Thomas Hill
Math 4200
SelfQuiz #12
(3)
Prove Theorem 2.22.3. Suppose
continuous at
Let
that
g ( a ) , then
Next since
g
is continuous at
f (g ( x ) ) is continuous at
> 0 be arbitrary. Since
zg( a)< 1
g( x)
implies
f
is continuous at
x=a
and
f ( x)
is
Thomas Hill
Math 4200
SelfQuiz #16
Prove using the principle of mathematical induction:
n
i 2=
i=0
n(n+1)(2 n+1)
6
n=0
The base case for
0
i2=0=
i=0
0(0+1)(2(0)+1)
6
We can similarly verify
1
i2=1=
i=0
i 2=
Now we assume
i=0
n
We show that
n
n=1 just
Thomas Hill
Math 4200
SelfQuiz #22
Please prove that if a set is compact, then it is bounded.
If a set
S
is compact then it has a finite subcover. In other words:
S I U =U N
For
NI
S U N
which gives us the largest interval which covers the set
and since
Thomas Hill
Math 4200
Self Quiz #4
1. (4) Claim: assuming associative, distributive, and commutative laws hold for infinite sums
11+11+1=
we argue:
4
7
Consider the following infinite series:
4
7
11
14
18
21
1x x + x + x x x +
Now we notice that by facto
Thomas Hill
Math 4200
SelfQuiz #17
( x )= t x1 et dt . Prove that
(4) Define
( n )=(n1)!
0
for
n N and not 0.
You will need to use the principle of mathematical induction.
First we show that
( n )=(n1) !
is true for
n=1 .
( 1 )= t e dt= et d t=et0 =
Thomas Hill
Math 4200
SelfQuiz 14
Note: I found a cool proof on Wikipedia that uses the BolzanoWeierstrass theorem
that I thought was pretty cool. I tried to explain it here in my own words, but Im
not sure that it is correct.
(2)
If
f :[a ,b ] R
[a ,b
Thomas Hill
Math 4200
Self Quiz #11
f ( x )=sin
Consider (again) the function
( 1x ) .
It is not continuous at
x=0 , but it is
continuous at every other real number. Try to prove it. First try to use the definition
x 0 , remembering that
of continuity at
Thomas Hill
Math 4200
SelfQuiz #18
Use the NewtonRaphson method to find (or at least approximate) all roots of
f ( x )=2 x 4 +24 x 3 +61 x 216 x +1 .
Using our methods in excel:
xi
x i+1
8.000000000000000000000000000000
8.13125000000000000000000000000
Thomas Hill
Math 4200
Homework #4
1. Prove that if we have an interval
I
around a root
x
and
r
and a positive number
<1
such that


f ' ' ()
r x
2 f ' ( x)
xI
For all
converge to
and
between
r , then the NewtonRaphson Method must
r .
Firstly, let us
Thomas Hill
Math 4200
Self Quiz #1
1. (2) Approximate
, pretending you are ignorant to what its value is. First, argue that
1
= 1t 2 dt .
4 0
.
Second, use Riemann sums to approximate
t=sin u
Consider the substitution of variables, let
This substitution
Thomas Hill
Math 4200
Homework 2
1. How much precision are you guaranteed in using
when
f ( x )=x
x3 x 5 x 7
+
3! 5! 7!
for
f ( x )=sin x
x [2,2] .
First we observe that:
f ( x )=sin x=x
( 9)
x 3 x 5 x 7 f ( ) 9
+ +
x
3! 5! 7!
9!
The Lagrange remainder t
Thomas Hill
Math 4200
Self Quiz #13
Please prove that if
f ( x)
is continuous from
[ 0,1 ] [0,1]
, then there is an
x 0 [0,1] such that f ( x 0 ) =x0 .
g ( x ) =f ( x )x . Since
Let
g is the sum of two continuous functions on [0,1], it is
continuous on th
Thomas Hill
Math 4200
SelfQuiz #21
a , b , c ,d R
Suppose that
(a , b)
the cardinality of
with
a<b , c< d , a c , and
b d . Please prove that
is the same as the cardinality of
( c , d) .
