14
Further Integration Techniques
and Applications of the Integral
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14.3
Averages and Moving Averages
Copyright © Cengage Learning. All rights reserved.
Averages
3
Averages
To find the average of, say, 20 numbers, we simply add
them up and divide by 20. More generally, if we want to find
the average, or mean, of the n numbers y1, y2, y3, . . . yn,
we add them up and divide by n. We write this average as
(“y-bar”).
Average, or Mean, of a Collection of Values
Quick Example
The average of {0, 2, –1, 5} is
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Example 1 – Average Speed
Over the course of 2 hours, my speed varied from 50 miles
per hour to 60 miles per hour, following the function
v(t) = 50 + 2.5t 2, 0 ≤ t ≤ 2. What was my average speed
over those two hours?
Solution:
Recall that average speed is simply the total distance
traveled divided by the time it took. Recall, also, that we
can find the distance traveled by integrating the speed:
Distance traveled
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Example 1 – Solution
cont’d
It took 2 hours to travel this distance, so the average speed
was
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Example 1 – Solution
cont’d
In general, if we travel with velocity v(t) from time t = a to
time t = b, we will travel a distance of
in time b – a,
which gives an average velocity of
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Averages
Average Value of a Function
The average, or mean, of a function f(x) on an interval
[a, b] is
Quick Example
The average of f(x) = x on [1, 5] is
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Interpreting the Average of a
Function Geometrically
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Interpreting the Average of a Function Geometrically
The average of a function has a geometric interpretation.
We can compare the graph of y = f(x) with the graph of
y = 3, both over the interval [1, 5] (Figure 8).
We can find the area under the graph
of f(x) = x by geometry or by calculus;
it is 12. The area in the rectangle under
y = 3 is also 12.
Figure 8
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Interpreting the Average of a Function Geometrically
In general, the average of a positive function over the
interval [a, b] gives the height of the rectangle over the
interval [a, b] that has the same area as the area under the
graph of f(x) as illustrated in Figure 9.
The equality of these areas follows
from the equation
Figure 9
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Example 2 – Average Balance
A savings account at the People’s Credit Union pays 3%
interest, compounded continuously, and at the end of the
year you get a bonus of 1% of the average balance in the
account during the year. If you deposit $10,000 at the
beginning of the year, how much interest and how large a
bonus will you get?
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Example 2 – Solution
We can use the continuous compound interest formula to
calculate the amount of money you have in the account at
time t:
A(t) = 10,000e 0.03t
where t is measured in years.
At the end of 1 year, the account will have
A(1) = $10,304.55
so you will have earned $304.55 interest.
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Example 2 – Solution
cont’d
To compute the bonus, we need to find the average amount
in the account, which is the average of A(t) over the interval
[0, 1].
Thus,
The bonus is 1% of this, or $101.52.
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Moving Averages
15
Moving Averages
Suppose you follow the performance of a company’s stock
by recording the daily closing prices. The graph of these
prices may seem jagged or “jittery” due to random
day-to-day fluctuations. To see any trends, you would like a
way to “smooth out” these data.
The moving average is one common way to do that.
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Example 3 – Stock Prices
The following table shows Colossal Conglomerate’s closing
stock prices for 20 consecutive trading days.
Plot these prices and the 5-day moving average.
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Example 3 – Solution
The 5-day moving average is the average of each day’s
price together with the prices of the preceding 4 days.
We can compute the 5-day moving averages starting on
the fifth day. We get these numbers:
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Example 3 – Solution
cont’d
The closing stock prices and moving averages are plotted
in Figure 10.
Figure 10
As you can see, the moving average is less volatile than
the closing price.
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Example 3 – Solution
cont’d
Because the moving average incorporates the stock’s
performance over 5 days at a time, a single day’s fluctuation
is smoothed out.
Look at day 9 in particular. The moving average also tends
to lag behind the actual performance because it takes past
history into account. Look at the downturns at days 6 and 18
in particular.
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Moving Averages
The period of 5 days for a moving average, as used in
Example 3, is arbitrary.
Using a longer period of time would smooth the data more
but increase the lag.
For data used as economic indicators, such as housing
prices or retail sales, it is common to compute the 4-quarter
moving average to smooth out seasonal variations.
It is also sometimes useful to compute moving averages of
continuous functions.
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Moving Averages
We may want to do this if we use a mathematical model of
a large collection of data.
Also, some physical systems have the effect of converting
an input function (an electrical signal, for example) into its
moving average.
By an n-unit moving average of a function f(x) we mean
the function for which
is the average of the value of
f(x) on [x – n, x].
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Moving Averages
Using the formula for the average of a function, we get the
following formula.
n-Unit Moving Average of a Function
The n-unit moving average of a function f is
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Moving Averages
Quick Example
The 2-unit moving average of f(x) = x2 is
The graphs of f(x) and
are shown in Figure 11.
Figure 11
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