107年 - [無官方正解]107 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#147052

(7). (2%) The limit \( \lim_{x \to 0} f(x) \) does not exist for \( f(x) = x^2 \), if \( x \in \text{rational} \); \( f(x) = x^4 \), if \( x \in \text{irrational} \).(A)O(B)X

(1). (8%) Find the second difference for the function \( f''(x) \) for \( f(x) = 0 \), if \( x = 0 \); \( f(x) = \frac{1 - \cos x}{x} \), if \( x \neq 0 \).

(2). (8%) Find the limit for \( f(x) = \lim_{x \to \infty} \frac{x^4 + 5\cos(x^2)}{x^4} \), where \( k \) is some positive finite constant.

(3). (8%) Find the integral \( \int_{x + \sqrt{1 + x + x^2}} dx \).

(4). (8%) Find the definite integral \( \iint_{R} \ln(x^2 + y^2) \, \mathrm{d}A \), \( R = \{(x, y) | 0 \leq x^2 + y^2 \leq 9, x \geq 0, y \geq 0\} \).

(5). (8%) Prove or disprove whether \( f(x, y) = \frac{2x^2y}{x^2 + y^2} \), if \( (x, y) \neq (0, 0) \), and \( f(0, 0) = 0 \) is differentiable at \( (0, 0) \) or not?

(6). (8%) Find the limit \( \lim_{n\to \infty}\sum_{k = 1}^{n}\ln \sqrt[n]{(1 + \frac{2k}{n})} \).

(a) Write down the characteristic polynomial of \( A \) and use it to find the eigenvalues. (5%)

(b) Find the eigenspaces of A. (10%)

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

(2). Let B, C be two bases for P₂. If B = {x, 1 + x, 1 - x + x²} and the change-of-basis matrix from B to C is
$P_{C \leftarrow B} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 1 \\ -1 & 1 & 1 \end{bmatrix},$
please find C. (10%)

(a) Please show that \( \{A\vec{v}_1, A\vec{v}_2, \dots, A\vec{v}_n\} \) is also a basis for \( R^n \). (12%)

(b) Is (a) still true if A is singular? If not, please give a counter example. (3%)