error term of taylor series












1












$begingroup$


I've been posed the following questions and i'm struggling to solve (b) and (c):
enter image description here



I'm not exactly sure how to use taylor's theorem in this case, any suggestions?






calculus







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$endgroup$












  • $begingroup$
    Last question: which form does the remainder have in the version of the Taylor theorem you learned?
    $endgroup$
    – Raskolnikov
    Jan 16 at 22:51










  • $begingroup$
    $(x-c)^n$ for a taylor series centered around $c$
    $endgroup$
    – lohboys
    Jan 16 at 23:02
















1












$begingroup$


I've been posed the following questions and i'm struggling to solve (b) and (c):
enter image description here



I'm not exactly sure how to use taylor's theorem in this case, any suggestions?






calculus







share|cite|improve this question











$endgroup$












  • $begingroup$
    Last question: which form does the remainder have in the version of the Taylor theorem you learned?
    $endgroup$
    – Raskolnikov
    Jan 16 at 22:51










  • $begingroup$
    $(x-c)^n$ for a taylor series centered around $c$
    $endgroup$
    – lohboys
    Jan 16 at 23:02














1












1








1





$begingroup$


I've been posed the following questions and i'm struggling to solve (b) and (c):
enter image description here



I'm not exactly sure how to use taylor's theorem in this case, any suggestions?






calculus







share|cite|improve this question











$endgroup$




I've been posed the following questions and i'm struggling to solve (b) and (c):
enter image description here



I'm not exactly sure how to use taylor's theorem in this case, any suggestions?






calculus




calculus






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 16 at 21:33







lohboys

















asked Jan 16 at 4:49









lohboyslohboys

958




958












  • $begingroup$
    Last question: which form does the remainder have in the version of the Taylor theorem you learned?
    $endgroup$
    – Raskolnikov
    Jan 16 at 22:51










  • $begingroup$
    $(x-c)^n$ for a taylor series centered around $c$
    $endgroup$
    – lohboys
    Jan 16 at 23:02


















  • $begingroup$
    Last question: which form does the remainder have in the version of the Taylor theorem you learned?
    $endgroup$
    – Raskolnikov
    Jan 16 at 22:51










  • $begingroup$
    $(x-c)^n$ for a taylor series centered around $c$
    $endgroup$
    – lohboys
    Jan 16 at 23:02
















$begingroup$
Last question: which form does the remainder have in the version of the Taylor theorem you learned?
$endgroup$
– Raskolnikov
Jan 16 at 22:51




$begingroup$
Last question: which form does the remainder have in the version of the Taylor theorem you learned?
$endgroup$
– Raskolnikov
Jan 16 at 22:51












$begingroup$
$(x-c)^n$ for a taylor series centered around $c$
$endgroup$
– lohboys
Jan 16 at 23:02




$begingroup$
$(x-c)^n$ for a taylor series centered around $c$
$endgroup$
– lohboys
Jan 16 at 23:02










1 Answer
1






active

oldest

votes


















1












$begingroup$

So, the third order term $R_2(x)$ should be of the form $$frac{f'''(z)}{3!}(x-16)^3$$ for some $z$ in between $x$ and $16$ by Taylor's theorem.



The third derivative of $x^{1/2}$ can be evaluated to $$f'''(z) = frac{3}{8}z^{-5/2} ; .$$



So you need to look for which values of $x$, $|R_2(x)|<0.01$. Can you take it from here with the hint in (c)?



EDIT: To complete the answer, since OP already solved the problem,



$$left|frac{f'''(z)}{3!}(x-16)^3right|=left|frac{3}{3! 8}z^{-5/2}(x-16)^3right|leqleft|frac{1}{2^{14}}p^3right|$$



where in the last step I've made use of the fact that for $z>16$,



$$z^{-5/2}<16^{-5/2} = 2^{-10}$$



and where I also put $x=16+p$. Then requesting $|R_2(x)|<0.01$ amounts to



$$p<sqrt[3]{0.01 cdot 2^{14}} approx 5.4719 ; .$$






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    for part (c), is the domain $16<x<21.47$?
    $endgroup$
    – lohboys
    Jan 17 at 0:07











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1 Answer
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1












$begingroup$

So, the third order term $R_2(x)$ should be of the form $$frac{f'''(z)}{3!}(x-16)^3$$ for some $z$ in between $x$ and $16$ by Taylor's theorem.



The third derivative of $x^{1/2}$ can be evaluated to $$f'''(z) = frac{3}{8}z^{-5/2} ; .$$



So you need to look for which values of $x$, $|R_2(x)|<0.01$. Can you take it from here with the hint in (c)?



