applications of integration 6. 6.5 average value of a function in this section, we will learn about:...
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APPLICATIONS OF INTEGRATIONAPPLICATIONS OF INTEGRATION
6
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6.5Average Value
of a Function
In this section, we will learn about:
Applying integration to find out
the average value of a function.
APPLICATIONS OF INTEGRATION
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It is easy to calculate the average
value of finitely many numbers
y1, y2 , . . . , yn :
1 2 nave
y y yy
n
AVERAGE VALUE OF A FUNCTION
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However, how do we compute the
average temperature during a day if
infinitely many temperature readings
are possible?
AVERAGE VALUE OF A FUNCTION
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This figure shows the graph of a
temperature function T(t), where: t is measured in hours T in °C Tave , a guess at the average temperature
AVERAGE VALUE OF A FUNCTION
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In general, let’s try to compute
the average value of a function
y = f(x), a ≤ x ≤ b.
AVERAGE VALUE OF A FUNCTION
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We start by dividing the interval [a, b]
into n equal subintervals, each with
length .( ) /x b a n
AVERAGE VALUE OF A FUNCTION
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Then, we choose points x1*, . . . , xn* in
successive subintervals and calculate the
average of the numbers f(xi*), . . . , f(xn*):
For example, if f represents a temperature function and n = 24, then we take temperature readings every hour and average them.
AVERAGE VALUE OF A FUNCTION
( *) ( *)i nf x f x
n
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Since ∆x = (b – a) / n, we can write
n = (b – a) / ∆x and the average value
becomes:
1
1
1
( *) ( *)
1( *) ( *)
1( *)
n
n
n
ii
f x f xb a
x
f x x f x xb a
f x xb a
AVERAGE VALUE OF A FUNCTION
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If we let n increase, we would be
computing the average value of a large
number of closely spaced values.
For example, we would be averaging temperature readings taken every minute or even every second.
AVERAGE VALUE OF A FUNCTION
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By the definition of a definite integral,
the limiting value is:
1
1 1lim ( *) ( )
n b
i ani
f x x f x dxb a b a
AVERAGE VALUE OF A FUNCTION
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So, we define the average value of f
on the interval [a, b] as:
1( )
b
ave af f x dx
b a
AVERAGE VALUE OF A FUNCTION
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Find the average value of
the function f(x) = 1 + x2 on
the interval [-1, 2].
AVERAGE VALUE Example 1
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With a = -1 and b = 2,
we have:
2 2
1
23
1
1( )
11
2 ( 1)
12
3 3
b
ave af f x dx
b a
x dx
xx
Example 1AVERAGE VALUE
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If T(t) is the temperature at time t,
we might wonder if there is a specific time
when the temperature is the same as
the average temperature.
AVERAGE VALUE
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For the temperature function graphed here,
we see that there are two such times––just
before noon and just before midnight.
AVERAGE VALUE
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In general, is there a number c at which
the value of a function f is exactly equal
to the average value of the function—that is,
f(c) = fave?
AVERAGE VALUE
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The mean value theorem for
integrals states that this is true
for continuous functions.
AVERAGE VALUE
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MEAN VALUE THEOREM
If f is continuous on [a, b], then there exists
a number c in [a, b] such that
that is,
1( ) ( )
b
ave af c f f x dx
b a
( )b
af x dx f c b a
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The Mean Value Theorem for Integrals is
a consequence of the Mean Value Theorem
for derivatives and the Fundamental Theorem
of Calculus.
MEAN VALUE THEOREM
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The geometric interpretation of the Mean
Value Theorem for Integrals is as follows.
For ‘positive’ functions f, there is a number c such that the rectangle with base [a, b] and height f(c) has the same area as the region under the graph of f from a to b.
MEAN VALUE THEOREM
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Since f(x) = 1 + x2 is continuous on the
interval [-1, 2], the Mean Value Theorem for
Integrals states there is a number c in [-1, 2]
such that:
2 2
11 ( )[2 ( 1)]x dx f c
Example 2MEAN VALUE THEOREM
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In this particular case, we can
find c explicitly.
From Example 1, we know that fave = 2.
So, the value of c satisfies f(c) = fave = 2.
Therefore, 1 + c2 = 2.
Thus, c2 = 1.
MEAN VALUE THEOREM Example 2
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So, in this case, there happen to be
two numbers c = ±1 in the interval [-1, 2]
that work in the Mean Value Theorem for
Integrals.
Example 2MEAN VALUE THEOREM
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Examples 1 and 2 are illustrated
here.
MEAN VALUE THEOREM
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Show that the average velocity of a car
over a time interval [t1, t2] is the same
as the average of its velocities during
the trip.
Example 3MEAN VALUE THEOREM
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If s(t) is the displacement of the car at time t,
then by definition, the average velocity of
the car over the interval is:
2 1
2 1
( ) ( )s t s ts
t t t
MEAN VALUE THEOREM Example 3
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On the other hand, the average value of thevelocity function on the interval is:
2 2
11
2 1 2 1
2 12 1
2 1
2 1
by the Net Change Theorem
average velocity
1 1( ) '( )
1( ) ( )
( ) ( )
t t
ave t tv v t dt s t dt
t t t t
s t s tt t
s t s t
t t
Example 3MEAN VALUE THEOREM