Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/50

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26
DIFFERENTIAL CALCULUS

Assuming for the initial value of fixes as the initial value of .

Suppose increases to , that is, ;
then increases to , and .
Suppose decreases to , that is, ;
then increases to , and .

It may happen that as increases, decreases, or the reverse; in either case and will have opposite signs.

It is also clear (as illustrated in the above example) that if is a continuous function and is decreasing in numerical value, then also decreases in numerical value.

27. Comparison of increments. Consider the function

(A)
.

Assuming a fixed initial value for , let take on an increment . Then will take on a corresponding increment , and we have

  , Subtracting (A)
or, .
Subtracting (A),  
(B)  

we get the increment in terms of and .

To find the ratio of the increments, divide (B) by , giving

If the initial value of is , it is evident that

Let us carefully note the behavior of the ratio of the increments of and as the increment of diminishes.

Initial
value of
New
value of
Increment
Initial
value of
New
value of
Increment
4 5.0 1.0 16 25. 9. 9.
4 4.8 0.8 16 23.04 7.04 8.8
4 4.6 0.6 16 21.16 5.16 8.6
4 4.4 0.4 16 19.36 3.36 8.4
4 4.2 0.2 16 17.64 1.64 8.2
4 4.1 0.1 16 16.81 0.81 8.1
4 4.01 0.01 16 16.0801 0.0801 8.01