Find the third taylor polynomial p3(x) for the function f(x) = (x-1)ln(x) about x0 = 1.

(a) use p3(0.5) to approximate f(0.5). find an upper bound for error |f(0.5) - p3(0.5)| using the error formula, and compare it to the actual error.
(b) find a bound for the error |f(x) - p3(x)| in using p3(x) to approximate f(x) on the interval [0.5,1.5].
(c) approximate \int_{0.5}^{1.5}f(x)dx using \int_{0.5}^{1.5}p_{3}(x)dx .
(d) find an upper bound for the error in (c) using \int_{0.5}^{1.5}|r_{3}(x)dx| , and compare the bound to the actual error.

A) The actual error |f(0.5) - P3(0.5)| is 0.0340735. The error using Taylor's error formula R₃(0.5) is 0.2917.

B) Te error in the interval [0.5,1.5] is 0.0340735.

C) The approximate integral using P₃(x) is 1/12

D) The real error between the two integrals is 4.687·10^(-3) while the error using \int_{0.5}^{1.5}|R_{3}(x)dx| is 0.018595

Step-by-step explanation:

A) To determine the error we first have to write the third grade Taylor polynomial P3(x):

with N=3

The error for the third grade Taylor polynomial R₃(x) is represented as the next term non-written in the polynomial expression near to the point x:

Therefore the real error is:

The error of the Taylor polynomial R₃(0.5) is:

B) The error in the interval [0.5,1.5] is the maximum error in that interval.

This is found in the extremes of the intervals. We analyze what happens in X=1.5:

Both of these errors are smaller than Err(0.5) and R₃(0.5). Therefore the error in this interval is Err[0.5,1.5] is Err(0.5).

C) The approximation of the integral and the real integral is:

D) The error in the integrals is:

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