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台灣聯合大學系統 103 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
甲、計算、證明題：共 2 題，每題 10 分，共 20 分，須詳細寫出計算及證明過程，答對不予計分。
第 1 題10 分
The production of QQ units of a commodity is related to the amount of labor xx and the amount of capital yy (in suitable units) expended by the equation Q=f(x,y)=x3/4y1/4Q = f(x, y) = x^{3/4}y^{1/4}. If an expenditure of 100 units is available for production, how should it be apportioned between labor and capital so that QQ is maximized?
第 2 題10 分
A manufacturer of aquariums wants to make a large rectangular box-shaped (an open rectangular box) aquarium (開口的長方體水族箱) that will hold 64 cubic feet of water. If the material for the base costs 20persquarefootandthematerialforthesidescosts20 per square foot and the material for the sides costs 10 per square foot, find the dimension for which the cost of the materials will be the least.
乙、填充題：共 10 題，每題 8 分，共 80 分，請將答案依題號填入答案卷上，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 limit. lim⁡x→−1x(x2−1)∣x2−1∣\lim\limits_{x \to -1} \frac{x(x^2 - 1)}{|x^2 - 1|}
第 3 題8 分
Suppose that the first derivative of y=f(x)y = f(x) is y′=6(x+1)(x−2)3y' = 6(x + 1)(x - 2)^3. At what points does the graph of ff have a point of inflection?
第 4 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.
第 5 題8 分
Suppose that xx and yy are related by the equation x=∫0y11+4t2dtx = \int_0^y \frac{1}{\sqrt{1 + 4t^2}} dt. Find dydx\frac{dy}{dx}.
第 6 題8 分
Suppose that the edge lengths x,yx, y, and zz of a rectangular box are changing at the following rates: dxdt=1\frac{dx}{dt} = 1 m/sec, dydt=−2\frac{dy}{dt} = -2 m/sec, dzdt=1\frac{dz}{dt} = 1 m/sec. Find the rate at which the box's volume is changing at instant when x=4x = 4, y=3y = 3, and z=2z = 2.
第 7 題8 分
Evaluate ∬Rex2dA\iint_R e^{x^2} dA; RR is bounded by x=0x = 0, y=2xy = 2x, and y=2y = 2.
第 8 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 9 題8 分
Evaluate the integral. ∫01x4−5xdx\int_0^1 \frac{x}{\sqrt{4 - 5x}} dx
第 10 題8 分
Find f′(2)f'(2) if f(x)=eg(x)f(x) = e^{g(x)} and g(x)=∫2xt1+t4dtg(x) = \int_2^x \frac{t}{1 + t^4} dt.