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成功大學 104 年度 微積分B

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計算題：每計算題各不給分

第 1 題18 分
Find the following limits:
(a)6 分
lim⁡x→1(1−x)tan⁡πx2\lim\limits_{x \to 1} (1 - x) \tan \frac{\pi x}{2}
(b)6 分
lim⁡x→∞(x−1x+3)x+2\lim\limits_{x \to \infty} \left(\frac{x - 1}{x + 3}\right)^{x+2}
(c)6 分
lim⁡x→0x2−S(x)xx2\lim\limits_{x \to 0} \frac{x^2 - S(x)\sqrt{x}}{x^2}, where S(x)=∫0xsin⁡(t2)dtS(x) = \int_0^{\sqrt{x}} \sin(t^2)dt
第 2 題12 分
Evaluate the following integrals:
(a)6 分
∫0π4sec⁡2x(4−tan⁡2x)12dx\int_0^{\frac{\pi}{4}} \sec^2 x(4 - \tan^2 x)^{\frac{1}{2}} dx
(b)6 分
∫2+x2−2−x24−x4dx\int \frac{\sqrt{2 + x^2} - \sqrt{2 - x^2}}{\sqrt{4 - x^4}} dx
第 3 題10 分
If u=xyxu = x^{y^x}, find the partial derivatives ∂u∂x\frac{\partial u}{\partial x}, ∂u∂y\frac{\partial u}{\partial y} at (x,y)=(2,2)(x, y) = (2, \sqrt{2}).
第 4 題10 分
Evaluate ∫03∫x34−y2e−x2−y2dxdy\int_0^{\sqrt{3}} \int_{\frac{x}{\sqrt{3}}}^{\sqrt{4-y^2}} e^{-x^2-y^2} dx dy
第 5 題10 分
Find the derivative of the function f(x)=tan⁡−1(xa)+ln⁡x−ax+af(x) = \tan^{-1}\left(\frac{x}{a}\right) + \ln\sqrt{\frac{x - a}{x + a}}, simplify where possible.
第 6 題10 分
Find the area of the polar region R, where R is bounded by the smaller loop of the curve r=1+2cos⁡θr = 1 + 2\cos\theta.
第 7 題10 分
Find the interval of convergence of the series ∑n=0∞2n(x−3)nn+3\sum\limits_{n=0}^{\infty} \frac{2^n(x-3)^n}{\sqrt{n+3}}. Then identify the values of xx for which the series converges absolutely or conditionally.
第 8 題10 分
In which direction from the point (1,1)(1, 1) is the directional derivative of f(x)=x2+y2f(x) = x^2 + y^2 zero?
第 9 題10 分
Determine and classify the stationary points of the function f(x,y)=x2+y3+6xy−7x−6yf(x, y) = x^2 + y^3 + 6xy - 7x - 6y.
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