106年 - 106 國立臺灣大學轉學生招生考試試題_理工科聯招：微積分(B)#110859

選擇題 (10)

1. (A) (B)(tan π)=π (C) If f(R) = , then f(x) = x. (D) .

2. (A) If f(x) is continuous on (a,b), then there is c (a,6) such that f(c) =(B) Suppose that f(x) is continuous but not differentiable at a = 0. The function g(x) = xf(x) must be differentiable at x = 0. (C) If f(2) is a differentiablo function, then . (D) If f(x) is continuous on [a, b] and differentiable on (a,b), then there is a unique c (a,b) such that f(b)-f(a)=f(c)(b-a).

3. Consider the function(A) f(x) does not have any horizontal asymptotes and any vertical asymptotes. (B) f(xc) has a horizontal asymptote, but f(x) does not have any vertical asymptotes. (C) f(x) has a vertical asymptote, but f(x) does not have any horizontal asymptotes. (D) f(x) has a horizontal asymptote and a vertical asymptote.

複選題

4. (A) sin me sin ne de = 0 for all m,n N. (B) sin me cos na de = 0 for all m,n N. (C) cos me cos nae do = 0 for all m,n N. (D) sin^m t cosⁿ a dx = 0 for all m,n N.

5. Consider the improper integral , where p is a postive aumber. (A) Ip is convergent if p ＞ 1 and divergent if 0 ＜ p ≤ 1. (B) Ip is convergent if 0 ＜ p <1 and divergent if p ≥ 1. (C) Ip is convergent for all p ＞ 0. (D) Ip is divergent for all p ＞ 0.

6. The enclosed area of the curve (x² + y²)2 = x² - y²can be written in the integral form as(A)(B)(C) (D)

7. (A) If both and m are abolitely cnvergemt, than(B) If is convergent and is divergent, then must be divengent. (C)Sugpose that f(x) is a positive and continuous function on [1,∞), and the improper integral

f(x) de is convergent. Let a_n= f(n), then a_n is convergent.

n=1 (D) Ifa_n≤ b_nfor all n is convergent, thena_n must be convergent.

8. (A) is absoiutely convergent. (B) is convergent. (C)is cnvergent. (D) is divergent.

9. Consider the function f(x,y) = . (A) f(x,y) is not continuous at (0,0), (B) f(x,y) is continuous at (0,0), but partial derivatives fa(0,0) and fy(0,0) do not exist. (C) Both partial derivatives fx(0,0) and fy(0,0) exist, but f(x,y) is not differentiable at (0,0). (D) f(x,y) is differentiable at (0,0).

10. Consider the spherical coordinates systemn a = psin dcosO,y = psin osin 6, and z = pcoso. The volume element dV = dedyda can be changed as (A) ρsinΦdpdθdΦ. (B) ρcosΦdρdθdΦ. (C) p²sinΦdρdθdΦ. (D) p²cosΦdρdθdΦ.