PastExamLabPastExamLab

臺灣綜合大學系統 113 年度 微積分A

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答題時需詳述計算過程，否則不予計分。
第 1 題10 分單側極限・絕對值
Find lim⁡x→1+∣x2−1∣x−1\lim\limits_{x \to 1^+} \frac{|x^2-1|}{x-1} and lim⁡x→1−∣x2−1∣x−1\lim\limits_{x \to 1^-} \frac{|x^2-1|}{x-1}.
第 2 題10 分微積分基本定理・鏈鎖律
Find the derivative f′(0)f'(0) for the function f(x)=∫tan⁡xsec⁡x11+t4dt\displaystyle f(x) = \int_{\tan x}^{\sec x} \frac{1}{\sqrt{1+t^4}} dt.
第 3 題10 分閉區間最大最小值
Find the absolute maximum and absolute minimum values of f(x)=xx2−x+1f(x) = \frac{x}{x^2-x+1} on the interval [−2,0][-2, 0].
第 4 題10 分定積分・三角代換（或幾何）
Evaluate the integral ∫01/21−4x2dx\int_0^{1/2} \sqrt{1-4x^2} dx.
第 5 題10 分瑕積分・分部積分
Evaluate the improper integral ∫01xln⁡xdx\displaystyle \int_0^1 x \ln x dx.
第 6 題10 分冪級數收斂域
Find the interval of convergence of the power series ∑n=1∞xnn\sum\limits_{n=1}^\infty \frac{x^n}{\sqrt{n}}.
第 7 題10 分梯度・方向導數
Find the direction in which the function f(x,y)=ln⁡(xy)f(x,y) = \ln(xy) decreases fastest at the point (1,2)(1,2)? What is the rate of decrease?
第 8 題10 分空間曲線弧長
Find the arc length for the circular helix with vector function r(t)=cos⁡2ti+sin⁡2tj+tk\mathbf{r}(t) = \cos 2t \mathbf{i} + \sin 2t \mathbf{j} + t \mathbf{k} from the point (1,0,0)(1, 0, 0) to the point (1,0,π)(1, 0, \pi).
第 9 題10 分二重積分・極座標換元
Evaluate the integral ∫01∫0x−x2x2+y2 dydx\displaystyle \int_0^1 \int_0^{\sqrt{x-x^2}} \sqrt{x^2 + y^2} \, dy dx.
第 10 題10 分線積分・直接計算
Evaluate the line integral ∫C−yx2+y2dx−xx2+y2dy\displaystyle \int_C \frac{-y}{x^2+y^2} dx - \frac{x}{x^2+y^2} dy, where CC is the counterclockwise oriented ellipse 4x2+9y2=364x^2 + 9y^2 = 36.