97年 - 97 淡江大學轉學考高等微積分#55794

科目：研究所、轉學考(插大)-高等微積分 | 年份：97年 | 選擇題數：0 | 申論題數：10

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 (10)

【已刪除】1. Show that t(x) =

is not uniformly continuous on (0,1). (10 points).

【已刪除】2.

【已刪除】3. Iff: R is differentiable and i = f(x-y), show that

=0.(10 points)

4. Let K be a compact subset of R and let f be a real —vaiued function on K. Prove that if f is continuous on K, then f (K) is compact. (10 points)

(1). Show that g=f^-1 exists and is differentiable in some nonempty open set containing (2,5). (10 points)

(2). Find Dg(2，5)(the total derivative ofg at(2,5)). (10 points)

【已刪除】6. Let f be continuous on [a, b]. Show that

A in [a, b] if and only iff(x) = 0 for all x in[a，b]. (15 points)

【已刪除】7. Let t : [0, l]→[0，1] be continuous. Prove that there is c

[0，11 sucli that f(c)=c. (10 points)

【已刪除】8.

(5 points)

9. Let {f_n} be a sequence of real-vaiued functions on [0，l] and f_n converges uniformly to a function f. Prove that if each f_n is continuous on [0,1], then f is continuous on [0,1]. (10 points)