Section 5.2

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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}}

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