課程名稱：高等微積分(二)
課程性質：必修
課程範圍：課本ch8~ch9
開課教師：蔡紋琦
開課學院：商學院
開課系級：統計二
考試日期(年月日)：2009/04/14
考試時限(Mins)：二234
試題本文：
1.(40 pts) For each of the following statements, determine whether it is true
or false.
(a)___The Taylor series expansion of the function f:R→R define by f(x) =
e^x^2+e^(-x^2) about the point x0 = 0 is 2+2x^4/2!+2x^8/4!+2x^12/6!+...
for all x in R.
(b)___The equality ln(1+x) = x-x^2/2+x^3/3-x^4/4+...holds only when -1 < x < 1
(c)___If a sequence of functions {|fn|} converges pointwise to a function |f|
on D, then {f} also converges pointwise to f on D.
(d)___Suppose that the function f:R→R is continuous. Then for each positive
number ε, there is a polynomial p:R→R such that |f(x)-p(x)| <ε for all
points x in R.
∞ ∞
(e)___Suppose that Σak and Σbk are series of positive numbers such that
k=1 k=1
∞
lim(bk/ak) = l > 0. Then the series Σak diverges if and only if the series
k=1
∞
Σbk diverges.
k=1
(f)___Suppose that both the sequences {fn:D→R} and {gn:D→R} are uniformly
Cauchy on D. Then the sequence{fn-gn:D→R} is also uniformly Cauchy.
(g)___If fn is a sequence of continuous functions on D converging to a
continuous function f on D, then the convergence is uniform.
(h)___Suppose that {fn:(0,1)→R} is a sequence of continuously differentiable
function that converges uniformly to the function f:(0,1)→R. Then the
limit function f is also continuously differentiable and fn'(x)→f'(x) for
all x in (0,1).
∞
(i)___If the power seriesΣakx^k diverges when x = -1/3, then it also diverges
k=0
when x = 2/3.
∞
(j)___Consider the power series Σakx^k. Suppose that lim |ak|^(1/k) = α for
k=0 k→∞
∞
some α> 0. Then the radius of convergence of the seriesΣakx^k is 1/α.
k=0
2.(8pts)Please compute the third Taylor polynomial for the function f:R→R
x
defined by f(x) =∫e^(-t^2) dt at the point x0 = 0. Show your work.
0
3.(8pts)Please state the Cauchy integral remainder theorem.
4.(10pts)Let n be an odd natural number and x0 is a point in R. Suppose that
(n+1)
f:R→R has n+1 derivatives and that f :R→R is continuous. Assume that
(k) (n+1)
f (x0) = 0 if 1≦k≦n and that f (x0) > 0. Please use the Lagrange
remainder theorem to verify that x0 is a local minimizer.
5.For each number n and each x in[0,1], define fn(x) = 1/(nx+1).
(a)(6pts)Find the function f:[0,1]→R to which the sequence{fn:[0,1]→R}
converges pointwise.
(b)(8pts)Prove the convergence in (a) is not uniform.
6.(10pts)Please give an example of a sequence{fn:[0,1]→R} where the sequence
{fn} converges pointwise to f on [0,1] but sup fn(x) does not converge to
0≦x≦1
sup f(x).
0≦x≦1
7.(10pts)For each natural number n, let the function fn:R→R be bounded.
Suppose that the sequence{fn} converges uniformly to f on R. Prove that the
limit function f:R→R also is bounded.
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※ 編輯: linda780531 來自: 114.45.67.69 (07/01 00:19)