getting the area of a set using the supremum and infinum concept
$begingroup$
Prove that the area between the $x$-axis and the function $y=e^x$ in the interval $0 < x < c$ is $e^c - 1$. You're not allowed to use integrals.
I have started to calculate the area by using the right and left Riemann sums. I got the following expressions:
Right Riemann sum:
$$frac{c}{n} cdot frac{e^{frac{c}{n}}}{e^{frac{c}{n}}-1}cdot(e^c - 1)tag{1}$$
and by using L'Hospital's rule it is easy to show that the sum is $e^c -1$ as $n rightarrow infty$
Similarly the left Riemann sum:
$$frac{c}{n} cdot frac{1}{e^{frac{c}{n}}-1}cdot(e^{frac{c}{n}} - 1)tag{2}$$
again I can show that the limit as n approches infinity is $e^c - 1$.
Now I would like to use the concept of supremum and infinimum to show that the expression (1) and (2) lead to the same conclusion that is the area is: $e^c - 1$.
As for the infimum I have also concluded that
$$frac{c}{n}cdotfrac{1}{e^{frac{c}{n}}-1}cdot(e^{frac{c}{n}} - 1) < left(1-frac{c}{n}right)(e^{frac{c}{n}} - 1)$$ but how do I apply that
$$inf left(frac{c}{n}cdotfrac{1}{e^{frac{c}{n}}-1}(e^{frac{c}{n}} - 1)right) = e^{frac{c}{n}} - 1?$$
The second question is: how on earth can I use the sup to get to the same conclusion?
Can anybody give me a step by step solution for this problem using the sup and inf concept?
elementary-set-theory exponential-function supremum-and-infimum
edited Dec 16 '18 at 21:40
Cameron Buie
85.2k772156
asked Dec 16 '18 at 21:16
manman
33
$endgroup$
add a comment |
$begingroup$
Prove that the area between the $x$-axis and the function $y=e^x$ in the interval $0 < x < c$ is $e^c - 1$. You're not allowed to use integrals.
I have started to calculate the area by using the right and left Riemann sums. I got the following expressions:
Right Riemann sum:
$$frac{c}{n} cdot frac{e^{frac{c}{n}}}{e^{frac{c}{n}}-1}cdot(e^c - 1)tag{1}$$
and by using L'Hospital's rule it is easy to show that the sum is $e^c -1$ as $n rightarrow infty$
Similarly the left Riemann sum:
$$frac{c}{n} cdot frac{1}{e^{frac{c}{n}}-1}cdot(e^{frac{c}{n}} - 1)tag{2}$$
again I can show that the limit as n approches infinity is $e^c - 1$.
Now I would like to use the concept of supremum and infinimum to show that the expression (1) and (2) lead to the same conclusion that is the area is: $e^c - 1$.
As for the infimum I have also concluded that
$$frac{c}{n}cdotfrac{1}{e^{frac{c}{n}}-1}cdot(e^{frac{c}{n}} - 1) < left(1-frac{c}{n}right)(e^{frac{c}{n}} - 1)$$ but how do I apply that
$$inf left(frac{c}{n}cdotfrac{1}{e^{frac{c}{n}}-1}(e^{frac{c}{n}} - 1)right) = e^{frac{c}{n}} - 1?$$
The second question is: how on earth can I use the sup to get to the same conclusion?
Can anybody give me a step by step solution for this problem using the sup and inf concept?
elementary-set-theory exponential-function supremum-and-infimum
edited Dec 16 '18 at 21:40
Cameron Buie
85.2k772156
asked Dec 16 '18 at 21:16
manman
33
$endgroup$
$begingroup$
I've cleaned up your layout to the best of my ability, though I think you may have made a few mistakes. Specifically, your left Riemann sum and the last two equations should probably have $e^c$ instead of $e^{frac{c}{n}}$ in at least one place each.
$endgroup$
– Cameron Buie
Dec 16 '18 at 21:44
add a comment |
$begingroup$
Prove that the area between the $x$-axis and the function $y=e^x$ in the interval $0 < x < c$ is $e^c - 1$. You're not allowed to use integrals.
I have started to calculate the area by using the right and left Riemann sums. I got the following expressions:
Right Riemann sum:
$$frac{c}{n} cdot frac{e^{frac{c}{n}}}{e^{frac{c}{n}}-1}cdot(e^c - 1)tag{1}$$
and by using L'Hospital's rule it is easy to show that the sum is $e^c -1$ as $n rightarrow infty$
Similarly the left Riemann sum:
$$frac{c}{n} cdot frac{1}{e^{frac{c}{n}}-1}cdot(e^{frac{c}{n}} - 1)tag{2}$$
again I can show that the limit as n approches infinity is $e^c - 1$.
Now I would like to use the concept of supremum and infinimum to show that the expression (1) and (2) lead to the same conclusion that is the area is: $e^c - 1$.
As for the infimum I have also concluded that
$$frac{c}{n}cdotfrac{1}{e^{frac{c}{n}}-1}cdot(e^{frac{c}{n}} - 1) < left(1-frac{c}{n}right)(e^{frac{c}{n}} - 1)$$ but how do I apply that
$$inf left(frac{c}{n}cdotfrac{1}{e^{frac{c}{n}}-1}(e^{frac{c}{n}} - 1)right) = e^{frac{c}{n}} - 1?$$
The second question is: how on earth can I use the sup to get to the same conclusion?
Can anybody give me a step by step solution for this problem using the sup and inf concept?
elementary-set-theory exponential-function supremum-and-infimum
edited Dec 16 '18 at 21:40
Cameron Buie
85.2k772156
asked Dec 16 '18 at 21:16
manman
33
$endgroup$
Prove that the area between the $x$-axis and the function $y=e^x$ in the interval $0 < x < c$ is $e^c - 1$. You're not allowed to use integrals.
I have started to calculate the area by using the right and left Riemann sums. I got the following expressions:
Right Riemann sum:
$$frac{c}{n} cdot frac{e^{frac{c}{n}}}{e^{frac{c}{n}}-1}cdot(e^c - 1)tag{1}$$
and by using L'Hospital's rule it is easy to show that the sum is $e^c -1$ as $n rightarrow infty$
Similarly the left Riemann sum:
$$frac{c}{n} cdot frac{1}{e^{frac{c}{n}}-1}cdot(e^{frac{c}{n}} - 1)tag{2}$$
again I can show that the limit as n approches infinity is $e^c - 1$.
Now I would like to use the concept of supremum and infinimum to show that the expression (1) and (2) lead to the same conclusion that is the area is: $e^c - 1$.
As for the infimum I have also concluded that
$$frac{c}{n}cdotfrac{1}{e^{frac{c}{n}}-1}cdot(e^{frac{c}{n}} - 1) < left(1-frac{c}{n}right)(e^{frac{c}{n}} - 1)$$ but how do I apply that
$$inf left(frac{c}{n}cdotfrac{1}{e^{frac{c}{n}}-1}(e^{frac{c}{n}} - 1)right) = e^{frac{c}{n}} - 1?$$
The second question is: how on earth can I use the sup to get to the same conclusion?
Can anybody give me a step by step solution for this problem using the sup and inf concept?
elementary-set-theory exponential-function supremum-and-infimum
elementary-set-theory exponential-function supremum-and-infimum
edited Dec 16 '18 at 21:40
Cameron Buie
85.2k772156
asked Dec 16 '18 at 21:16
manman
33
edited Dec 16 '18 at 21:40
Cameron Buie
85.2k772156
asked Dec 16 '18 at 21:16
manman
33
edited Dec 16 '18 at 21:40
Cameron Buie
85.2k772156
edited Dec 16 '18 at 21:40
Cameron Buie
85.2k772156
edited Dec 16 '18 at 21:40
Cameron Buie
85.2k772156
85.2k772156
asked Dec 16 '18 at 21:16
manman
33
asked Dec 16 '18 at 21:16
manman
33
asked Dec 16 '18 at 21:16
manman
33
33
$begingroup$
I've cleaned up your layout to the best of my ability, though I think you may have made a few mistakes. Specifically, your left Riemann sum and the last two equations should probably have $e^c$ instead of $e^{frac{c}{n}}$ in at least one place each.
$endgroup$
– Cameron Buie
Dec 16 '18 at 21:44
add a comment |
$begingroup$
I've cleaned up your layout to the best of my ability, though I think you may have made a few mistakes. Specifically, your left Riemann sum and the last two equations should probably have $e^c$ instead of $e^{frac{c}{n}}$ in at least one place each.
$endgroup$
– Cameron Buie
Dec 16 '18 at 21:44
$begingroup$
I've cleaned up your layout to the best of my ability, though I think you may have made a few mistakes. Specifically, your left Riemann sum and the last two equations should probably have $e^c$ instead of $e^{frac{c}{n}}$ in at least one place each.
