PastExamLabPastExamLab

台灣聯合大學系統 105 年度 微積分 A2

原始 PDF
壓縮檔內：微積分(A2 951).pdf
中央大學圖書館考古題頁

甲、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
Sketch the region of integration, reverse the order of integration, and evaluate the integral.
∫02∫04−x2xe2y4−ydydx.\int_0^2 \int_0^{4-x^2} \frac{xe^{2y}}{4-y} dy dx.
第 2 題12 分
Determine whether the series is convergent or divergent. Give your reason. (a) ∑n=1∞n3n2+5\sum\limits_{n=1}^{\infty} \frac{n}{3n^2 + 5} (b) ∑n=2∞1nln⁡n\sum\limits_{n=2}^{\infty} \frac{1}{n\sqrt{\ln n}}.
(a)6 分
∑n=1∞n3n2+5\sum\limits_{n=1}^{\infty} \frac{n}{3n^2 + 5}
(b)6 分
∑n=2∞1nln⁡n\sum\limits_{n=2}^{\infty} \frac{1}{n\sqrt{\ln n}}
第 3 題12 分
Find the points on the sphere x2+y2+z2=4x^2 + y^2 + z^2 = 4 that are closest to and farthest from the point (3,1,−1)(3, 1, -1).

乙、填充題

共 8 題，每題 8 分，共 64 分。請將答案依題號順序寫在答案卷上，不必寫演算過程。
第 1 題8 分
Suppose that the first derivative of y=f(x)y = f(x) is y′=6x(x+1)(x−2)y' = 6x(x + 1)(x - 2). At what points does the graph of ff have a point of inflection? Answer: ________
第 2 題8 分
Evaluate the integral. ∫26x2x−3dx\int_2^6 x\sqrt{2x - 3} dx. Answer: ________
第 3 題8 分
Find the slope of the tangent line to the graph of 2xy+ex+y−2=02xy + e^{x+y} - 2 = 0 at the point P(0,ln⁡2)P(0, \ln 2). Answer: ________
第 4 題8 分
Find the limit: lim⁡h→01h∫xx+h1+t2dt\lim\limits_{h \to 0} \frac{1}{h} \int_x^{x+h} \sqrt{1 + t^2} dt. Answer: ________
第 5 題8 分
Find the limit: lim⁡n→∞∑j=1nn2−j2n2\lim\limits_{n \to \infty} \sum\limits_{j=1}^n \frac{\sqrt{n^2 - j^2}}{n^2}. Answer: ________
第 6 題8 分
Find the area under the curve y=ln⁡xx2y = \frac{\ln x}{x^2} from x=1x = 1 to x=∞x = \infty. Answer: ________
第 7 題8 分
A new drug is introduced through an advertising campaign to a population of 1 million potential customers? The rate at which the population hears about the drug is assumed to be proportional to the number of people who are not yet aware of the drug? By the end of 1 year) half of the population has heard of the drug? How many will have heard of it by the end of 2 years? Answer: ________
第 8 題8 分
Find the absolute maximum value of the function f(x,y)=x2−2xy+2yf(x, y) = x^2 - 2xy + 2y on the rectangle D={(x,y)∣0≤x≤3,0≤y≤2}D = \{(x, y)| 0 \leq x \leq 3, 0 \leq y \leq 2\}. Answer: ________