PastExamLabPastExamLab

台灣聯合大學系統 112 年度 微積分 A2

原始 PDF
壓縮檔內：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。
第 1 題8 分
Find the limit lim⁡x→0+(1x−1ex−1)\lim\limits_{x \to 0^+} \left(\frac{1}{x} - \frac{1}{e^x - 1}\right).
第 2 題8 分
Find d2023dx2023(xsin⁡x)\displaystyle \frac{d^{2023}}{dx^{2023}}(x \sin x).
第 3 題8 分
Evaluate the integral ∫−14dx∣x∣\displaystyle\int_{-1}^{4} \frac{dx}{\sqrt{|x|}}.
第 4 題8 分
Evaluate the integral ∫08∫x321y4+1dydx\displaystyle\int_0^8 \int_{\sqrt[3]{x}}^2 \frac{1}{y^4 + 1} dy dx.
第 5 題8 分
Consider the region bounded by the graphs of y=ln⁡xy = \ln x, y=0y = 0, and x=ex = e. Find the volume of the solid formed by revolving the region about the xx-axis.
第 6 題8 分
Let y=(sin⁡x)xy = (\sin x)^x, sin⁡x>0\sin x > 0. Find dydx\frac{dy}{dx}.
第 7 題8 分
Assume that constants aa and bb are positive. Find equations for all horizontal asymptotes for the graph of y=ax2+4x−by = \frac{\sqrt{ax^2 + 4}}{x - b}.
第 8 題8 分
We say that the two commodities are substitute commodities if a decrease in the demand for one results in an increase in the demand for the other. Conversely, two commodities are referred to as complementary commodities if a decrease in the demand for one results in a decrease in the demand for the other as well. Suppose that 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)=q2q+p2x = f(p, q) = \frac{q^2}{q + p^2} and y=g(p,q)=e−2q+py = g(p, q) = e^{-2q+p}. Are A and B substitute, complementary or neither?

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
Consider the function f(x)={xsin⁡(1x),x≠00,x=0f(x) = \begin{cases} x \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}.
(a)4 分
Show that ff is continuous at x=0x = 0.
(b)4 分
Find f′(x)f'(x) for x≠0x \neq 0.
(c)4 分
Use the limit definition of the derivative to show that ff is not differentiable at x=0x = 0.
第 2 題12 分
(a)6 分
Use the integral test to determine if the series ∑n=1∞nn2+4\sum\limits_{n=1}^{\infty} \frac{n}{n^2 + 4} converges or diverges.
(b)6 分
Find all values of xx for which ∑n=1∞(n+1−n)(x−3)n\sum\limits_{n=1}^{\infty} \left(\sqrt{n+1} - \sqrt{n}\right)(x - 3)^n converges absolutely.
第 3 題12 分
Suppose xx units of labor and yy units of capital are required to produce
f(x,y)=100x3/4y1/4f(x, y) = 100x^{3/4}y^{1/4}
units of a certain product. If each unit of labor costs $200, each unit of capital costs $300, and a total of $60,000 is available for production, determine how many units of labor and how many units of capital should be used to maximize production.