109年 - 109 國立臺北大學_日間學士班暨進修學士班轉學生招生考試試題_經濟學系日間學士班暨進修學士班2年級：微積分#147830

1.1      (3%)      True or False: The function \( f(x) \) is an increasing function.

1.2      (3%)      True or False: The function \( f(x) \) is a continuous function.

1.3      (3%)      True or False: The function \( f(x) \) is a concave function.

1.4      (3%)      True or False: The function \( f(x) \) is differentiable at \( x = \frac{1}{2} \).

1.5      (5%)      Let \( n \) be a strictly positive integer. Find out \( \int_1^n f(x)dx \), if exists; if the integral does not exist, explain why.

1.6      (3%)      True or False: Given any \( a, b \in \mathbb{R} \) such that \( a < b \), if there exists a function \( g(x) \) such that $$\int_a^b f(x)dx = \int_a^b g(x)dx, \text{ then } f(x) = g(x) \text{ for all } x \in [a, b].$$

2.1 If \( \int_0^\pi |x(x - 2)|dx = 0 \) , then \( a = \underline{\quad (5\%)} \)

2.2 \( \int x^{3}e^{1 - x}dx = \underline{\quad (5\%)} \)

2.3 \( \int_{-2}^{1}x(x^{2} - 1)^{4}dx = \underline{\quad (5\%)} \)

2.4 If \( f(x) = \sin [\sin (\sin x)] \), then \( \frac{df(x)}{dx} = \underline{\quad (5\%)} \) at \( x = 0 \).

2.5 If \( f(x) = (\ln x)^{1 / x} \) for \( x > 0 \), then \( \frac{df(x)}{dx} = \underline{\quad (5\%)} \), and \( \lim_{x\to \infty}f(x) = \underline{\quad (5\%)} \).

2.6 Given the function \( f(x) = \ln (x - 1), x > 1 \), if we use the Taylor polynomial with degree 10 at the point \( x = 2 \) to evaluate \( f(3) \), the approximate value is \( \underline{\quad (5\%)} \).

2.7 For the function \( f(x, y) = \sin x + \cos y \), the Taylor polynomial of degree 2 centered at \( (x, y) = (0, 0) \) is \( \underline{\quad (10\%)} \).

2.8 Suppose that the three variables \( x, y \), and \( z \) satisfy \( x^3 + y^2 + z = 3 \) and \( x \cdot y^2 \cdot z^3 = 1 \). Then \( \frac{dx}{dz} = \underline{\quad (5\%)} \) and \( \frac{dy}{dz} = \underline{\quad (5\%)} \) at the point \( (x, y, z) = (1, 1, 1) \).

3.1 .      (5%)      \( f(x) = x - e^{x} \) , where \( 0\leq x\leq 10 \)

3.2      (5%)      \( f(x,y) = x\cdot \sin y \) , where \( x,y\in \mathbb{R} \)

4.      (10%)      Consider the infinite series $\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2}$, where $a > 0$. Find the necessary and sufficient condition of $a$ such that the series is convergent, or argue that no such condition exists.