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台灣聯合大學系統 113 年度 微積分 A2

原始 PDF
壓縮檔內：轉學考/微積分.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 the integral:
∫0∞∫y∞y2e−x2dxdy.\int_0^\infty \int_y^\infty y^2 e^{-x^2} dx dy.
第 2 題8 分
Find the inflection points of the function
f(x)=exsin⁡xf(x) = e^x \sin x
over the interval [0,2π][0, 2\pi]. Remind you that the inflection point is where concavity changes and it is a point on the graph.
第 3 題8 分
Evaluate
lim⁡x→0xsin⁡x1−cos⁡x\lim\limits_{x \to 0} \frac{x \sin x}{1 - \cos x}
第 4 題8 分
Find the interval of convergence of the following power series:
∑n=0∞(−1)n(n+2)3n(x−3)n.\sum\limits_{n=0}^\infty \frac{(-1)^n(n+2)}{3^n}(x-3)^n.
第 5 題8 分
Let
f(x)={sin⁡(2x),if x≤0;2x2+ax+b−3,if x>0.f(x) = \begin{cases} \sin(2x), & \text{if } x \leq 0; \\ 2x^2 + ax + b - 3, & \text{if } x > 0. \end{cases}
Find aa and bb such that ff is differentiable at 00. Write your answer as (a,b)(a,b).
第 6 題8 分
Find the volume of the right circular cone of base radius rr and height hh as shown in the following figure. Write you answer in terms of hh and rr.
第 6 題圖表
第 7 題8 分
Find all the extrema of the function f(x,y)=x3−12xy+8y3f(x,y) = x^3 - 12xy + 8y^3 on (−∞,∞)×(−∞,∞)(-\infty,\infty) \times (-\infty,\infty).
第 8 題8 分
Suppose that x2+xy=sin⁡xx^2 + xy = \sin x. Find dy/dxdy/dx in terms of xx and yy.

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
Determine for which p≥0p \geq 0 the series
∑n=3∞1n(ln⁡n)p\sum\limits_{n=3}^\infty \frac{1}{n(\ln n)^p}
converges or diverges.
第 2 題12 分
A tank initially contains 20 gallons of pure water. Brine (high-concentration solution of salt) containing 2 pounds of salt per gallon flows into the tank at a rate of 4 gallons per minute, and the well-stirred mixture flows out of the tank at the same rate. How much slat is present at the end of 10 minutes?
第 3 題12 分
Find the area of the region RR that is completely enclosed by the graphs of the functions
y=f(x)=ex−e−x+3andy=g(x)=2ex+5e−x−2.y = f(x) = e^x - e^{-x} + 3 \quad \text{and} \quad y = g(x) = 2e^x + 5e^{-x} - 2.