10-y), y(0)=y0y(0) = y_0, where kk, vv and y0y_0 are positive constants with y<10y < 10. Find lim⁡t→∞y\lim\limits_{t \to \infty} y.
第 8 題8 分
Find the minimum of the function f(x,y,z)=xy+2yz+2xzf(x,y,z) = xy + 2yz + 2xz subject to the constraint xyz=108xyz = 108.

乙、計算、證明題

共 3 大題，每大題 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 airplane is flying on a flight path that will take it directly over a radar tracking station. The distance ss is decreasing at a rate of 640 kilometers per hour when s=16s = 16 km. What is the speed of the plane?
第 2 題12 分
Determine if the given series converges or diverges. Explain your reasoning.
(a.)6 分
(6 分)∑n=1∞e2/nn2(6\text{ 分}) \sum\limits_{n=1}^{\infty} \frac{e^{2/n}}{n^2}
(b.)6 分
(6 分)∑n=1∞n3n2+5(6\text{ 分}) \sum\limits_{n=1}^{\infty} \frac{n}{\sqrt{3n^2 + 5}}
第 3 題12 分
Consider the function f(x,y)={ke−(x+y)/a,if x≥0,y≥00,elsewhere.f(x,y) = \begin{cases} ke^{-(x+y)/a}, & \text{if } x \geq 0, y \geq 0 \\ 0, & \text{elsewhere.} \end{cases} Find the relationship between the positive constants aa and kk such that ff is a joint probability density function of the continuous random variables xx and yy.