Final Review 3: Other Gadgets and Properties
• Sigma Notation
n
X
ai
i=m
– Definition:
n
X
ai = am + am+1 + · · · + an−1 + an
i=m
–
Constant multiple:
n
X
cai = c
i=m
Sum and difference rule:
n
X
n
X
ai
i=m
(ai + bi ) =
i=m
n
X
ai +
i=m
n
X
i=m
bi ,
n
X
(ai − bi ) =
i=m
n
X
i=m
ai −
n
X
bi
i=m
– No product or quotient rule.
n
n
n
X
X
X
– Special sums:
i,
i2 and
i3 . The formulaes will be given in the final. Notice that the
i=1
i=1
i=1
sums in the formulaes are taken from the first term to the nth term.
Z b
Z c
Z b
• Interval Additivity:
f (x) dx =
f (x) dx +
f (x) dx regardless the order of a, b and c on
the real number line.
a
a
c
• Intermediate Value Theorem: Suppose f (x) is a continuous function on the closed interval [a, b]
and A is any real number between f (a) and f (b). Then there is at least one real number c between
a and b such that f (c) = A.
• Mean Value Theorem
– of Derivatives: Suppose f (x) is a differentiable function on the close interval [a, b] (i.e. f 0 (x)
is defined everywhere between [a, b]). Then there is at least one real number c between a and
b such that
f (b) − f (a)
f 0 (c) =
b−a
Average slope on the interval [a, b].
– of Integrals: Suppose f (x) is a continuous function on the closed interval [a, b], then there is
at least one real number c between a and b such that
Z b
1
f (c) =
f (t) dt
b−a a
Average value on the interval [a, b].