2.2 Average Value of a Function
the
average ofa finite setof points y,, yz, ys...., yn is
Yave in(y, +yz yz
=
+
..
+
.
yn)
+
In,yx
=
eg. Average temperature in a
day=?
for a continuous function f on Ca, bI
pick n points and take the average f(x,*), f(x,*).
of ..., f(xn*):
'n,f(Xi*)
the more data points, the better the estimate
the
average, value off over Ca, bI
is
eln, f(x) Limitofa Riemann sum ...
definite integral
=4 ya
Ax 1 =
=
· f*)=f(x)
-b =,f(X,*)
Ax
a
mannsum:Rn
-
fave=b!a Rn
=b a(if(x,*)Ax)
-b !a(af(x)dx
(af(x)dx= (b a)fave
-
example:
find fare off(x) ex
=
on S-1, 11
fave =
IS, e*dx
=ex) (e- e) =
example:
ang velocity (E) over - interval (a,b]
->S(t) position/displacement
=
by our definition:Vare=
Ia Svitdt=aSa S'(t) dt
*
S(b) S(a) =
I (S(t) =
b -
a
G
is fl
generally, "ang rate of
change inf"
fave=p!af'(x)dx p!af(x)( f(b) f(a)
= -
=
/
b -
a
the
average ofa finite setof points y,, yz, ys...., yn is
Yave in(y, +yz yz
=
+
..
+
.
yn)
+
In,yx
=
eg. Average temperature in a
day=?
for a continuous function f on Ca, bI
pick n points and take the average f(x,*), f(x,*).
of ..., f(xn*):
'n,f(Xi*)
the more data points, the better the estimate
the
average, value off over Ca, bI
is
eln, f(x) Limitofa Riemann sum ...
definite integral
=4 ya
Ax 1 =
=
· f*)=f(x)
-b =,f(X,*)
Ax
a
mannsum:Rn
-
fave=b!a Rn
=b a(if(x,*)Ax)
-b !a(af(x)dx
(af(x)dx= (b a)fave
-
example:
find fare off(x) ex
=
on S-1, 11
fave =
IS, e*dx
=ex) (e- e) =
example:
ang velocity (E) over - interval (a,b]
->S(t) position/displacement
=
by our definition:Vare=
Ia Svitdt=aSa S'(t) dt
*
S(b) S(a) =
I (S(t) =
b -
a
G
is fl
generally, "ang rate of
change inf"
fave=p!af'(x)dx p!af(x)( f(b) f(a)
= -
=
/
b -
a
