102年 - 102 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：基礎數學#146283

1. Evaluate the limit: = ? (5%)

(a)\( y = \ln(x \tan y) \)

(b)\[ y = \int_{x^2}^{\sqrt{x}} \sin(t^3) dt \]

3.Find the area of the region bounded by the curve \( y=\ln x \), the \( x \) axis and the lines \( x=1 \) and \( x=e \). (10%)

4. Given the function . Find: (a) relative extrema; (b) point(s) of inflection (c) sketch the graph. (10%)

5. Show that the volume of a sphere (球) of radius is . (10%)

二、線性代數部分：

1. Let \( A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \). Find \( A^2, A^3, A^n \). (8%)

2. Show that \( \text{tr}(AB) = \text{tr}(BA) \) for any \( n \)-square matrices \( A \) and \( B \), where \( \text{tr}(AB) \) and \( \text{tr}(BA) \) denote the trace of \( AB \) and \( BA \), respectively.
(7%)

3.

Find the dimension and a basis of the solution space \( W \) of the system (10%)

\[ \begin{aligned} x + 2y + z - 3t &= 0 \\ 2x + 4y + 4z - t &= 0 \\ 3x + 6y + 7z + t &= 0 \end{aligned} \]

(a)Find all the eigenvalues. (5%)

(b)Find all eigenvectors. (5%)

5. Determine whether the matrix is invertible, and if it is, please compute its inverse. (15%)