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107年 - 107 國立臺灣大學轉學生招生考試試題_理工科聯招：微積分(B)#110857

科目：研究所、轉學考(插大)-微積分 | 年份：107年 | 選擇題數：5 | 申論題數：11

試卷資訊

所屬科目：研究所、轉學考(插大)-微積分

選擇題 (5)

1. Let f be a smooth function on R. 
(a) If f" ＞ 0 on R, then f' must.be increasing on R.
 (b) If f" ＞ 0 on R, then f must be concave upward on R.
 (c) If f"(a) = 0, then the point (a, f(a)) must be an infection point. 
Among the above three statements, how many of them are true? (A)Only one.(B) Only two. (C)All of them. (D) None of them.
複選題

2. A function f on R is odd if f(-x)"=-f(x) for all xR while it is even if f(-x) = f(x) for all x  R. Suppose that g and h are two functions on R. 

(a) If g is an even function, then the composite function h o g must be even.
(b) If g is an odd function and , then the limit must be -3.(c) If g is an even function and , then the limit must be 3.

Among the above three statements, how many of them are true? (A)Only one.(B) Only two. (C)All of them. (D) None of them. 

3. Consider the integral dA, where D is the region bounded by two concentric circles centred at the origin with radi r and 1 respectively, 0＜" ＜ 1. Let . 
(a)Jpn is convergent if p ＜ 1.
(b) Jp is convergent if 1 ＜p ＜2.
(c) Jp is convergent if 2 ＜ p. 

Among the above three statements, how many of them are true? (A)Only one.(B) Only two. (C)All of them. (D) None of them.

4. Let (a)(b)(c)Among the above three staterents, how many of them are true? (A)Only one.(B) Only two. (C)All of them. (D) None of them. 

5. (a) The series is absolutely convergent.

 (b) The series is absolutely convergent.
 (c) The series is absolutely convergent.

  Among the above three statements, how many of them are true? (A)Only one.(B) Only two. (C)All of them. (D) None of them. 

申論題 (11)

6.63294fb1382d9.jpg=__(6)__.

7. Let f(x)= 63294fe20c52c.jpg sin(y+xcosy) dy. It follows that f'(0) = __(7)__.  

8. Let f(x) = arctan 63295030698b8.jpg.The graph of f has a horizontal asymptote represented by the equation __(8)__ and the global minimum value of f is__(9)__.

9.d63295063422fc.jpg= __(10)__. 
10. The third term of the Maclaurin series (i:e. the 'Taylor series centred at x = 0) of arcsin(3x) is __(11)__ (note that the answer should be a monomial in a； the term of ,' is counted as the0-th term). The radius of convergence of the series is __(12)__.
11. If y = y(x) satisfies the differential equation x2y' + 3xy = 2ln(x) for x ＞ 0 with y(1) = 2, then y(x)= __(13)__.

12. Let C be a variable path in the ay-plane of arc-length 1 starting at the point (632951765115a.jpg,1) and ending at the point (a,b). Suppose that G(a,b) := 63295197d9d18.jpg .dr, where f(x,y) := arctan 632951b5f362c.jpg, is a function in a and b. Then, G attains its maximum at a =__(14)__ and b =__ (15)__ and the maxinum value of G is __(16)__.

13. If a and b are positive constants and if max{p,a} denotes the maximum between the numbers p and g, the iterated integral63295218eabeb.jpg dy dx= __(17)__.
(a) =__ (18)__.

(b) If S1is the boundary surface of E (including all faces) endowed with the outward orientation, one has 632953349ed29.jpg curl Fㆍ dS = __(19)__

(c) If S2 is the surface obtained from S1by removing the face in the xy-plane while keeping the orientation from S1 on all other faces, one then has 6329537e98dcb.jpgcurlF. dS =__ (20)__.