« Section 5.1
Section 5.3 »

Section 5.2

Please post comments if you find any errors, or if you wish to contribute answers for additional problems. You may find information on using \LaTeX here.

1. (a) \displaystyle{P_5(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!}}
\sin(0. <img src='http://s.wordpress.com/wp-includes/images/smilies/icon_cool.gif' alt='8)' class='wp-smiley' /> \approx P_5(0. <img src='http://s.wordpress.com/wp-includes/images/smilies/icon_cool.gif' alt='8)' class='wp-smiley' /> = 0.717397 , rounded to 6 decimal places
\displaystyle{|\sin(0. <img src='http://s.wordpress.com/wp-includes/images/smilies/icon_cool.gif' alt='8)' class='wp-smiley' /> - P_5(0.8)| \le \frac{0.8^7}{7!} = 0.000041610}, rounded to 9 decimal places

(c) \displaystyle{P_5(x) = 2x - \frac{4}{3}x^3 + \frac{4}{15}x^5}
\sin(-1) \approx P_5(-0.5) = -0.841667 , rounded to 6 decimal places
\displaystyle{|\sin(-1) - P_5(-0.5)| \le \frac{2^7(0.5)^7}{7!} = 0.000198413}, rounded to 9 decimal places

(e) \displaystyle{P_5(x) = 3 + \frac{1}{6}(x - 9) - \frac{1}{216}(x - 9)^2 + \frac{1}{3888}(x - 9)^3}
\displaystyle{- \frac{5}{279,936}(x - 9)^4 + \frac{7}{5,038,848}(x - 9)^5}
\sqrt{10} \approx P_5(10) = 3.1622778 , rounded to 7 decimal places
\displaystyle{|\sqrt{10} - P_5(10)| \le \frac{945}{(64)(3^{11})(6!)} = 0.000000115767}, rounded to 12 decimal places

(g) \displaystyle{P_5(x) = 1 - (x - 1) + (x - 1)^2 - (x - 1)^3 + (x - 1)^4 - (x - 1)^5}
\displaystyle{\frac{1}{0.8} \approx P_5(0. <img src='http://s.wordpress.com/wp-includes/images/smilies/icon_cool.gif' alt='8)' class='wp-smiley' /> = 1.24992}, rounded to 5 decimal places
\displaystyle{\left|\frac{1}{0.8} - P_5(0.8)\right| \le \frac{720(0.2)^6}{(0.8)^7(61)} = 0.000305176}, rounded to 9 decimal places

3. \sin(-1.3) \approx P_9(-1.3) = -0.963559 , rounded to 6 decimal places

5. \displaystyle{P_9(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!}}

7. (a) \displaystyle{P_{10}(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \frac{x^6}{6!} + \frac{x^7}{7!} + \frac{x^8}{8!}}
\displaystyle{+ \frac{x^9}{9!} + \frac{x^{10}}{10!}}

(b) E(1) \approx P_{10}(1) = 2.7182818011 , rounded to 10 decimal places

(c) \displaystyle{|E(1) - P_{10}(1)| \le \frac{3}{11!} = 0.00000007515633}, rounded to 14 decimal places

(d) \displaystyle{P_{11}(x) = 1 + x + \frac{x^2}{2} \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \frac{x^6}{6!} + \frac{x^7}{7!} + \frac{x^8}{8!}}
\displaystyle{+ \frac{x^9}{9!} + \frac{x^{10}}{10!} + \frac{x^{11}}{11!}}

8. \displaystyle{\int_0^3 \frac{\sin(x)}{x} \ dx \approx \int_0^3 \frac{P_9(x)}{x} \ dx}
\displaystyle{= \int_0^3 \left(1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \frac{x^8}{9!}\right)dx = 1.849037}, rounded to 6 decimal places

9. (a) \displaystyle{P_6(x) = x^2 - \frac{x^6}{6}}

(b) \displaystyle{P_7(x) = \frac{x^3}{3} - \frac{x^7}{42}}

This entry was posted on 26 July 26 2007 at 3:51 pm and is filed under Section 5.2. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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