Mixing
Consider the polynomial $f_kleft ( x right )=sum_i=0^kx ^i$. Find $f_k'left ( 1 right )$ in terms of $k$.
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
Consider the polynomial $f_kleft ( x right )=sum_i=0^kx ^i$. Find $f_k'left ( 1 right )$ in terms of $k$.
If I think of $f_kleft ( x right )=sum_i=0^kx ^i$ as $x^0+x^1+x^3+... +x^k$, it's derivative would be $1+3x^2+4x^3+ ... +kx^k-1$ and $f_k'left ( 1 right )$ would equal $frack^2+k2$.
However, if I think of $f_kleft ( x right )=sum_i=0^kx ^i$ as $frac1-x^k+11-x$, it's derivative would be $frac(x-1)(k+1)x^k+(1-x^k+1)(1-x)^2$ (quotient rule) and $f_k'left ( 1 right )$ would be ... $frac00$?
calculus sequences-and-series derivatives
edited Jul 26 at 0:33
Shaun
7,31592972
asked Jul 26 at 0:27
Jennifer Lin
141
add a comment |Â
up vote
0
down vote
favorite
Consider the polynomial $f_kleft ( x right )=sum_i=0^kx ^i$. Find $f_k'left ( 1 right )$ in terms of $k$.
If I think of $f_kleft ( x right )=sum_i=0^kx ^i$ as $x^0+x^1+x^3+... +x^k$, it's derivative would be $1+3x^2+4x^3+ ... +kx^k-1$ and $f_k'left ( 1 right )$ would equal $frack^2+k2$.
However, if I think of $f_kleft ( x right )=sum_i=0^kx ^i$ as $frac1-x^k+11-x$, it's derivative would be $frac(x-1)(k+1)x^k+(1-x^k+1)(1-x)^2$ (quotient rule) and $f_k'left ( 1 right )$ would be ... $frac00$?
calculus sequences-and-series derivatives
edited Jul 26 at 0:33
Shaun
7,31592972
asked Jul 26 at 0:27
Jennifer Lin
141
3
You dropped $x^2$ in your first version. In the other version one would have to take the limit as $x to 1$ and doing that would likely get rid of the zero divide.
– coffeemath
Jul 26 at 0:34
1
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Shaun
Jul 26 at 0:34
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider the polynomial $f_kleft ( x right )=sum_i=0^kx ^i$. Find $f_k'left ( 1 right )$ in terms of $k$.
If I think of $f_kleft ( x right )=sum_i=0^kx ^i$ as $x^0+x^1+x^3+... +x^k$, it's derivative would be $1+3x^2+4x^3+ ... +kx^k-1$ and $f_k'left ( 1 right )$ would equal $frack^2+k2$.
However, if I think of $f_kleft ( x right )=sum_i=0^kx ^i$ as $frac1-x^k+11-x$, it's derivative would be $frac(x-1)(k+1)x^k+(1-x^k+1)(1-x)^2$ (quotient rule) and $f_k'left ( 1 right )$ would be ... $frac00$?
calculus sequences-and-series derivatives
edited Jul 26 at 0:33
Shaun
7,31592972
asked Jul 26 at 0:27
Jennifer Lin
141
Consider the polynomial $f_kleft ( x right )=sum_i=0^kx ^i$. Find $f_k'left ( 1 right )$ in terms of $k$.
If I think of $f_kleft ( x right )=sum_i=0^kx ^i$ as $x^0+x^1+x^3+... +x^k$, it's derivative would be $1+3x^2+4x^3+ ... +kx^k-1$ and $f_k'left ( 1 right )$ would equal $frack^2+k2$.
However, if I think of $f_kleft ( x right )=sum_i=0^kx ^i$ as $frac1-x^k+11-x$, it's derivative would be $frac(x-1)(k+1)x^k+(1-x^k+1)(1-x)^2$ (quotient rule) and $f_k'left ( 1 right )$ would be ... $frac00$?
calculus sequences-and-series derivatives
edited Jul 26 at 0:33
Shaun
7,31592972
asked Jul 26 at 0:27
Jennifer Lin
141
edited Jul 26 at 0:33
Shaun
7,31592972
edited Jul 26 at 0:33
Shaun
7,31592972
edited Jul 26 at 0:33
Shaun
7,31592972
7,31592972
asked Jul 26 at 0:27
Jennifer Lin
141
asked Jul 26 at 0:27
Jennifer Lin
141
asked Jul 26 at 0:27
Jennifer Lin
141
141
3
You dropped $x^2$ in your first version. In the other version one would have to take the limit as $x to 1$ and doing that would likely get rid of the zero divide.
– coffeemath
Jul 26 at 0:34
1
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Shaun
Jul 26 at 0:34
add a comment |Â
3
You dropped $x^2$ in your first version. In the other version one would have to take the limit as $x to 1$ and doing that would likely get rid of the zero divide.
– coffeemath
Jul 26 at 0:34
1
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Shaun
Jul 26 at 0:34
3
3
You dropped $x^2$ in your first version. In the other version one would have to take the limit as $x to 1$ and doing that would likely get rid of the zero divide.
– coffeemath
Jul 26 at 0:34
You dropped $x^2$ in your first version. In the other version one would have to take the limit as $x to 1$ and doing that would likely get rid of the zero divide.
