课程名称︰微积分乙上
课程性质︰共同必修
课程教师︰李聪成
开课学院：管院
开课系所︰
考试日期（年月日）︰2016/11/10
考试时限（分钟）：160分钟 (10:20-13:00)
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
请将每一步骤表达清楚,不可以只写答案.
1. Is there a number a such that
3x^2+ax+a+3
lim ──────
x→2 x^2+x-2
exists? If so, find the value of a and the value of the limit.
2.Evaluate the limit.
lim (√(4x^2+3x)-2x ) 注:根号是整个4x^2+3x
x→+∞
3.Express the limit as the derivate of some function f at some point a.
Then evaluate the limit.
cosθ-0.5
lim ──────
θ→2 θ-(π/3)
4.Show the the function f(x) = x|x| has an inflection point at (0,0) but
f"(0) does NOT exist.
5.For what values of a and b is the line 2x+y=b tangent to the parabola
y = ax^2 when x=2?
6.Sketch the graph of the given function, making use of any suitable
information you can obtain from the fuction and its first and second
derivatives.
2x^2
f(x) = ──────
x^2 - 1
7.Find the absolute maximum and absolute minimum values of
f(x) = 2cos x + sin 2x
on thee interval [0, π/2].
8.Use implict differentiation to find an equation of the tangent line (切线)
to the curve
x^2 + y^2 = ( 2x^2 +2y^2 -x )^2
at the point (0, 1/2).
9.Two cars start moving from the same point. One travels south at 30km/h
and the other travels west at 72km/h. At what rate is the distance between
the cars increasing two hours later?
10.Find the points on the ellipse (椭圆) 4x^2 + y^2 =4 that are farthest
away from (离最远) the point (1,0).