110年 - 110 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#147046

(a) \( (6\%) \) Prove that \( f(x) \) is continuous at 0.

(b) \( (7\%) \) Prove that \( f(x) \) is differentiable at 0 and find \( f^{\prime}(0) \).

(c) \( (6\%) \) Find \( f^{\prime}(x), x > 0 \).

(d) \( (6\%) \) Find \( \lim_{x\to \infty}f'(x) \)

2. (15%) Find the relative maxima, relative minima, and saddle points of
$f(x, y) = 2x^3 - 3x^2y + 2y^2 - 6y$

3. (10%) Find the area of R, where R is the region bounded by the curves $y = \sqrt{x}, \ y = \sqrt{5x}, \ y = 2x^2 \text{ and } y = 8x^2.$

二、線性代數 (10% each, 50% in total)
1. (Proof) Let \( V \) be a finite-dimensional vector space with \( \dim(V) = n \). If a set has more than \( n \) vectors, then it is linearly independent.

2. Let \( W \) be the subspace of \( R^5 \) spanned by the vectors \( w_1 = (2, 2, -1, 0, 1) \), \( w_2 = (-1, -1, 2, -3, 1) \), \( w_3 = (1, 1, -2, 0, -1) \), \( w_4 = (0, 0, 1, 1, 1) \). Find a basis for the orthogonal complement of \( W \).

3. If \( V \) is a finite-dimensional vector space, and \( T: V \to V \) is a linear operation, then nullity \( (T) = 0 \) if the range of \( T \) is \( V \).

4. Show that the matrix \( A = \begin{bmatrix} 2\sin^2 \alpha & 2\sin^2 \beta & 2\sin^2 \gamma \\ 3\cos^2 \alpha & 3\cos^2 \beta & 3\cos^2 \gamma \\ 1 & 1 & 1 \end{bmatrix} \)is not invertible for any value $\alpha, \beta, \gamma$.

5. Let $A = \begin{bmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3 \end{bmatrix}$, please express $A^3$ as $PDP^{-1}$ with D a diagonal matrix and P an invertible matrix.