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台灣聯合大學系統 103 年度 微積分 A3/A4/A5/A7

本頁整理這份微積分考卷的題目、解答與詳解步驟，可直接對照每題內容與答案重點。

原始 PDF
壓縮檔內：轉學考/微積分 A3 A4 A5 A7.pdf
中央大學圖書館考古題頁

這份試題整理

依本站已整理題目自動彙整題數、分數與題型，方便先判斷這份考卷的出題結構。

PastExamLab Summary
題數
11
題
總分
100
分
已整理詳解
0%
0 / 11 題

微積分 A3/A4/A5/A7 其他年度考古題

本科目共 2 個年度已整理題目，可直接切換查閱。

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104103

甲、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
For what value of the constant mm is f(x)={cos⁡3x,x≤0mx,x>0f(x) = \begin{cases} \cos 3x, & x \leq 0 \\ mx, & x > 0 \end{cases} differentiable at x=0x = 0?
第 2 題8 分
Find the horizontal asymptote of the graph of y=xtan⁡(1x)y = x \tan(\frac{1}{x}).
第 3 題8 分
Find the interval of convergence of the power series ∑n=1∞(−1)nn(x−5)n\sum\limits_{n=1}^{\infty} \frac{(-1)^n}{n}(x-5)^n.
第 4 題8 分
The function f(x)=ex+xf(x) = e^x + x, being differentiable and one-to-one, has a differentiable inverse f−1(x)f^{-1}(x). Find the value of df−1/dxdf^{-1}/dx at the point f(ln⁡2)f(\ln 2).
第 5 題8 分
A passenger ship and an oil tanker left port sometime in the morning; the former headed north, and the latter headed east. At noon, the passenger ship was 40 miles from port and sailing at 30 mph, while the oil tanker was 30 miles from the port and sailing at 20 mph. How fast was the distance between the two ships changing at that time?
第 6 題8 分
Evaluate the integral ∫261x4x+1dx\int_2^6 \frac{1}{x\sqrt{4x+1}} dx.
第 7 題8 分
Find the volume under the surface f(x,y)=e−x2f(x,y) = e^{-x^2}, bounded by the xzxz-plane and the planes y=xy = x and x=1x = 1.
第 8 題8 分
Find the work done by the force F=xyi+(y−x)j\mathbf{F} = xy\mathbf{i} + (y - x)\mathbf{j} over the straight line from (1,1)(1,1) to (2,3)(2,3).

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
Suppose that xx and yy are related by the equation x=∫0y11+4t2dtx = \int_0^y \frac{1}{\sqrt{1+4t^2}} dt. Show that d2y/dx2d^2y/dx^2 is proportional to yy and find the constant of proportionality.
第 2 題12 分
Find the directional derivative of the function h(x,y,z)=cos⁡xy+eyz+ln⁡zxh(x,y,z) = \cos xy + e^{yz} + \ln zx at (1,0,1/2)(1,0,1/2) in the direction of u=i+2j+2k\mathbf{u} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k}.
第 3 題12 分
Find the critical point of f(x,y)=xy+2x−ln⁡x2yf(x,y) = xy + 2x - \ln x^2 y in the open first quadrant (x>0,y>0)(x > 0, y > 0) and show that ff takes on a minimum there.