(a)First determine the powers of x needed for evaluating the polynomial.
\(\displaystyle{x}={1.37}{N}{S}{K}{7}{x}^{{2}}={13.1383}\)
Chopping to 3 digit arithmetic yields \(\displaystyle{x}^{{2}}={13.1}{N}{S}{K}{8}{x}={10.96}\)
Chopping to 3 digit arithmetic yields \(\displaystyle{8}{x}={10.9}{N}{S}{K}{x}^{{3}}={2.571353}\)
Chopping to 3 digit arithmetic yields \(\displaystyle{x}^{{3}}={2.57}\)
Insert these values in the given polynomial to find \(\displaystyle{y}{\left({1.37}\right)}.{N}{S}{K}{y}{\left({1.37}\right)}={2.57}−{13.1}+{10.9}−{0.35}{N}{S}{K}={0.02}\)
Whereas the exact result is,
\(\displaystyle{y}{\left({1.37}\right)}={2.571353}−{13.1383}+{10.96}−{0.35}{N}{S}{K}={0.043053}\)
Hence, the percent relative error is,
\(\displaystyle\xi={\frac{{{0.043053}-{0.02}}}{{{0.043053}}}}\times{100}\%{N}{S}{K}={53.54}\%\)
(b)
\(\displaystyle{y}={\left({\left({x}-{7}\right)}{x}+{8}\right)}{x}-{0.35}\)
Now, when x = 1.37,

(x - 7)x = (1.37 - 7)(1.37)

=-7.7131ZSK Chopping to 3 digit arithmetic yields, \(\displaystyle{\left({x}-{7}\right)}{x}=-{7.71}{N}{S}{K}{\left({\left({x}-{7}\right)}{x}+{8}\right)}{x}={\left(-{7.71}+{8}\right)}{1.37}{N}{S}{K}={0.047}\) Here, the percent relative error is, \(\displaystyle\xi={\frac{{{0.043053}-{0.047}}}{{{0.043053}}}}\ \times{100}\%{N}{S}{K}=-{9.16}\%\) The percent relative error is smaller when compared to part a.

(x - 7)x = (1.37 - 7)(1.37)

=-7.7131ZSK Chopping to 3 digit arithmetic yields, \(\displaystyle{\left({x}-{7}\right)}{x}=-{7.71}{N}{S}{K}{\left({\left({x}-{7}\right)}{x}+{8}\right)}{x}={\left(-{7.71}+{8}\right)}{1.37}{N}{S}{K}={0.047}\) Here, the percent relative error is, \(\displaystyle\xi={\frac{{{0.043053}-{0.047}}}{{{0.043053}}}}\ \times{100}\%{N}{S}{K}=-{9.16}\%\) The percent relative error is smaller when compared to part a.