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台灣聯合大學系統 103 年度 微積分 A3/A4/A5/A7

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

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.