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Math 120
Homework Assignment 2
due date: Sept. 23, 2008
1. Find the natural domains of
(a)
p
√
1 − 4 − x2
(b)
q
x−
1
x
(c)
q
sin(x)( 21 − sin(x))
2. Suppose that |x − 4| < 2. Find upper and lower bounds for
(a) |x + 1|
(b) |x − 3|
(c) |x2 − 2x − 3|
(d)
|x − 3|
.
|x + 1|
For parts (a) and (b) give both a geometric argument and an algebraic argument using
the triangle inequality. (They really are the same argument and we want to practice
seeing that this is true.) You can use both the upper bound and lower bound parts of
the triangle inequality in your arguments.
3. Suppose that we know that |x − 6| < δ where δ is some positive number.
5x
(a) Find an upper bound (in terms of the unknown number δ) for − 3.
x+4
5x
(b) If |x−6| < 4, what does your answer in (a) give as an upper bound for − 3?
x+4
5x
1
(c) Find a value of δ so that if |x − 6| < δ then − 3 < .
x+4
2
5x
1
(d) For any positive integer m, show how to pick δ so that − 3 <
whenever
x+4
m
|x − 6| < δ.
(i.e., give a rule for picking δ so that the above inequality is true; the particular δ will
depend on m. Part (c) is the case m = 2).
Note1 : The symbol δ is pronounced “delta”.
Note2 : You may assume that δ < 8 if it helps make any steps of the argument less
troublesome.
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4. Suppose that A and B are any real numbers. The goal of this question is to prove
the lower bound part of the triangle inequality:
|A| − |B| ≤ |A + B|.
Throughout the question you are allowed to assume the part of the triangle inequality
that we did prove in class.
(a) For any two numbers C and D, explain why the inequality |C| ≤ D is equivalent
to the inequalities C ≤ D and −C ≤ D.
(This means that you should show that if |C| ≤ D, then C ≤ D and −C ≤ D, and
on the other hand, if both C ≤ D and −C ≤ D you should show that |C| ≤ D.
This question isn’t hard – but it does take a little organizing.)
(b) Use the equation A = (A+B)−B and the upper bound triangle inequality (the one
from class) to prove that |A| ≤ |A+B|+|B|, and therefore that |A|−|B| ≤ |A+B|.
Note: Just because both the triangle inequality from class and this question
use symbols called “A” and “B” it doesn’t mean that when you go to apply the
triangle inequality that you should use the A of the question as the A of the
triangle inequality (and the same for the B’s).
(c) Similarly, show that |B| − |A| ≤ |A + B|.
(d) Finally, prove the lower bound part of the triangle inequality:
|A| − |B| ≤ |A + B|.
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