110年 - 110 國立清華大學碩士班考試入學試題_計算與建模科學研究所:數學分析#105592

(i)(7分)

(ii) (10 分)

P.S. For (i) you can use the fact

5.(8分)Calculate
where C is the path enclosing the annular region R shown in Figure 1.

(i) (10 分) Show that the Taylor series of f(x) at x = 0 is

(ii) (8 分) By integrating f(-x²) over a suitable region and using the formula of verify the value of
 Here you need only calculate it without the rigorous analysis.

(iii) (8 分) Let Find a matrix B such that = I + A. 

 Hint: (i) is a key for obtaining a B, where you shall notice16 = 2⁴. 

7. (8 分) Show that the area of a triangle with the vertices (x₁,y₁), (x₂,y₂),and (X₃,y₃) is
 where0＜x₁＜x₃2and 0＜y₁＜y₂＜y₃. 

(ii) (8 分) If B = is an orthonormal basis for V, then the coordinate representation of a vector w relative to B is w =(w,น₁)u₁ + (w,u₂)u₂ +...+(w,u_n)u_n.

(i) (8分)ρ(AB) ≤ρ(A)p(B).

(i) (10 分) Prove the Cauchy-Schwarz Inequality
  

(ii) (9 分) If v is not a zero vector, show that if and only if there exists a scalarsuch that u = σv.