所屬科目：研究所、轉學考(插大)-微積分
1. The value of \( \lim_{x\to 1}\frac{x^2 - x^2}{x^2 - 1} \) is _(1)_
2. The tangent line to the curve \( x^{2} + xy - y^{2} = 1 \) at point (2, 3) is _(2)_ .
3. If \( y = \frac{2\sqrt{x}}{3(1 + \sqrt{x})} \), then \( \frac{dy}{dx} = \) _(3)_.
4. Let \( f(x) = \frac{2x - 1}{8 - 7x} \), then \( f^{-1}(1) = \) _(4)_.
5. Let \( f(x) = 2x^{3/2} - 5x^{4/2} \), then \( f \) is decreasing on the interval of _(5)_.
6. The absolute maximum value of \( f(x) = \frac{x + 1}{x^2 + x + 9} \) on the interval of \( [0, \infty) \) is _(6)_.
7. The value of \( \int_0^1 x\sqrt{1 - x} dx \) is _(7)_.
8. The value of \( \int_0^\infty \frac{dx}{1 + e^x} = \) _(8)_ .
9.Does series \( \sum_{n = 1}^{\infty}\frac{5^n - 2^n}{7^n - 6^n} \) converge or diverge? _(9)_
10. Is \( A = \sum_{n=1}^{\infty} \frac{(-1)^n}{2^n} \) or \( B = \sum_{n=1}^{\infty} \frac{1}{n2^n} \) smaller? _(10)_
1. Use the first three terms of the Taylor series to approximate \( f(x) = \cos(5x) \) at \( x = 0 \). Then, the result is _(1)_.
2. If \( y \) is a differentiable function of \( x \) satisfying equation \( \int_0^{x + y} e^{-t^2} dt = xy \), then \( \left. \frac{d^2y}{dx^2} \right|_{(x,y) = (0,0)} = \) _(2)_.
3. The value of \( \lim_{x\to 0}\left(\frac{\sin x}{x}\right)^{1 / x^2} \) is _(3)_.
4. Let \( u = x + y + z \), \( v = xy + yz + zx \), \( w = xyz \), and suppose that \( f(u, v, w) \) is a differentiable function satisfying \( f(u, v, w) = x^4 + y^4 + z^4 \) for all \( (x, y, z) \). Then, \( f_u(2, -1, -2) = \) _(4)_ .
5. The value of \( \int_0^2\int_0^{\sqrt{2x - y^2}}\frac{xy}{x^2 + y^2} dxdy \) is _(5)_.
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