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台灣聯合大學系統 106 年度 微積分 A2

原始 PDF
壓縮檔內：106t/微積分A2.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。請在答案卷上列出題號依序作答。請注意：本（甲、）部分，共 8 題，命題型態為填充題，不必詳列計算過程，倘若答案被包含在演算過程，將被視為試算流程，無法計分。
第 1 題8 分
Find lim⁡h→01h∫11+h1+t2dt\lim\limits_{h \to 0} \frac{1}{h} \int_1^{1+h} \sqrt{1 + t^2} dt. Answer: _______
第 2 題8 分
How many critical points does the function f(x)=∣x2−1∣f(x) = |x^2 - 1| have? Answer: _______
第 3 題8 分
Find the number aa and bb such that lim⁡x→0ax+b−2x=1\lim\limits_{x \to 0} \frac{\sqrt{ax + b} - 2}{x} = 1. Answer: _______
第 4 題8 分
Market has two commodities A and B. The demand equations that relate the quantities demanded xx and yy to the unit prices pp and qq of the commodities A and B respectively are given by x=f(p,q)=qp+qx = f(p, q) = \frac{q}{p + q}, y=g(p,q)=e−(2q+p2)y = g(p, q) = e^{-(2q+p^2)}. Are A and B substitute, complementary or neither? Answer: _______
第 5 題8 分
Find the slope of the tangent line to the graph of the function f(x)=(tan⁡x)2+exf(x) = (\tan x)^2 + e^x at (0,1)(0, 1). Answer: _______
第 6 題8 分
Find the volume of solid bounded above by z=f(x,y)=e−x2z = f(x, y) = e^{-x^2} and below by the plane region RR which is bounded by y=xy = x, x=1x = 1 and y=0y = 0. Answer: _______
第 7 題8 分
The Cobb-Douglas production function for a software manufacturer is given by f(x,y)=100x3/4y1/4f(x, y) = 100x^{3/4}y^{1/4} where xx represents the units of labor (at 150perunit)and150 per unit) and yrepresentstheunitsofcapital(at represents the units of capital (at 250 per unit). The total cost of labor and capital is limited to 50,000.Find50,000. Find xand and y$ that will yield the maximum production level for this manufacturer. Answer: _______
第 8 題8 分
Suppose f′′f'' is continuous on [0,1][0, 1], f(1)=2f(1) = 2, f′(1)=2f'(1) = 2 and the average value of ff on [0,1][0, 1] is 22. Evaluate ∫01x2f′′(x)dx\int_0^1 x^2 f''(x) dx. Answer: _______

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
Find any extrema of the function f(x,y)=e−xy/4f(x, y) = e^{-xy/4} subject to the constraint x2+y2≤1x^2 + y^2 \leq 1.
第 2 題12 分
Evaluate ∫03∫x/32xy3+1dydx\int_0^3 \int_{x/3}^2 x\sqrt{y^3 + 1} dy dx
第 3 題12 分
Determine whether the series ∑n=1∞n!nn\sum\limits_{n=1}^{\infty} \frac{n!}{n^n} converges absolutely or conditionally, or diverges.