112年 - 112 國立政治大學_碩士班暨碩士在職專班招生考試試題_統計學系：基礎數學#130257

1. State and prove the mean value theorem.

2. Use ε-δ argument to show that = 0.

(a)

(b)

(c)

(d)

(e) ∫ sin⁵x cos² xdx.

4. Find if tan= x + y.

5. Find the extreme value of the function f(x, y, z) = x + 2y + 3z subject to constraints x - y + z = 1 and x² + y² = 1.

(a) Find possible eigenvalues of A.

(b) Determine the rank of A.

7. Prove that if Q is an orthogonal matrix, then det(Q) = 1.

8. Let V and W be two vector spaces and let T : V → W denote a linear transformation. Suppose that N(T) and R(T) are null space and range of T, respectively. Show that N(T) and R(T) are subspaces of V and W, respectively.

9. Let A =. Compute for any positive integer n.