Magnitude

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Problem 3.10

Using equation (1,8) in Table 2.2a, show that the fractional change is not necessarily small when , , and are all small.

Solution

Equation (1,8) in Table 2.2a is

Because does not enter into this equation, it has no effect upon . The fractions , , and are of the form which suggests that we use logs [since . Taking logs of both sides of the above equation, we get

.

Differentiation gives

Thus,

Since , the product of the two -factors varies between (when ) and (when ). Therefore, even though is small (being the difference between two small quantities), the right-hand side can be large.

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