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Exercise 2.13

Let’s take two intervals defined by their center and percentage tolerance: - interval \(A\) with center \(c_1\) and percentage \(p_2\) - interval \(B\) with center \(c_2\) and percentage \(p_2\)

We can easily find the lower bound and upper bound of each interval using the ideas of the make-center-percent procedure defined in the previous exercise:

\(A=(c_1-,c_1+)B=(c_2-,c_2+)\)

Assuming that the bounds are positives (as proposed by the exercise), we can determine the product of the two intervals :

\(A.B=((c_1-)(c_2-),(c_1+)(c_2+))\)

\(\ \ \ \ =(c_1c_2+--,c_1c_2+++)\)

Now we can easily see the center and the with of our new interval.\ Next we can evaluate the percentage of our new interval following the ideas of our percent procedure:\ \[(AB)=\] You can verify that this percentage is correct with substitution using the percent procedure.

If the percentages \(p_1\) and \(p_2\) are small then \(\) is negligible and we get: \[(AB)p_1p_2\]

We just proved that the percentage tolerance of the product of two intervals is approximately the sum of the two percentage tolerance of the factors if these percentages are small.