– coffeemath
Jul 26 at 0:34
1
1
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Shaun
Jul 26 at 0:34
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Shaun
Jul 26 at 0:34
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
$$f_kleft ( x right )=sum_i=0^kx ^i=1+x+x^2+x^3+...+x^k$$
$$ f'_kleft ( x right )=1+2x+3x^2+4x^3+...+kx^k-1=sum_i=1^k ix ^i-1 $$
$$f'_kleft ( 1 right )=1+2+3+4+...+k= frac k(k+1)2$$
You can not write your function as $$frac1-x^k+11-x$$ around $x=1$ because it is not defined at $x=1$
Same with your derivative,which is not defined as $x=1.$
Have you tried to simplify your derivative and see what is the result after removing the $(1-x)^2 $ from the top and bottom?
answered Jul 26 at 0:53
Mohammad Riazi-Kermani
27.3k41851
please read the whole question
– Jennifer Lin
Jul 26 at 0:59
@JenniferLin Check my edited solution please.
– Mohammad Riazi-Kermani
Jul 26 at 1:08
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
$$f_kleft ( x right )=sum_i=0^kx ^i=1+x+x^2+x^3+...+x^k$$
$$ f'_kleft ( x right )=1+2x+3x^2+4x^3+...+kx^k-1=sum_i=1^k ix ^i-1 $$
$$f'_kleft ( 1 right )=1+2+3+4+...+k= frac k(k+1)2$$
You can not write your function as $$frac1-x^k+11-x$$ around $x=1$ because it is not defined at $x=1$
Same with your derivative,which is not defined as $x=1.$
Have you tried to simplify your derivative and see what is the result after removing the $(1-x)^2 $ from the top and bottom?
answered Jul 26 at 0:53
Mohammad Riazi-Kermani
27.3k41851
please read the whole question
– Jennifer Lin
Jul 26 at 0:59
@JenniferLin Check my edited solution please.
– Mohammad Riazi-Kermani
Jul 26 at 1:08
add a comment |Â
up vote
0
down vote
$$f_kleft ( x right )=sum_i=0^kx ^i=1+x+x^2+x^3+...+x^k$$
$$ f'_kleft ( x right )=1+2x+3x^2+4x^3+...+kx^k-1=sum_i=1^k ix ^i-1 $$
$$f'_kleft ( 1 right )=1+2+3+4+...+k= frac k(k+1)2$$
You can not write your function as $$frac1-x^k+11-x$$ around $x=1$ because it is not defined at $x=1$
Same with your derivative,which is not defined as $x=1.$
Have you tried to simplify your derivative and see what is the result after removing the $(1-x)^2 $ from the top and bottom?
answered Jul 26 at 0:53
Mohammad Riazi-Kermani
27.3k41851
please read the whole question
– Jennifer Lin
Jul 26 at 0:59
@JenniferLin Check my edited solution please.
– Mohammad Riazi-Kermani
Jul 26 at 1:08
add a comment |Â
up vote
0
down vote
up vote
0
down vote
$$f_kleft ( x right )=sum_i=0^kx ^i=1+x+x^2+x^3+...+x^k$$
$$ f'_kleft ( x right )=1+2x+3x^2+4x^3+...+kx^k-1=sum_i=1^k ix ^i-1 $$
$$f'_kleft ( 1 right )=1+2+3+4+...+k= frac k(k+1)2$$
You can not write your function as $$frac1-x^k+11-x$$ around $x=1$ because it is not defined at $x=1$
Same with your derivative,which is not defined as $x=1.$
Have you tried to simplify your derivative and see what is the result after removing the $(1-x)^2 $ from the top and bottom?
answered Jul 26 at 0:53
Mohammad Riazi-Kermani
27.3k41851
$$f_kleft ( x right )=sum_i=0^kx ^i=1+x+x^2+x^3+...+x^k$$
$$ f'_kleft ( x right )=1+2x+3x^2+4x^3+...+kx^k-1=sum_i=1^k ix ^i-1 $$
$$f'_kleft ( 1 right )=1+2+3+4+...+k= frac k(k+1)2$$
You can not write your function as $$frac1-x^k+11-x$$ around $x=1$ because it is not defined at $x=1$
Same with your derivative,which is not defined as $x=1.$
Have you tried to simplify your derivative and see what is the result after removing the $(1-x)^2 $ from the top and bottom?
answered Jul 26 at 0:53
Mohammad Riazi-Kermani
27.3k41851
edited Jul 26 at 1:07
answered Jul 26 at 0:53
Mohammad Riazi-Kermani
27.3k41851
answered Jul 26 at 0:53
Mohammad Riazi-Kermani
27.3k41851
answered Jul 26 at 0:53
Mohammad Riazi-Kermani
27.3k41851
27.3k41851
please read the whole question
– Jennifer Lin
Jul 26 at 0:59
@JenniferLin Check my edited solution please.
– Mohammad Riazi-Kermani
Jul 26 at 1:08
add a comment |Â
please read the whole question
– Jennifer Lin
Jul 26 at 0:59
@JenniferLin Check my edited solution please.
– Mohammad Riazi-Kermani
Jul 26 at 1:08
please read the whole question
– Jennifer Lin
Jul 26 at 0:59
please read the whole question
– Jennifer Lin
Jul 26 at 0:59
@JenniferLin Check my edited solution please.
– Mohammad Riazi-Kermani
Jul 26 at 1:08
@JenniferLin Check my edited solution please.
– Mohammad Riazi-Kermani
Jul 26 at 1:08
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2862962%2fconsider-the-polynomial-f-k-left-x-right-sum-i-0kx-i-find-f%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
3
You dropped $x^2$ in your first version. In the other version one would have to take the limit as $x to 1$ and doing that would likely get rid of the zero divide.
– coffeemath
Jul 26 at 0:34
1
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
– Shaun
Jul 26 at 0:34