MATH 409.501
Final Exam
May 8, 2009
Name:
ID#:
The exam consists of 11 questions, the first 5 of which are multiple choice. The point
value for a question is written next to the question number. There is a total of 100 points.
No aids are permitted.
For questions 1 to 5 circle the correct answer.
1. [4] Which of the following is false?
(a) Every bounded sequence in R has a convergent subsequence.
(b) Every continuous function on [0, 1] is uniformly continuous on [0, 1].
(c) Every absolutely convergent series of real numbers is convergent.
(d) Every bounded function on [0, 1] is the uniform limit of a sequence of continuous
functions on [0, 1].
Z
x
|t| dt for all x ∈ R. Which of the following is false?
2. [4] Define g(x) =
−1
(a) g 0 (0) exists
(b) g 00 (0) exists
Z 1
g(x) dx ≥ 0
(c)
0
(d) g is continuous on [0, 1]
1
3. [4] (a) For each n ∈ N define the function fn (x) = cos(x/n). Which of the following
functions f is the pointwise limit on R of the sequence {fn }∞
n=1 ?
(a) f (x) = 1 for all x ∈ R
(b) f (x) = 0 for all x ∈ R
(c) f (x) = cos x for all x ∈ R
(d) f (x) = x for all x ∈ R
4. [4] Let f and g be differentiable functions on R such that f (0) = 2, f 0 (0) = 3, and
g 0 (2) = 2. Then (g ◦ f )0 (0) is equal to
(a) 0
(b) 2
(c) 3
(d) 6
5. [4] Which of the following functions is continuous on [0, 1]?
(a) the pointwise limit as n → ∞ of the sequence of functions fn given by fn (x) = xn
x, x 6= 1/2
(b) f (x) =
0, x = 1/2
Z ∞
1
(c) f (x) =
dt
x+2
t
1
P∞ −x−1
converges
1, if the series
k=1 k
(d) f (x) =
0, otherwise
2
6. [12] (a) State the completeness axiom for the real numbers.
(b) Let E be a nonempty subset of R. State what it means for a function f to be
uniformly continuous on E.
(c) State what it means for a series
∞
X
ak of real numbers to converge.
k=1
(d) Let f be a function defined on an open interval I and let a ∈ I. State the definition
of the limit of f (x) as x → a.
3
7. [14] Determine with explanation whether the series converges or diverges.
(a)
(b)
∞
X
k+3
k2 + 2
k=1
∞
X
k=1
(c)
1
(3 + (−1)k )k
∞
X
cos(k 3 − 4)
k=1
k3 − 4
4
Z
x2
2
et dt. Determine F 0 .
8. [14] (a) Define F : R → R by F (x) =
0
Z
1
(b) Evaluate
−1
4|x|
dx.
x2 + 3
Z
(c) Does the improper integral
tion.
0
∞
e−x
dx converge or diverge? Provide justificax2 + 3
5
9. [14] (a) Find the radius of convergence and interval of convergence of the power series
∞
X
(−2)k k
√
x .
k2 + 1
k=1
(b) Give an example of a power series whose interval of convergence is [0, 2].
6
10. [12] (a) State what it means for a sequence {fn }∞
n=1 of functions on [0, 1] to converge
uniformly to a function f on [0, 1].
(b) For each n ∈ N define the function fn : [0, 1] → R by fn (x) = xex/n , and define
f : [0, 1] → R by f (x) = x. Prove that fn → f uniformly on [0, 1].
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11. [14] (a) Define what it means for a bounded function f on a closed interval [a, b] to be
integrable on [a, b].
(b) Prove directly from the definition of integrability that the function f : [0, 1] → R
given by
x, 0 < x ≤ 1
f (x) =
1, x = 0
is integrable on [0, 1].
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