110年 - 110 國立清華大學碩士班考試入學試題_計量財務金融學系/乙組(財務工程組):微積分#105286

1. The coefficient of the x² term in the Tayler series of expanded at 2 = 0 is(A) (In2)²/2² (B) (In2)²/2 (C)1n²/2 (D) In²/2²

2. lim(x,y )→(0,0)= (A) 2 (B) 1 (C)0 (D) ㆀ

3.= (A) 1 (B) ㆀ (C) exp (1) (D) In(1)

Part 1: Fill in blanks. 
1. =______

2. =______

(a) For p ＞ 0, f (x) = exp(-px), show that

(b) For p ＞ 0, f(x) = exp , show that

2. Given a natural logarithmic function Iny, for variables y ＞ 0 and x ＞ 0,
calculate the right-hand side of the equation below is equal to the left-hand side.

where z⁺ = max f{z, 0].