MATH 409.200
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]
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3. [4] (a) Which of the following is true?
(a) There exists a function on [0, 1] which fails to be continuous at every point except
0.
(b) Every bounded function on [0, 1] which is zero except at countably many points
is integrable on [0, 1].
(c) The set of irrational numbers is countable.
(d) Every bounded function on [0, 1] is continuous at at least one point.
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
dt
(c) f (x) =
x+2
t
1
P∞ −x−1
1, if the series
converges
k=1 k
(d) f (x) =
0, otherwise
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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.
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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
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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].
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10. [12] (a) Let f : R → R be a continuous function and let {an }∞
n=1 be a sequence in R
which converges to 0. For each n ∈ N define fn : R → R by fn (x) = f (x + an ) for all
x ∈ R. Prove that fn → f uniformly on every bounded subset of R.
7
P
(b) Let ∞
k=1 ak be a (not necessarily convergent) series with nonnegative terms. For
each n ∈ N define a function fn : R → R by
Pn
P
1,
ak ≤ x ≤ n+1
k=1
k=1 ak
fn (x) =
0, otherwise.
Prove thatPthe sequence {fn }∞
n=1 converges pointwise on R. Also, find a condition on
∞
the series k=1 ak that is equivalent to the uniform convergence of {fn }∞
n=1 on R, and
justify your answer.
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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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