[考題] 103微積分
1. True or false.(If it is false, explain why or give an example that disproves
the statement)
(a) If f(x) is continuous at a, then f(x) is differentiable at a.
(b) If {a_n } and {b_n } are divergent, then {a_n+b_n } is divergent.
(c) If f(x) is a continuous, decreasing function at [1,∞),
∞
and limf(x)=0, then ∫f(x)dx is convergent.
x→∞ 1
(d) If f(x) and g(x) are continuous on [a,b] then
b b b
∫[f(x)g(x)]dx=[∫f(x)dx][∫g(x)dx]
a a a
2. Evaluate ∫e^x/x dx? as an infinite series.
π/4
3. Evaluate ∫(1+tant)^3 sec^2 tdt?.
0
4. Evaluate the integral
1
∫(x-1)/√x dx
0
or show that it is divergent.
5.Determine whether the series,Sigma(n=1到∞)n^3/5^n,is convergent or divergent.
6.Evaluate lim(e^4x-1-4x)/x^2.
x→0
7.Use polar coordinates to evaluate
3 √(9-x^2 )
∫ ∫(x^3+xy^2 )dydx
0 -√(9-x^2 )
8.Evaluate ∫y^3 dx-x^3 dy, where C is the circle, x^2+y^2=4.
c
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