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

科目：研究所、轉學考(插大)-微積分 | 年份：109年 | 選擇題數：0 | 申論題數：17

試卷資訊

所屬科目：研究所、轉學考(插大)-微積分

選擇題 (0)

申論題 (17)

1. Consider the function $f(x) = [x] - [x]$, where $[x]$ denotes the smallest integer that is greater than or equal to $x$, and $[x]$ denotes the greatest integer that is smaller than or equal to $x$. (That is, $[x] = n \in \mathbb{Z}$ such that $n \geq x > n - 1$, and $[x] = m \in \mathbb{Z}$ such that $m \leq x < m + 1$.)

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. Answer the following questions:

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. Given the specified range of the variables, find out the all the critical points of the following functions, and identify the global maxima and minima, if exist:

3.1 \((5\%)\) \(f(x) = x - e^{x}\) , where \(0\leq x\leq 10\)
3.2 \((10\%)\) \(f(x,y) = x\cdot \sin y\) , 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.