Department: ID #: Name:
Write your answers for the problems (problems 2, 4, 6, 8, 10, 12) in the boxes provided. For the other problems, justify your answers with details.
주의: 객관식 문제(2, 4, 6, 8, 10, 12 번 문제)의 답은 주어진 네모 칸 안에 써야 합니다. 주관식 문제 (1, 3, 5, 7, 9, 11 번 문제)는 풀이 과정과 답을 써야 합니다.
1. (10 points) Evaluate the integral.
Z xp4x2− 4x + 3 dx 2. (5 points) ㄱ. lim n→∞sinh(ln n) ㄴ. lim n→∞ Z n 1 1 xp dx, p > 1 ㄷ. limn→∞ (ln n)10 n ㄹ. ∞ X n=1 1 − 1 n n ㅁ. ∞ X n=1 ln n n + 1
Which of the above sequences and series are convergent? Answer: ⇐ 답은 네모 칸 안에 쓰시오.
A ㄱ, ㄴ B ㄱ, ㄷ C ㄱ, ㄹ D ㄱ, ㅁ E ㄴ, ㄷ F ㄴ, ㄹ
G ㄴ, ㅁ H ㄷ, ㄹ I ㄷ, ㅁ J ㄹ, ㅁ K ㄱ, ㄴ, ㄷ L ㄱ, ㄴ, ㄹ
M ㄱ, ㄴ, ㅁ N ㄱ, ㄷ, ㄹ O ㄱ, ㄷ, ㅁ P ㄱ,ㄹ,ㅁ Q ㄴ, ㄷ, ㄹ R ㄴ, ㄷ, ㅁ
3. (12 points) Evaluate the integrals. (a) Z 1 0 x 1 − x 1/2 dx (b) Z ∞ 2 x2 x4− 1dx 4. (5 points) ㄱ. a1 = 1, an+1= 5an− 3 ㄴ. a1 = 2, an+1= 1 3 − an ㄷ. a1 = 2, an+1= an 1 + an ㄹ. a1 = 6, an+1= an n
Which of the above sequences is/are convergent? Answer: ⇐ 답은 네모 칸 안에 쓰시오.
A ㄱ B ㄴ C ㄷ D ㄹ E ㄱ, ㄴ
F ㄱ, ㄷ G ㄱ, ㄹ H ㄴ, ㄷ I ㄴ, ㄹ J ㄷ, ㄹ
5. (12 points)
(a) Find the real values of p for which the integral converges. Z ∞
0
1
xp(1 + x3)pdx
(b) Find the real values of p for which the integral converges. Z ∞
1
(sin(1/x))p dx
(c) Determine if the integral converges or diverges. If the integral converges, determine its value. Z ∞
0
(ln x)3 1 + x2 dx
6. (5 points) Find the area of the region bounded by the curves
y = sin2x, y = sin3x, 0 ≤ x ≤ π. Answer: ⇐ 답은 네모 칸 안에 쓰시오. A π 2 − 1 3 B π 2 − 2 3 C π 2 − 1 D π 2 − 4 3 E π − 5 3 F π − 2
7. (10 points) Find the area of the surface generated by rotating the curve x = cosh t, y = sinh t from (1, 0) to (√2, 1) about the x-axis.
8. (5 points) ㄱ. Z ∞ 0 x x3+ 1dx ㄴ. Z ∞ 1 1 + sin2x √ x dx ㄷ. Z ∞ 1 x + 1 √ x4− xdx ㄹ. Z ∞ 0 arctan x 2 + ex dx ㅁ. Z 1 0 sec2x x√x dx
Which of the above integrals is/are convergent? Answer: ⇐ 답은 네모 칸 안에 쓰시오.
A ㄱ B ㄴ C ㄷ D ㄹ E ㅁ F ㄱ, ㄴ
G ㄱ, ㄷ H ㄱ, ㄹ I ㄱ, ㅁ J ㄴ, ㄷ K ㄴ, ㄹ L ㄴ, ㅁ
M ㄷ, ㄹ N ㄷ, ㅁ O ㄹ, ㅁ P ㄱ, ㄴ, ㄷ Q ㄱ, ㄴ, ㄹ R ㄱ, ㄴ, ㅁ
S ㄱ, ㄷ, ㄹ T ㄱ, ㄷ, ㅁ U ㄱ,ㄹ,ㅁ V ㄴ, ㄷ, ㄹ W ㄴ, ㄷ, ㅁ X ㄴ, ㄹ, ㅁ
9. (14 points)
(a) What is the value for the following integral?
I := Z ∞
0
u2 (1 + u3)2du.
(b) Derive a polar equation r = f (θ) for the folium of Decartes, which is given by x3+ y3 = 3xy.
In particular, express f (θ) in terms of tan θ and sec θ.
(c) Find equations for the tangent lines to the polar curve r = f (θ) obtained in (b) when r = 0 (that is, find equations for the tangent lines to the folium of Decartes at the origin).
(d) Let J be the area of the region enclosed by the folium of Decartes. Find J/I, where I is given in (a).
10. (5 points) Find the length of the arc of the curve y = x
3 12 + 1 x, 1 ≤ x ≤ 2. Answer: ⇐ 답은 네모 칸 안에 쓰시오. A 2 3 B 3 4 C 5 6 D 11 12 E 1 F 13 12
11. (12 points)
(a) Graph the polar curve r = 1 + sin(2θ) and r = cos(2θ).
(b) Find the area of the region that lies inside r = 1 + sin(2θ) and outside r = cos(2θ).
12. (5 points) Find Z 3 √ 3/2 0 x3 (4x2+ 9)3/2 dx Answer: ⇐ 답은 네모 칸 안에 쓰시오. A 1 32 B 1 16 C 3 32 D 1 8 E 5 32 F 3 16
