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台灣聯合大學系統 109 年度 微積分 A3/A4/A6

原始 PDF
壓縮檔內：轉學考/微積分A3A4A6.pdf
中央大學圖書館考古題頁

甲、簡答題

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，命題型態為簡答題，請勿列出計算過程，倘若答案被包含在演算過程，將被視為試算流程，不予另行批出計分。
第 1 題8 分
Find the limit: lim⁡x→∞x+cos⁡xx−cos⁡x\lim\limits_{x \to \infty} \frac{x + \cos x}{x - \cos x}
第 2 題8 分
Find the smallest positive (x>0)(x > 0) inflection point of F(x)=∫0xcos⁡(t3/2)dtF(x) = \int_0^x \cos(t^3/2) dt.
第 3 題8 分
How many local extreme values does the function f(x,y)=10xye−(x2+y2)f(x,y) = 10xy e^{-(x^2+y^2)} have?
第 4 題8 分
Let CC be the curve of intersection of the two surfaces x2+y2+z2=3x^2 + y^2 + z^2 = 3 and (x−2)2+(y−2)2+z2=3(x - 2)^2 + (y - 2)^2 + z^2 = 3. Find parametric equations of the tangent line to CC at P=(1,1,1)P = (1, 1, 1).
第 5 題8 分
Evaluate ∬Rln⁡x2+y2dA\iint_R \ln \sqrt{x^2 + y^2} dA where RR is the unit disk x2+y2≤1x^2 + y^2 \leq 1.
第 6 題8 分
Find the volume between the two spheres: x2+y2+z2=1x^2 + y^2 + z^2 = 1, x2+y2+z2=2x^2 + y^2 + z^2 = 2 and inside the cone z2=x2+y2z^2 = x^2 + y^2.
第 7 題8 分
Calculate ∬Rex2+4y2dA\iint_R e^{x^2+4y^2} dA where RR is the interior of the ellipse (x2)2+(y3)2≤1\left(\frac{x}{2}\right)^2 + \left(\frac{y}{3}\right)^2 \leq 1.
第 8 題8 分
Find the area of the region enclosed by the limaçon x=2cos⁡t−cos⁡2tx = 2\cos t - \cos 2t, y=2sin⁡t−sin⁡2ty = 2\sin t - \sin 2t, 0≤t≤2π0 \leq t \leq 2\pi
第 8 題圖表

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
Let
f(x,y)={(xy)px2+y4if (x,y)≠(0,0)0,if (x,y)=(0,0)f(x, y) = \begin{cases} \frac{(xy)^p}{x^2+y^4} & \text{if } (x,y) \neq (0, 0) \\ 0, & \text{if } (x,y) = (0, 0) \end{cases}
Use polar coordinates to show that f(x,y)f(x, y) is continuous at all (x,y)(x, y) if p>2p > 2, but discontinuous at (0,0)(0, 0) if p≤2p \leq 2.
第 2 題12 分
(a.)6 分
Determine whether the series ∑n=1∞(−1)nln⁡(1+1n)\sum\limits_{n=1}^{\infty} (-1)^n \ln\left(1 + \frac{1}{n}\right) diverges or converges conditionally or converges absolutely and give reasons for your answer.
(b.)6 分
Show that if ∑n=1∞an\sum\limits_{n=1}^{\infty} a_n converges, then ∑n=1∞(3+sin⁡(an)5)n\sum\limits_{n=1}^{\infty} \left(\frac{3 + \sin(a_n)}{5}\right)^n converges.
第 3 題12 分
Goods 1 and 2 are available at prices (in dollars) of p1p_1 per unit of good 1 and p2p_2 per unit of good 2. A utility function U(x1,x2)U(x_1, x_2) is a function representing the utility or benefit to consuming xjx_j units of good jj. The marginal utility of the jjth good is ∂U/∂xj\partial U/\partial x_j, the rate of increase in utility per unit increase in the jjth good. Prove the following law of economics: Given a budget of LL dollars, utility is maximized at the consumption level (a,b)(a, b) where the ratio of marginal utility is equal to the ratio of prices:
Marginal utility of good 1Marginal utility of good 2=∂U/∂x1∂U/∂x2=p1p2\frac{\text{Marginal utility of good 1}}{\text{Marginal utility of good 2}} = \frac{\partial U/\partial x_1}{\partial U/\partial x_2} = \frac{p_1}{p_2}