average - Average
of
a
func+on


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Unformatted text preview: Average
of
a
func+on
 Defini&on.
Let
f
be
a
func+on
which
is
con+nuous
 on
the
closed
interval
[a,
b].

 The
average
value
of
f
from
x
=
a
to
x
=
b
is
the
 integral
 b 1 b−a ∫ a f ( x ) dx (this
is
a
number!)
 € An
example
of
average
 Find
the
average
of
the
func+on
f(x)=x2+1
over
[‐1,2]
 f ave 1 = b−a b ∫ a f ( x ) dx = 2 1 = ( x 2 + 1) dx = ∫ 2 − (−1) −1 1 x = x + = 2 3 3 −1 32 Another
example
of
average
 Find
the
average
of









































over
[‐1,5/2]
 f(x)
 b‐a
 Geometric
meaning
of
average
(for
f≥0)
 Suppose
that
f
is
a
con+nuous
func+on
on
[a,b],
 and
that
f
is
non‐nega+ve.
 f ave fdfdfdffffdd
 BvhvghM
 1 = b−a b ∫ a f ( x ) dx fave
 b € b‐a
 















vcnvvcvxv







 f ave (b − a) = ∫ a f ( x ) dx € fave
 Geometric
meaning
of
average
(for
f≥0)
 The
green
and
the
 pink
region
must
 compensate
each

 other
 fave
 fave
 Geometric
meaning
of
average
for
f≥0
 Example:
f(x)=1+
x2
over
[‐1,2]
 (b − a) f ave =
 b‐a
 b ∫ a f ( x ) dx € € Geometric
meaning
of
average
 for
a
(possibly
non‐posi+ve)
f
 Example:
 fave=
 fave=‐9/5
 Geometric
meaning
of
average
for
all
f
con+nuous
 The
signed
area
under
the
graph
on
[a,b]
 is
equal
to
the
signed
area
of
a
rectangle

 of
basis
(b‐a)
and
“height”
fave

 f ave
 Mean
Value
Theorem
for
Integrals
 Let
f(x)
be
a
con+nuous
func+on
on
the
closed
interval
[a,
b].
 Then
there
exists
a
point
c
in
[a,
b]
such
that
 b ∫ a f ( x ) dx = f (c ) (b − a) € 1 f (c ) = (b − a) b ∫ a f ( x ) dx average
 If
f(x)
is
con+nuous
on
a
closed
interval
[a,
b],
then
there
 exists
a
point
€
in
[a,
b]
such
that
f(c)=
fave
 c Mean
Value
Theorem
for
Integrals
 If
f

is
a
con+nuous
func+on
on
[a,b],
then
 somewhere
in
[a,b]
the
func+on
will
take
on

 its
average
value.
 The
rectangle
and
the

 shaded
region
have
the
 same
area
 f(c)=fave
 The continuous function f(x)=1+x2 has average 2 over [-1,2]. fave=2 By the MVT, there is a point c in [-1,2] such that f( c ) = fave. Find c c ? You need to set f( c )=fave 1+x2=2 x2=1 x = ±1 . Question Let f be continuous function whose graph is: ff f ff The average of f over the closed interval [-3,3] is = 0. A) TRUE B) FALSE REVIEW Think of a function f as a machine that, for each given input x, produces an output f(x). input (x) x f f(x) output •Domain of f ={all possible inputs x} •Range of f ={all possible outputs f(x)} A function f is one-to-one if any two different points in the domain have different images: a≠b f(a)≠f(b) one-to-one not one-to-one Horizontal Line Test. If f is one-to-one, then each horizontal line meets the graph of f in at most one point. f(a)=f(b) a b Not one-to-one One-to-one Not one-to-one f one-to-one a1 domain of f = A a2 a3 b1 b2 b3 range of f = B If f is one-to-one, every b in the range comes from a unique a in the domain. So we can invert the function. f inverse b1 domain of f- 1 =B b2 b3 a1 a2 a3 Range of f -1= A = domain of f = range of f f a1 a2 a3 Domain of f b b11 bb2 2 b3 f- 1 a1 a2 a3 Domain of f Range of f The composition fo(f-1) is the identity on the domain of f. The composition (f-1)of is the identity on the range of f. The graph of f-1 is obtained from the graph of f by flpping about the line y=x ...
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This note was uploaded on 12/07/2010 for the course MATH MATH 2B taught by Professor Famiglietti,c during the Fall '09 term at UC Irvine.

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