EDIT: To complete the answer, since OP already solved the problem,



$$left|frac{f'''(z)}{3!}(x-16)^3right|=left|frac{3}{3! 8}z^{-5/2}(x-16)^3right|leqleft|frac{1}{2^{14}}p^3right|$$



where in the last step I've made use of the fact that for $z>16$,



$$z^{-5/2}<16^{-5/2} = 2^{-10}$$



and where I also put $x=16+p$. Then requesting $|R_2(x)|<0.01$ amounts to



$$p<sqrt[3]{0.01 cdot 2^{14}} approx 5.4719 ; .$$






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    for part (c), is the domain $16<x<21.47$?
    $endgroup$
    – lohboys
    Jan 17 at 0:07
















1












$begingroup$

So, the third order term $R_2(x)$ should be of the form $$frac{f'''(z)}{3!}(x-16)^3$$ for some $z$ in between $x$ and $16$ by Taylor's theorem.



The third derivative of $x^{1/2}$ can be evaluated to $$f'''(z) = frac{3}{8}z^{-5/2} ; .$$



So you need to look for which values of $x$, $|R_2(x)|<0.01$. Can you take it from here with the hint in (c)?



EDIT: To complete the answer, since OP already solved the problem,



$$left|frac{f'''(z)}{3!}(x-16)^3right|=left|frac{3}{3! 8}z^{-5/2}(x-16)^3right|leqleft|frac{1}{2^{14}}p^3right|$$



where in the last step I've made use of the fact that for $z>16$,



$$z^{-5/2}<16^{-5/2} = 2^{-10}$$



and where I also put $x=16+p$. Then requesting $|R_2(x)|<0.01$ amounts to



$$p<sqrt[3]{0.01 cdot 2^{14}} approx 5.4719 ; .$$






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    for part (c), is the domain $16<x<21.47$?
    $endgroup$
    – lohboys
    Jan 17 at 0:07














1












1








1





$begingroup$

So, the third order term $R_2(x)$ should be of the form $$frac{f'''(z)}{3!}(x-16)^3$$ for some $z$ in between $x$ and $16$ by Taylor's theorem.



The third derivative of $x^{1/2}$ can be evaluated to $$f'''(z) = frac{3}{8}z^{-5/2} ; .$$



So you need to look for which values of $x$, $|R_2(x)|<0.01$. Can you take it from here with the hint in (c)?



EDIT: To complete the answer, since OP already solved the problem,



$$left|frac{f'''(z)}{3!}(x-16)^3right|=left|frac{3}{3! 8}z^{-5/2}(x-16)^3right|leqleft|frac{1}{2^{14}}p^3right|$$



where in the last step I've made use of the fact that for $z>16$,



$$z^{-5/2}<16^{-5/2} = 2^{-10}$$



and where I also put $x=16+p$. Then requesting $|R_2(x)|<0.01$ amounts to



$$p<sqrt[3]{0.01 cdot 2^{14}} approx 5.4719 ; .$$






share|cite|improve this answer











$endgroup$



So, the third order term $R_2(x)$ should be of the form $$frac{f'''(z)}{3!}(x-16)^3$$ for some $z$ in between $x$ and $16$ by Taylor's theorem.



The third derivative of $x^{1/2}$ can be evaluated to $$f'''(z) = frac{3}{8}z^{-5/2} ; .$$



So you need to look for which values of $x$, $|R_2(x)|<0.01$. Can you take it from here with the hint in (c)?



EDIT: To complete the answer, since OP already solved the problem,



$$left|frac{f'''(z)}{3!}(x-16)^3right|=left|frac{3}{3! 8}z^{-5/2}(x-16)^3right|leqleft|frac{1}{2^{14}}p^3right|$$



where in the last step I've made use of the fact that for $z>16$,



$$z^{-5/2}<16^{-5/2} = 2^{-10}$$



and where I also put $x=16+p$. Then requesting $|R_2(x)|<0.01$ amounts to



$$p<sqrt[3]{0.01 cdot 2^{14}} approx 5.4719 ; .$$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Jan 17 at 16:38

























answered Jan 16 at 23:21









RaskolnikovRaskolnikov

12.6k23571




12.6k23571








  • 1




    $begingroup$
    for part (c), is the domain $16<x<21.47$?
    $endgroup$
    – lohboys
    Jan 17 at 0:07














  • 1




    $begingroup$
    for part (c), is the domain $16<x<21.47$?
    $endgroup$
    – lohboys
    Jan 17 at 0:07








1




1




$begingroup$
for part (c), is the domain $16<x<21.47$?
$endgroup$
– lohboys
Jan 17 at 0:07




$begingroup$
for part (c), is the domain $16<x<21.47$?
$endgroup$
– lohboys
Jan 17 at 0:07


















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