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标题[试题] 103-1 洪立昌 微积分甲上 第三次小考
时间Fri Dec 19 10:24:37 2014
课程名称︰微积分甲上
课程性质︰大一必修
课程教师︰洪立昌
开课学院：工学院
开课系所︰土木系
考试日期（年月日）︰2014/11/18
考试时限（分钟）：60 mins
试题：
1. (55 points) Solve the following problems.
(a) (20 points) Find the limit
lim_{x→0} (1/x) ∫(from 0 to x) [1-tan(2t)^(1/t)] dt.
(b) (20 points) Find the limit
lim_{x→0} [1/(x^4)] ∫(from 0 to x^2) [t/sqrt(1+t^3)] dt.
(c) (15 points) Find a function f and number a such that
6 + ∫ (from a to x) [f(t)/(t^2)] dt = 2 sqrt(x)
for all x > 0.
2. (45 points) [Fundamental Theorem of Calculus]
(a) (25 points) State without proof the Fundamental Theorem of Calculus,
specifying all conditions required of the function
concerned to guarantee the conclusion of the theorem.
(b) (20 points) If f is continuous and g and h are dierentiable
functions, find a fomula for
d/dx ∫ (from h(x) to g(x)) f(t) dt.
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