SOLUTIONS:
(a) The derivatives of ln x at x = 1 are given in the table:
From the table we have(b) Figure N10–4 shows the graphs of ln x and the four Taylor polynomials above, in [0,2.5] ×
[−1,1].
FIGURE N10–4
(c) ln 1.3 ≈ P 2 (1.3) = (1.3 − 1) − − 0.045 = 0.255.
For x = 1.3 the Taylor series converges by the Alternating Series Test, so the error is less than
the magnitude of the first omitted term:EXAMPLE 47
For what positive values of x is the approximate formula
ln (1 + x) =
correct to three decimal places?
SOLUTION: We can use series (4) of Common Maclaurin Series: