作者lisasweet (小如)
看板NTU-Exam
标题[试题] 105上 李聪成 微积分乙上 期末考
时间Wed Jan 25 00:29:42 2017
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）︰2017/01/12
考试时限（分钟）：160分钟 (10:20-13:00)
试题 :
请将每一步骤表达清楚,不可以只写答案.
2
1.Evaluate the definite integral ∫ x √(x-1) dx
1
2.Find the area (面积) of the region (区域) bounded by the curves
y = tan x, y = sin 2x with 0 <= x <= π/3
3.Evaluate the indefinite integral ∫ 1/ (√(x^2-4)) dx
4.Let f:(0,+∞)→R be continuous such that
x x
∫f(t)dt = xsinx + ∫f(t)/(1+t^2) dt
1 1
for all x belongs to (0, +∞), find an explict formula for f(x)
5. Evaluate the limit.
lim ( 1/(n+1) + 1/(n+2) + 1/(n+3) + .... + 1/(n+n) )
n→+∞
6.Use logarithmic differentiation to find the derivative of the function
x (x^2+1)^2
y = ─────── , where x > 1.
√(2x^3-1)
7.Show that f is one-to-one on (0, + ∞) and (f^(-1))'(3), where f^(-1) is
the inverse function (反函数) of f and
f(x) = 3 + x + tan(πx/2) with -1 < x < 1.
8.Find the volume of a right circular cone (圆锥) with height h and base
radius r.
9.For what values of a and b is the following equation true?
sin(2x) b
lim (──── + a + ────) = 0.
x→3 x^3 x^2
10.Find an equation of the tangent line to the curve y = F(x) at the point
with x-coordinate (x轴) π, where
x
F(x) = ∫ (cos(t)/t) dt.
π
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