Then prove that
card ( ( 0,1 ) =card (R) .
card ( a ,b )=card (b ,
Thomas Hill
Math 4200
SelfQuiz #8
(4) For what values of
does
lim
n0
1cos x
x
exist and what are those values?
We first notice that
(
1cos x =1 1
x2 x4
x2 x 4
+
O ( x 5 ) =
+O ( x 5 )
2! 4!
2! 4 !
)
And we see that
2
4
1cos x x
x
=
+O ( x 5 )
2!
4
Thomas Hill
Math 4200
SelfQuiz #7
Without looking it up attempt to prove Future Theorem 2.1.4 and write up your proof
or an explanation of your thought process. After you have completed that, look up a
proof online or from a text and write it down annot
Thomas Hill
Math 4200
Self Quiz #10
(3) Consider the function
cfw_
f ( x )= x sin (1/x ) , x 0
0 x=0
Please explain why Cauchys attempt at a proof of the Mean Value Theorem is
invalid for this function on the interval [0,1]. Please note this function has
Thomas Hill
Math 4200
SelfQuiz #19
DRAFT: Prove the statement, If every Cauchy sequence converges, then the Nested
Interval Principle holds.
Consider the nested intervals,
[ a0 , b0 ] [ a1 , b1 ] [ a n ,b n ]
lim ( bn an ) =0
Where
n
By construction of
Thomas Hill
Math 4200
Self Quiz #3
1. (4) Use Brook Taylors method for a generic function
expansion about
Suppose
f , center the
x=c .
f (x) C (R) , then
f (x)
can be written as some polynomial, say:
xc
xc
f ( x )=a 0+ a1 ( xc )+a2
Notice that since
f (x
Thomas Hill
Math 4200
Homework 1
1. Some Nepsilonics. Please prove, using only the definition of convergence of a sequence
L
to a value
that:
2
lim
a.
n
n +100 n
=0
52
n
2
n + 100 n 1 100
= +
52
n
n n n .
Proof: Firstly we notice that
Applying Theorem
Thomas Hill
Math 4200
SelfQuiz #20
Prove that the intersection of any finite collection of open sets is open.
Suppose
Let
S 1 , S2 , S 3 ,
xS
Since each
are open sets and let
be arbitrary, then by construction
Si
is open for each
i
S= i=1 n S i .
x S i f
Thomas Hill
Math 4200
SelfQuiz #15
f is continuous on
Prove the Mean Value Theorem, that if
differentiable on
f ' ( )=
Since
points
(a ,b)
such that
f ( b ) f (a)
.
ba
f
is continuous on
(a , f ( a )
h ( x )=
points is
We let
(a , b) , then there exists
Thomas Hill
Math 4200
Homework #3
1. Define
f :RR
values for which
Claim:
f
f ( x )= x +
by
10 x 10 x
+0.1 . Determine with justification the
10
x
f is continuous and those for which it is not continuous.
cfw_0,1,2,3,4,5,6,7,8,9 .
is discontinuous at
(t
Thomas Hill
Math 4200
Homework #5
R0=R \ cfw_0 . Prove that this set (together with the usual axioms of a
1. Consider the set
field) is not complete with respect to the metric
d ( x , y )=x y.
We will prove that
Develop a new metric
R0
defined by
with
Problem 1
a. Given, f (x) = cos x
I = ( , ) Then, f (I) [ 1, 1]
I = (0, 1) Then, f (I) ( 1, 1)
I = (1/2, 3/2) Then, f (I) [ 1, 0)
b. Given, f (x) = xex
f (x) = (1 x)ex = 0 Then the only critical point is at x = 1
where f (1) = (x 2)ex = e1 < 0 which impli