Graphics Reference
In-Depth Information
()
=
()
=
+
()
-
()
fa
lim
fx
and
fa
lim
fx
.
xaxa
Æ≥
,
xaxa
Æ£
,
D.1.1. Theorem.
(The Intermediate Value Theorem) Let f : [a,b] Æ
R
be a continu-
ous function. Assume that c Œ [f(a),f(b)]. Then there exists an aŒ[a,b] so that f(a) = c.
Proof.
See [Buck78].
D.1.2. Theorem.
(The Mean Value Theorem) Let f : [a,b] Æ
R
be continuous and
assume that f is differentiable on (a,b). Then there exists an aŒ(a,b) so that
()
-
()
=-
(
)
¢
(
a .
fb
fa
b af
Proof.
See [Buck78].
D.1.3. Theorem.
(The Leibnitz Formula) Suppose that h(x) = f(x)g(x), where f(x)
and g(x) are n-times differentiable function. Then the product rule for the derivative
generalizes to
n
n
i
Ê
Ë
ˆ
¯
Â
(
)
(
)
(
)
n
()
=
i
()
n i
-
()
hx
fxg
x
.
i
=
0
Proof.
Use induction.
Definition.
Let f be a real-valued function defined on all of
R
or an open interval
(a,b). The function f is said to be of
class C
k
if all the derivatives of f exist
and
are
continuous up to and including order k. If f is of class C
k
for all k, then we say that f
is of
class C
•
.
Definition.
A
partition
of an interval [a,b] is a sequence
=
(
)
P
t
,
t
,...,
t
,
where a
=
t
£
t
£ ◊◊◊£
t
=
b
.
01
k
0
1
k
Each interval [t
i
,t
i+1
] is called a
subinterval
of P. The
norm
of the partition P, denoted
by |P|, is defined by
{
}
P
=
max
t
-
t
i
=
12
, ,...,
k
.
i
i
-
1
A
refinement
of the partition P is a partition P¢=(s
0
,s
1
,...,s
m
) of [a,b] with the prop-
erty that {t
0
,t
1
,...,t
k
} Õ {s
0
,s
1
,...,s
m
}.
When it comes to integration, we assume that the reader is familiar with the
Riemann integral, but we recall a few basic definitions and facts. Given a bounded
function f : [a,b] Æ
R
the standard definition of the Riemann integral is in terms of a
limit of sums. More precisely, for each partition P = (x
0
,x
1
,...,x
n
) of [a,b] we look at
sums of the form
