PastExamLabPastExamLab

成功大學 106 年度 微積分B

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第 1 題10 分
Find the tangent line of the curve 3x2−2xy2+y4=93x^2 - 2xy^2 + y^4 = 9 at the point (2,1)(2, 1).
第 2 題10 分
Find the intervals on which f(x)=ln⁡xxf(x) = \frac{\ln x}{\sqrt{x}} (x>0)(x > 0) is increasing or decreasing.
第 3 題10 分
Let F(x)=∫x2x+12t2+t+3 dtF(x) = \int_{\frac{x}{2}}^{x+1} \sqrt{2t^2 + t + 3} \, dt (x>0)(x > 0). Evaluate F′(2)F'(2) and (F−1)′(0)(F^{-1})'(0).
第 4 題10 分
Evaluate the limit. lim⁡x→0+sin⁡(x)−xxx\lim\limits_{x \to 0^+} \frac{\sin(\sqrt{x}) - \sqrt{x}}{x\sqrt{x}}
第 5 題10 分
Evaluate the definite integral. ∫0ln⁡2ex4−e2t dt\int_0^{\ln 2} e^x \sqrt{4 - e^{2t}} \, dt
第 6 題10 分
Evaluate the improper integral. ∫01ln⁡xx dx\int_0^1 \frac{\ln x}{\sqrt{x}} \, dx
第 7 題10 分
Use 11−x=∑n=0∞xn\frac{1}{1-x} = \sum\limits_{n=0}^{\infty} x^n to express the value ∫01212+x3 dx\int_0^{\frac{1}{2}} \frac{1}{2+x^3} \, dx as an infinite series.
第 8 題10 分
Find the maximum rate of change of f(x,y)=xey2f(x,y) = xe^{\frac{y}{2}} at point (1,3)(1,3).
第 9 題10 分
Use the method of Lagrange multiplier to maximize the profit of a product 150xx+5+160yy+3\frac{150x}{x+5} + \frac{160y}{y+3} (x:(x: development, y:y: promotion)) subject to the budget x+y=100x + y = 100.
第 10 題10 分
Evaluate the double integral. ∫01∫0xy3(x+y2)2 dy dx\int_0^1 \int_0^x \frac{y^3}{(x+y^2)^2} \, dy \, dx
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