103年 - 103 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#146058

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

2. Find an equation for the tangent line to the curve y=F(x) at the point P where x=1 and (5%)

3. Calculate =? where [x] it the greatest integer n such that n≤x.(5%)

4. Find the points on the graph of y=4-x² that are closest to the point P(0, 2).(5%)

5. Show that the integral diverges if p≤1.(10%)

(a) relative extrema ；

(b)point(s) of inflection ；

(c)sketch the graph.

7. The region between the curve , , and the x-axis is revolved about the x-axis togenerate a solid. Find its volume.(10%)

二、線性代數部分：
1. Find the matrix A whose eigenvalues are 4 and 1 and whose eigenvectors are \( \begin{bmatrix} 2 \\ 1 \end{bmatrix} \)and\( \begin{bmatrix} 3 \\ 1 \end{bmatrix} \), respectively.(10%)

(1) Find an invertible matrix P such that P^-1AP is a triangular matrix.(8 %)

(2) Compute A¹⁰= ?(7 %)

3. Let T:→ be a linear transformation. Prove that T is one to one if and only ifT is onto.(10%)

4. Find the column space of the matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \) , the dimension of thecolumn space and the determinant of A, det(A).(15%)