18.014 Final Exam
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Dec 17, 2015
All problems are weighted equally. You may use any theorems or results that
have appeared in class, problem sets, or sections of the textbook covered in
this course. Just make it clear what you are using!
You may consult the textbook and your notes during this exam. All other
external resources (calculator, internet, etc) are forbidden.
Name:
1. Let f : [−π, π] → R be defined by
f (x) =
(a) Is f differentiable?
(b) Is f integrable?
√
1 − cos x.
18.014 Final Exam
Page 2 of 10
Dec 17, 2015
2. (a) Let x 6= 1 be a real number. Prove by induction that for any positive integer n,
n+1
1 − x2
.
(1 + x)(1 + x )(1 + x ) · · · (1 + x ) =
1−x
2
4
2n
(Here the exponents in the factors on the LHS are powers of 2.)
(b) Define a sequence {an } by
an =
1
1+
2
1
1
1 + 2 · · · 1 + 2n .
2
2
Does limn→∞ an exist? Determine its value if it does.
18.014 Final Exam
Page 3 of 10
Dec 17, 2015
3. Circle either T or F to indicate whether each statement is true or false. You do not need
to justify your answers.
T
F
1. Let S = { n1 | n ∈ N}. Then sup S = 1.
T
F
2. A function is integrable if and only if its upper integral is less than or equal
to its lower integral.
T
F
3. There are exactly 10 distinct functions f : {1, 2} → {1, 2, 3, 4, 5}.
T
F
4. Let f : R → R be continuous. Then
Z x
d
f (t)dt = f (x) − f (−x).
dx −x
T
F
5. Let S be the union of the intervals [0, 1] and [2, 4]. Then any continuous
function f : S → R is bounded.
T
F
6. Let f : R → R be a function such that limx→2 f (x) = 3. Then there exists
δ > 0 such that
2.9 < f (x) < 3.01 whenever 0 < |x − 2| < δ.
18.014 Final Exam
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Dec 17, 2015
4. Determine the minimum and maximum values of the function f (x) = x + x(log x)2 on
the interval [1/e2 , 1].
18.014 Final Exam
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Dec 17, 2015
5. Let f : R → R be differentiable. Prove that there exists a real number x such that
f (x) sin x − f 0 (x) cos x = 0.
18.014 Final Exam
6. (a) Compute the integral
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Z √
4 − x2 dx.
(b) Compute the integral
Z
0
2
√
x 4 − x2 dx.
Dec 17, 2015
18.014 Final Exam
7. (a) Compute the limit
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√
lim
x→0
1+x−
x
Dec 17, 2015
√
1−x
.
(b) Find real numbers A, B, C such that
√
√
1 + x − 1 − x = A + Bx + Cx2 + o(x2 ) as x → 0.
18.014 Final Exam
Page 8 of 10
Dec 17, 2015
8. Circle either T or F to indicate whether each statement is true or false. You do not need
to justify your answers.
T
F
1. Let f : [0, 1] → R be a continuous function. Then
Z 1 Z 1
Z 1
xf (x)dx =
f (t)dt dx.
0
T
F
F
x
2. If f is an antiderivative of the rational function
real number C such that
f (x) =
T
0
1
,
2x+1
then there exists a
1
1
log |2x + 1| + C for all x 6= − .
2
2
3. Let f, g : R → R be infinitely differentiable functions. Assume that the
fifth Taylor polynomial of f at the point 0 is x2 + 3x5 and the fifth Taylor
polynomial of g at the point 0 is x2 + 2x5 . Then
f (x)
3
= .
x→0 g(x)
2
lim
T
F
4. If {an } is an increasing sequence with limit L, then sup{an | n ∈ N} = L.
T
F
5. Any rearrangement of the infinite series
∞
X
n=1
1
1
1
1
1
(−1)n−1
=
−
+
−
+
− ···
(2n + (−1)n )2
12 52 52 92 92
converges to 1.
T
F
6. Let {fn } be a sequence of continuous functions on the interval [0, 1] such
that limn→∞ fn (x) = 1 for all x ∈ [0, 1]. Then there exists M ∈ R such
that fn is a nonnegative function for all n > M .
18.014 Final Exam
9. Consider the power series
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∞
X
n=0
Dec 17, 2015
2n
xn .
n log n
For which real numbers x does this series converge? For which x does it converge
absolutely?
18.014 Final Exam
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Dec 17, 2015
P
n
10. Let f be a function represented by a power series
an x on the interval (−1, 1). In
other words,
∞
X
f (x) =
an x n = a0 + a1 x + a2 x 2 + · · ·
n=0
for all x ∈ (−1, 1). Assume that f satisfies f 000 (0) = 4 and
f 0 (x) = f (x2 ) for all x ∈ (−1, 1).
Compute the first eight coefficients a0 , a1 , . . . , a7 of the power series.