18.014 Exam 2
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Oct 30, 2015
All four problems are weighted equally. You may use any theorems or results
that have appeared in class, problem sets, or sections of the textbook covered
thus far. Just make it clear what you are using!
Name:
1. Let f : [a, b] → R be a continuous function. Prove that there exists a positive real
number M such that
|f (x) − f (y)| < M
for all x, y ∈ [a, b].
18.014 Exam 2
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2. Let f : R≥0 → R be the function on the nonnegative reals given by
f (x) =
Oct 30, 2015
x
√ .
1+ x
Compute the derivative f 0 (x) for all x ≥ 0. Deduce from this that f is strictly increasing.
18.014 Exam 2
Page 3 of 4
Oct 30, 2015
3. For each of the following statements, f : [a, b] → R is a function on a closed interval.
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. If f is differentiable, then f is integrable.
T
F
2. If there is a differentiable function g : [a, b] → [a, b] such that f (x) =
(g(g(x)))2 , then f is differentiable and
f 0 (x) = 2g(g(x))g 0 (g 0 (x)).
T
F
3. If f is strictly increasing and f is differentiable, then f 0 (x) > 0 for every
x ∈ [a, b].
T
F
4. If f is continuous, f (a) ≥ a2 , and f (b) ≤ b2 , then there exists x ∈ [a, b]
such that f (x) = x2 .
T
F
5. If f is differentiable and |f 0 (x)| ≤ 1 for all x ∈ [a, b], then |f (x) − f (y)| ≤
|x − y| for all x, y ∈ [a, b].
T
F
6. If f is differentiable and the derivative f 0 is continuous, then
Z b
f (b) = f (a) +
f 0 (x)dx.
a
18.014 Exam 2
4. Let f : R → R be a function.
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(a) Suppose that f satisfies
|f (x) − f (y)| ≤ |x − y|
for all x, y ∈ R. Prove that f is continuous.
(b) Suppose that f satisfies
|f (x) − f (y)| ≤ (x − y)2
for all x, y ∈ R. Prove that f is constant.
Oct 30, 2015