108年 - 108 國立臺北大學_碩士班一般入學考試試題_統計學系：基礎數學#147021

1. Calculus( 50%)。清楚完整作答始得完全分數。
(1). Find the slant asymptotic(斜漸進線) for \( F(x, y) = x^3 + y^3 - 3xy = 0 \) (6%)

(2). If \( f_0(x) = \sqrt{x} \), and \( f_{n+1}(x) = \sqrt{2 + f_n(x)} \). Find the limit \( \lim_{x \to 4} \frac{f_2(x) - 2}{x - 4} = ? \) (6%)

(3). Find the limit \( \lim_{x\to 0}\frac{\cos(\sin x) + \sin(1 - \cos x) - 1}{x^2} = ? \) (6%)

(4). Find the integral \( \int \sqrt{1 + e^{2x}} dx = ? \) (8%)

(5). Find the integral \( \int \frac{1}{2 - \sin x + \cos x} dx = ? \) (8%)

(6). Find the area bound by \( y = 2\sin x \) and \( y = \sin 2x \) over \( 0 < x < 2\pi \). (8%)

(7). Find the integral value for \( \iint_{R} x^{2} + y^{2} dA \) for R: \( \frac{x^{2}}{4} + \frac{y^{2}}{9} < 1 \). (8%)

(a) Write down the characteristic polynomial of \( A \) and use it to find the eigenvalues. (8%)

(b) Find the eigenspaces of A. (6%)

(c) Orthogonally diagonalize the matrix \( A \). (You need to find out an orthogonal matrix \( P \) and a diagonal matrix \( D \) such that \( P^T AP = D \).) (10%)

(d) Let \( \vec{v} = (1,1,2,3) \). Find the projections of \( \vec{v} \) onto each of the eigenspaces. (6%)

(2). Let $T: V \to W$ be a linear transformation. Please show that T is one to one if and only if $\text{Ker}(T) = \{\vec{0}_V\}$. (10% 證明中零向量請用足標表明所在之空間，如 $\vec{0}_V$ 表示 $V$ 的零向量)

(3). Let $T: \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation and A be its standard matrix, that is, $T \vec{x} = A \vec{x}$ for each $\vec{x} \in \mathbb{R}^n$. Please show that the range of T is just the column space of A. (10%)