PastExamLabPastExamLab

成功大學 112 年度 微積分B

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Show all your work and justify all your answers.
第 1 題10 分
(a)5 分
Evaluate lim⁡x→2x2+5x3−2x\lim\limits_{x \to 2} \frac{x^2 + 5}{x^3 - 2x}.
(b)5 分
Evaluate ∫−116x5+3x2dx\int_{-1}^{1} 6x^5 + 3x^2 dx.
第 2 題10 分
Evaluate ∫121x2cos⁡(πx)dx\int_{1}^{2} \frac{1}{x^2} \cos\left(\frac{\pi}{x}\right) dx.
第 3 題10 分
Evaluate lim⁡x→∞ln⁡(8x+9x)3x\lim\limits_{x \to \infty} \frac{\ln(8^x + 9^x)}{3x}.
第 4 題10 分
Find an equation for the tangent line to the curve x2+4xy+y3+5=0x^2 + 4xy + y^3 + 5 = 0 at the point (2,−1)(2, -1).
第 5 題10 分
Let f(x)=x(x−1)2f(x) = \frac{x}{(x-1)^2}. Find the intervals on which ff is increasing and the intervals on which ff is decreasing.
第 6 題10 分
Evaluate ∫0112x+10x+12dx\int_{0}^{1} \frac{1}{2x + 10\sqrt{x + 12}} dx.
第 7 題10 分
Evaluate ∫0π/2∫y2ysin⁡xxdxdy+∫π/2π∫yπsin⁡xxdxdy\int_{0}^{\pi/2} \int_{y}^{2y} \frac{\sin x}{x} dxdy + \int_{\pi/2}^{\pi} \int_{y}^{\pi} \frac{\sin x}{x} dxdy.
第 8 題10 分
Use the method of Lagrange multipliers to find the minimum value of the function f(x,y,z)=x−y+zf(x, y, z) = x - y + z subject to the constraint 2x2+4y2+4z2=92x^2 + 4y^2 + 4z^2 = 9.
第 9 題10 分
Let f(x,y,z)=x2zey+xz2f(x, y, z) = x^2 ze^y + xz^2. In what direction does ff increase most rapidly at the point (1,ln⁡2,2)(1, \ln 2, 2)? What is the maximum rate of increase?
第 10 題10 分
For ∣x∣<32|x| < \frac{3}{2}, the function f(x)=2x516x4+81f(x) = \frac{2x^5}{16x^4 + 81} can be expressed as a power series ∑n=0∞anxn\sum\limits_{n=0}^{\infty} a_n x^n, where ana_n's are constants. Find the value of a25a_{25}.
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