試題 :
1. Let f be continuous on [a,b]. if G is any antiderivative function
of f on [a,b], prove that
b
∫ f(x)dx = G(b) - G(a)
a
2. If f is continuous on [a,b] and u is a differentiable function of x
with values in [a,b], to find derivative of
u
∫ f(t)dt
a
3. The base of a solid is the region between the
2 2
parabolas x = y and x = 3 - 2y
Find the volume of the solid given that cross section perpendicular
to the x-aixs are squares.
4. If a is positive and r is rational , show that
r
a a
∫ dt/t = r∫ dt/t
1 1
5. Prove that
x
as x →∞ , (1+1/x) → e
＿＿＿＿ 2 -1
6. Show that F(x) = x/2 √a^2 -x^2 +a /2 sin(x/a) ,a>0
＿＿＿＿
is an anitderivative for f(x)=√a^2 -x^2
7. Determine A,B and c so that y = Acoshcx + Bsinhcx satisfies the
condictions 4y" - y = 0 , y(0) = 1 , y'(0) = 2 Take c > 0 .
(1~6題 每題均15分 第7題10分)
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