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still much smaller than the value of the variation of x over which the function
changes appreciably.
For a systematic method to choose an upper limit on dx, you might want
to follow these simple steps:

1. Plot the function on the given interval and identify the point where
the derivative is largest.
2. Compute the derivative at that point using the sequence method
of Example 4.2, and determine the dx that would satisfy the desired
tolerance; then go ahead and use this value of dx in the above
routine to evaluate the derivative throughout the given interval.

In-Class Exercises

Plot the derivatives of the following functions on the indicated intervals:
x − 1
Pb. 4.11 ln on <2 x < 3
x + 1
1+ 1+ x 2
Pb. 4.12 ln on 1 < x < 2
x

Pb. 4.13 ln tanh( / )x 2 on 1 < x < 5
−1
Pb. 4.14 tan sinh( )x on <0 x < 10

Pb. 4.15 ln csc( ) tan( )x + x on 0 < x < π / 2

4.3 Infinite Sums

∞
∑
An infinite series is denoted by the symbol a . It is important not to con-
n
n=1
fuse the series with the sequence {a }. The sequence is a list of terms, while the
n
series is a sum of these terms. A sequence is convergent if the term a
n
approaches a finite limit; however, convergence of a series requires that the
N ∑
sequence of partial sums S = N a n approaches a finite limit. There are
n=1

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