18.01
Calculus
Jason
Starr
Fall
2005
Lecture
3.
September
13,
2005
Homework.
Problem
Set
1
Part
I:
(i)
and
(j).
Practice
Problems.
Course
Reader:
1E1,
1E3,
1E5.
1.
Another
derivative.
Use
the
3step
method
to
compute
the
derivative
of
f
(
x
)
=
1
/
√
3
x
+
1
is,
f
(
x
−
x
−
3
/
2
/
2
.
)
=
3(3 +
1)
Upshot:
Computing
derivatives
by
the
deFnition
is
too
much
work
to
be
practical.
We
need
general
methods
to
simplify
computations.
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18.01
Calculus
Jason
Starr
Fall
2005
2.
The
binomial
theorem.
For
a
positive
integer
n
,
the
factorial
,
n
!
=
n
×
(
n
−
1)
×
(
n
−
2)
× ··· ×
3
×
2
×
1
,
is
the
number
of
ways
of
arranging
n
distinct
objects
in
a
line.
For
two
positive
integers
n
and
k
,
the
binomial
coeﬃcient
,
n
n
!
n
(
n
−
1)
···
(
n
−
k
+
2)(
n
−
k
+
1)
,
=
=
k
k
!(
n
−
k
)!
1
3
·
2
·
k
(
k
−
1)
···
is
the
number
of
ways
to
choose
a
subset
of
k
elements
from
a
collection
of
n
elements.
A
funda
mental
fact
about
binomial
coeﬃcients
is
the
following,
n
n
n
+
1
+
=
.
k
k
k
−
1
This
is
known
as
Pascal’s
formula
.
This
link
is
to
a
webpage
produced
by
MathWorld
,
part
of
Wolfram
Research.
The
Binomial
Theorem
says
that
for
every
positive
integer
n
and
every
pair
of
numbers
a
and
b
,
(
a
+
b
)
n
equals,
n
n
n
a
+
na
n
−
1
b
+
···
+
k
a
n
−
k
b
k
+
···
+
nab
n
−
1
+
b .
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 Spring '08
 Moretti
 Derivative, Product Rule, Formulas, Mathematical Induction, Natural number, positive integer, induction hypothesis, Jason Starr

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