课程名称︰微积分甲下
课程性质︰理组必修
课程教师︰吕杨凯
开课学院：
开课系所︰
考试日期（年月日）︰
考试时限（分钟）：隔周助教课，每次约30-40分钟
试题 :
● 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.