課程名稱︰微積分甲下
課程性質︰理組必修
課程教師︰呂楊凱
開課學院：
開課系所︰
考試日期（年月日）︰
考試時限（分鐘）：隔週助教課，每次約30-40分鐘
試題 :
● Quiz 1
1. A saquence {a_n} is given by a_n=sqrt(6), a_(n+1)=sqrt(6+a_n).
(a) By introduction or otherwise, show that {a_n} is increasing and bounded
above by 6. Apply the Mpnptonic Sequence Theorem to show that
lim_a→n a_n exists.
(b) Find lim_a→n a_n.
2. Determine whether the series is convergent or divergent.
(a)Σ(n=1 to ∞) (-1)^(n-1)*(1+1/n)^(-n)
(b)Σ(n=2 to ∞) log_n(1+1/n)
● Quiz 2
1. Find the radius of convergence and interval of convergence of the series
Σ(n=1 to ∞)([(n!)^2](x-1)^n)/(2n)!
2. (a) Find a power series representation for (x^3)arctanx and its radius of
convergence.
(b) Use part (a) to approximate ∫(0 to 0.1)(x^3)arctanxdx correct to
within 10^-7.
● Quiz 3
1. Let γ(t)=(cost, sint, t) and vector_P = (-1,0,pi)
(a) Find the vectors T, N, B of the curve γ(t)at the point P.
(b) Find the equations of the normal plane and osculating plane of the
curve γ(t) at the point P.
2. Find the curvature and the osculating circle of xy=1 at the point (1,1).
● Quiz 6
1. Let H = ∫∫∫_E (9-x^2-y^2)dV, where the E is the soild hemisphere
x^2+y^2+z^2≦9, z≧0.
(a) Using the cylindrical coordinates to find H.
(b) Using the spherical coordinates to find H.
2. By making an appropriate change of variable, evaluate the integral
∫∫(x-2y)/(3x-y)dA, where R is the parallelogram enclosed by x-2y=0,
x-2y=4, 3x-y=1 and 3x-y=8.
● Quiz 7
1. Determine whether or not vector_F is a conservative vector field. If it is,
find a function f such that vector_F = deg(f).
(a) F = (y+ze^(xz))i + (xe^(xz)+2z)j + xk
(b) F = (y+ze^(xz))i + xj + (xe^(xz)+2z)k
2. Let F(x,y)=(-y/(x^2+y^2)+x)i + (x/(x^2+y^2)+y)j. Find the following line
integral ∫_C Fdr.
(a) C is boundary of the region enclosed by the parabolas y=x^2-1 and
y=1-x^2 and C is positively oriented.
(a) C is boundary of the region enclosed by the parabolas y-5=(x-5)^2-1 and
y-5=1-(x-5)^2 and C is positively oriented.