$endgroup$
– Cameron Buie
Dec 16 '18 at 21:44
$begingroup$
I've cleaned up your layout to the best of my ability, though I think you may have made a few mistakes. Specifically, your left Riemann sum and the last two equations should probably have $e^c$ instead of $e^{frac{c}{n}}$ in at least one place each.
$endgroup$
– Cameron Buie
Dec 16 '18 at 21:44
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
I would use the fact that $y=e^x$ is increasing to show that one of the two Riemann sums is always greater than the area between the $x$-axis and the curve (regardless of $n$), while the other is always less than the area between the $x$-axis and the curve. Consequently, the area between the $x$-axis and the curve is is at most the infimum of the greater sums, and at least the supremum of the lesser sums. Show that both the infimum of the greater sums and the supremum of the lesser sums are equal to $e^c-1,$ and you're done!
answered Dec 16 '18 at 21:30
Cameron BuieCameron Buie
85.2k772156
$endgroup$
$begingroup$
well the infinum I have figured it out but how do I show the supremum? can you please give me step by step solution?
$endgroup$
– man
Dec 16 '18 at 21:34
$begingroup$
Have you tried comparing the Riemann sums as $n$ increases? You should find that the greater sums are eventually decreasing, and that the lesser sums are eventually increasing. Thus, to find the supremum/infimum, we need only compute the limit as $n$ grows without bound.
$endgroup$
– Cameron Buie
Dec 16 '18 at 21:46
$begingroup$
what is the sup(1/(1-c/n)) as n increases?
$endgroup$
– man
Dec 16 '18 at 22:44
$begingroup$
That depends. If $c$ is a positive integer, then the supremum is $+infty$. However, if we concentrate only on those $n$ which are greater than $c,$ the supremum is $1.$
$endgroup$
– Cameron Buie
Dec 16 '18 at 23:56
add a comment |
$begingroup$
I have solved the problem but I'm not sure if the solution is correct. If you check this and comment my solution I would appreciate it. Be aware that instead of c the constant h is used :
solution
answered Dec 17 '18 at 1:35
manman
33
$endgroup$
$begingroup$
I'm all for benefit of the doubt, but I'm seriously impressed that you've ostensibly gone from having no apparent idea what LaTeX is to being able to render a publishable document in a few hours. However, it seems that you've figured out my hints, so feel free to accept my answer, if so. Welcome to the site!
$endgroup$
– Cameron Buie
Dec 17 '18 at 3:16
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I would use the fact that $y=e^x$ is increasing to show that one of the two Riemann sums is always greater than the area between the $x$-axis and the curve (regardless of $n$), while the other is always less than the area between the $x$-axis and the curve. Consequently, the area between the $x$-axis and the curve is is at most the infimum of the greater sums, and at least the supremum of the lesser sums. Show that both the infimum of the greater sums and the supremum of the lesser sums are equal to $e^c-1,$ and you're done!
answered Dec 16 '18 at 21:30
Cameron BuieCameron Buie
85.2k772156
$endgroup$
$begingroup$
well the infinum I have figured it out but how do I show the supremum? can you please give me step by step solution?
$endgroup$
– man
Dec 16 '18 at 21:34
$begingroup$
Have you tried comparing the Riemann sums as $n$ increases? You should find that the greater sums are eventually decreasing, and that the lesser sums are eventually increasing. Thus, to find the supremum/infimum, we need only compute the limit as $n$ grows without bound.
$endgroup$
– Cameron Buie
Dec 16 '18 at 21:46
$begingroup$
what is the sup(1/(1-c/n)) as n increases?
$endgroup$
– man
Dec 16 '18 at 22:44
$begingroup$
That depends. If $c$ is a positive integer, then the supremum is $+infty$. However, if we concentrate only on those $n$ which are greater than $c,$ the supremum is $1.$
$endgroup$
– Cameron Buie
Dec 16 '18 at 23:56
add a comment |
$begingroup$
I would use the fact that $y=e^x$ is increasing to show that one of the two Riemann sums is always greater than the area between the $x$-axis and the curve (regardless of $n$), while the other is always less than the area between the $x$-axis and the curve. Consequently, the area between the $x$-axis and the curve is is at most the infimum of the greater sums, and at least the supremum of the lesser sums. Show that both the infimum of the greater sums and the supremum of the lesser sums are equal to $e^c-1,$ and you're done!
answered Dec 16 '18 at 21:30
Cameron BuieCameron Buie
85.2k772156
$endgroup$
$begingroup$
well the infinum I have figured it out but how do I show the supremum? can you please give me step by step solution?
