107年 - 107 東吳大學_碩士班招生考試_數學系(A組)：高等微積分#107433

科目：研究所、轉學考(插大)-高等微積分 | 年份：107年 | 選擇題數：0 | 申論題數：9

試卷資訊

所屬科目：研究所、轉學考(插大)-高等微積分

選擇題 (0)

申論題 (9)

(a) If a sequence in Rⁿ is convergent, then it is Cauchy.

(b) The function f ( x ) = cos x + x is uniformly continuous on  .

2.(20%) Suppose that f is three times differentiable on an interval containing a . Show that

Can you find the generalization to higher derivatives?

(a) Prove that f is not continuous at (0, 0) .

(b) Find ∂_x f , ∂_y f and determine whether ∂ _xf (0,0) and ∂ _y f (0,0) exist or not.

4.(15%) Suppose that f is a homogeneous function of degree a on R ⁿ .

( f is twice differentiable)

(a)

(b)Let S = {( x, y ) : 0 ≤ y ≤ 2, y / 2 ≤ x ≤ 1} . EvaluatedA

(c) Let S = {( x, y ) : y ≥ 0, x 2 + y 2 ≤ 1} . EvaluatedA