a) Evaluate the polynomial y = x^3 - 7x^2 + 8x - 0.35 at x = 1.37 . Use 3-digit arithmetic with chopping. Evaluate the percent relative error. b) Repeat (a) but express y as y = ((x - 7) x + 8) x - 0.35 Evaluate the error and compare with part (a).

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asked 2020-11-01

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asked 2020-10-19

Given the following function:
$\displaystyle{f{{\left({x}\right)}}}={1.01}{e}^{{{4}{x}}}-{4.62}{e}^{{{3}{x}}}-{3.11}{e}^{{{2}{x}}}+{12.2}{e}^{{{x}}}$
a) Use three-digit rounding frithmetic, the assumption that $\displaystyle{e}^{{{1.53}}}={4.62}$, and the fact that $\displaystyle{e}^{{{n}{x}}}={\left({e}^{{{x}}}\right)}^{{{n}}}$ to evaluate $\displaystyle{f{{\left({1.53}\right)}}}$
b) Redo the same calculation by first rewriting the equation using the polynomial factoring technique
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asked 2020-12-29

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asked 2021-02-08

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asked 2021-02-21

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asked 2021-01-19

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asked 2020-10-20

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asked 2021-02-06

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asked 2020-12-05

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asked 2020-10-23

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Answer 1

(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.

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a) Evaluate the polynomial y = x^3 - 7x^2 + 8x - 0.35 at x = 1.37 . Use 3-digit arithmetic with chopping. Evaluate the percent relative error. b) Repeat (a) but express y as y = ((x - 7) x + 8) x - 0.35 Evaluate the error and compare with part (a).

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