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台灣聯合大學系統 114 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

一、填充題

共8題，每題8分，總計64分。請在答案卷上列出題號並依序作答。
第 1 題8 分
Evaluate
lim⁡n→∞n+1+nn(n+1−n)\lim\limits_{n \to \infty} \frac{\sqrt{n+1} + \sqrt{n}}{n(\sqrt{n+1} - \sqrt{n})}
第 2 題8 分
Define the function F(x)=f(ln⁡(2sin⁡x))F(x) = f(\ln(2 \sin x)), 0<x<π0 < x < \pi, where ff is differentiable on (−∞,∞)(-\infty, \infty) and f′(0)=2f'(0) = 2. Find F′(π/6)F'(\pi/6).
第 3 題8 分
Find the following integral
∫ex3x2dx.\int e^{x^3} x^2 dx.
第 4 題8 分
Find all critical points of the function
f(x,y)=x3−6xy−y2,−∞<x<∞and −∞<y<∞,f(x,y) = x^3 - 6xy - y^2, -\infty < x < \infty \quad and \: -\infty < y < \infty,
and classify each critical point as a location where a local maximum, local minimum, or saddle point occurs.
第 5 題8 分
Find the rational number that is represented by the repeating decimal 2.21‾=2.21212121...2.\overline{21} = 2.21212121...
第 6 題8 分
Calculate the area of the region that is completely enclosed by the graphs of the functions f(x)=x2f(x) = x^2 and g(x)=4−x2g(x) = 4 - x^2.
第 7 題8 分
Evaluate the double integral
∬R20x2e−ydA,\iint_R 20x^2 e^{-y} dA,
where R={(x,y):0≤x≤3,0≤y≤2}R = \{(x,y) : 0 \leq x \leq 3, 0 \leq y \leq 2\}.
第 8 題8 分
Find the domain of the following function:
f(x)=4−x2ln⁡(x−1).f(x) = \frac{\sqrt{4-x^2}}{\ln(x-1)}.

二、計算、證明題

共3題，每題12分，總計36分。請將題號標明清楚。
第 1 題12 分
Find the Taylor series of the function
f(x)=x3+2xf(x) = \frac{x}{3 + 2x}
and determine its radius of convergence.
第 2 題12 分
Use the separation of variables to solve the following differential equation:
dQdt=2(100−Q),t≥0,\frac{dQ}{dt} = 2(100 - Q), \quad t \geq 0,
with the initial condition Q(0)=3Q(0) = 3.
第 3 題12 分
To create an open box from a square piece of cardboard, we can cut out identical squares from each corner and then fold up the resulting flaps. If the cardboard measures 10 inches on each side, determine the dimensions of the box that will yield the maximum volume. Note: Points will be awarded only if calculus is used in the solution.