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台灣聯合大學系統 106 年度 微積分 A2

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：106t/微積分A2.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 A2 其他年度考古題

本科目共 11 個年度已整理題目，可直接切換查閱。

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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.