所屬科目：研究所、轉學考(插大)-微積分
1. The range of $f(x) = 1 - \sqrt{x}$ is __(1)__.
2. If $y^2 = x^2 + \sin xy$, then $\frac{dy}{dx} =$__(2)__.
3. The value of $\lim_{x \to -2^+} e^{x/(x+2)}$ is __(3)__.
4. Let $f(x) = \sqrt{x+1} + 1$, $x \ge 0$, then $f^{-1}(x) =$ __(4)__. (You need indicate $x$'s range too.)
5. Let $f$ be a differentiable one to one function with $f(1) = 2$, $f(2) = 1$, $f(3) = 0$, $f'(1) = -6$, $f'(2) = -2$ and $f'(3) = -1/2$. Then, $(f^{-1})'(0) =$ __(5)__.
6. The absolute minimum value of $f(x) = x^{4/3} - x - x^{1/3}$ on the interval of $[-1, 6]$ is __(6)__.
7. The value of $\int_0^1 \frac{x}{1+x^2} dx$ is __(7)__.
8. The value of $\int_{-1/2}^{1/2} \frac{dx}{x + \sqrt{1-x^2}}$ is __(8)__.
9. Does series $\sum_{n=1}^{\infty} \frac{1}{n(\ln n) \ln(\ln n)}$ converge or diverge?__ (9)__
10. Is $C = \sum{n=1}^{\infty} \frac{(-1)^{n+1}}{n!}$ or $D = \sum{n=1}^{\infty} \frac{n}{3^n}$ larger? __(10)__
1. Use the first three terms of the Taylor series to approximate $f(x) = x^8 + x^4 + 3$ at $x = 1$. Then, the result is ___(1)__.
2. Suppose that \( f \) is a continuous function satisfying \( f(x) = x \int_{0}^{x} f(t) dt + x^{3} \) for all \( x \), and \( c \) is a real number such that \( f(c) = 1 \). Then, expressing \( f'(c) \) in terms of \( c \) only is __________(2)__________.
3. For limit \( \lim_{x \to 0} \frac{\sin(x + ax^3) - x}{x^5} \) to exist, the value of constant \( a = \underline{\hspace{2cm}} (3) \underline{\hspace{2cm}}. \)
4. Assume that $z$ and $w$ are differentiable functions of $x$ and $y$, satisfying the equations $xw^3 + yz^2 + z^3 = -1$ and $zw^3 - xz^3 + y^2w = 1$. Then, at $(x, y, z, w) = (1, -1, -1, 1)$, $\frac{\partial z}{\partial x} =$ __(4)__.
5. The value of $\int0^\infty \int0^\infty \frac{dy dx}{(x^2 + y^2)^2 + 1}$ is __(5)__.
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