MthT 430 Problem Set 1
MthT 430 Problem Set 1
In class August 29, 2007 - Turn in September 5, 2007
Group Work Rules:
•
You are encouraged to work together!
•
Away from the group, do your own neat write up of the problems.
•
Acknowledge the group members and any other person/source you use.
1.
(Warmup) For all x, x
2
≥ 0 and x
2
= 0 iff x = 0.
2.
Spivak 1.17
(a)
Find the smallest possible value of 2 x
2
− 3x +4. Hint: Complete the square ....
(b)
Find the smallest possible value of x
2
− 3 x + 2 y
2
+4 y + 2.
(c)
Find the smallest possible value of x
2
+ 4 x y + 5 y
2
− 4 x − 6 y + 7. (A little harder)
Spivak p. 18
3.
(Spivak 1.20) Prove that if
|x − x
0
| <
ϵ
2
and
|y − y
0
| <
ϵ
2
,
then
|(x + y) − (x
0
+ y
0
)| < ϵ,
|(x − y) − (x
0
− y
0
)| < ϵ.
4.
(Spivak 1.21) Prove that if
|x − x
0
| <
min
⎛
⎝
ϵ
2|y
0
| + 1
,1
⎞
⎠
and
|y − y
0
| <
min
⎛
⎝
ϵ
2|x
0
| + 1
,1
⎞
⎠
,
then
|x y − x
0
y
0
| < ϵ.
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On 22 Aug 2014, 12:56.