$endgroup$
– man
Dec 16 '18 at 21:34
$begingroup$
Have you tried comparing the Riemann sums as $n$ increases? You should find that the greater sums are eventually decreasing, and that the lesser sums are eventually increasing. Thus, to find the supremum/infimum, we need only compute the limit as $n$ grows without bound.
$endgroup$
– Cameron Buie
Dec 16 '18 at 21:46
$begingroup$
what is the sup(1/(1-c/n)) as n increases?
$endgroup$
– man
Dec 16 '18 at 22:44
$begingroup$
That depends. If $c$ is a positive integer, then the supremum is $+infty$. However, if we concentrate only on those $n$ which are greater than $c,$ the supremum is $1.$
$endgroup$
– Cameron Buie
Dec 16 '18 at 23:56
add a comment |
$begingroup$
I would use the fact that $y=e^x$ is increasing to show that one of the two Riemann sums is always greater than the area between the $x$-axis and the curve (regardless of $n$), while the other is always less than the area between the $x$-axis and the curve. Consequently, the area between the $x$-axis and the curve is is at most the infimum of the greater sums, and at least the supremum of the lesser sums. Show that both the infimum of the greater sums and the supremum of the lesser sums are equal to $e^c-1,$ and you're done!
answered Dec 16 '18 at 21:30
Cameron BuieCameron Buie
85.2k772156
$endgroup$
I would use the fact that $y=e^x$ is increasing to show that one of the two Riemann sums is always greater than the area between the $x$-axis and the curve (regardless of $n$), while the other is always less than the area between the $x$-axis and the curve. Consequently, the area between the $x$-axis and the curve is is at most the infimum of the greater sums, and at least the supremum of the lesser sums. Show that both the infimum of the greater sums and the supremum of the lesser sums are equal to $e^c-1,$ and you're done!
answered Dec 16 '18 at 21:30
Cameron BuieCameron Buie
85.2k772156
answered Dec 16 '18 at 21:30
Cameron BuieCameron Buie
85.2k772156
answered Dec 16 '18 at 21:30
Cameron BuieCameron Buie
85.2k772156
answered Dec 16 '18 at 21:30
Cameron BuieCameron Buie
85.2k772156
85.2k772156
$begingroup$
well the infinum I have figured it out but how do I show the supremum? can you please give me step by step solution?
$endgroup$
– man
Dec 16 '18 at 21:34
$begingroup$
Have you tried comparing the Riemann sums as $n$ increases? You should find that the greater sums are eventually decreasing, and that the lesser sums are eventually increasing. Thus, to find the supremum/infimum, we need only compute the limit as $n$ grows without bound.
$endgroup$
– Cameron Buie
Dec 16 '18 at 21:46
$begingroup$
what is the sup(1/(1-c/n)) as n increases?
$endgroup$
– man
Dec 16 '18 at 22:44
$begingroup$
That depends. If $c$ is a positive integer, then the supremum is $+infty$. However, if we concentrate only on those $n$ which are greater than $c,$ the supremum is $1.$
$endgroup$
– Cameron Buie
Dec 16 '18 at 23:56
add a comment |
$begingroup$
well the infinum I have figured it out but how do I show the supremum? can you please give me step by step solution?
$endgroup$
– man
Dec 16 '18 at 21:34
$begingroup$
Have you tried comparing the Riemann sums as $n$ increases? You should find that the greater sums are eventually decreasing, and that the lesser sums are eventually increasing. Thus, to find the supremum/infimum, we need only compute the limit as $n$ grows without bound.
$endgroup$
– Cameron Buie
Dec 16 '18 at 21:46
$begingroup$
what is the sup(1/(1-c/n)) as n increases?
$endgroup$
– man
Dec 16 '18 at 22:44
$begingroup$
That depends. If $c$ is a positive integer, then the supremum is $+infty$. However, if we concentrate only on those $n$ which are greater than $c,$ the supremum is $1.$
$endgroup$
– Cameron Buie
Dec 16 '18 at 23:56
$begingroup$
well the infinum I have figured it out but how do I show the supremum? can you please give me step by step solution?
$endgroup$
– man
Dec 16 '18 at 21:34
$begingroup$
well the infinum I have figured it out but how do I show the supremum? can you please give me step by step solution?
$endgroup$
– man
Dec 16 '18 at 21:34
$begingroup$
Have you tried comparing the Riemann sums as $n$ increases? You should find that the greater sums are eventually decreasing, and that the lesser sums are eventually increasing. Thus, to find the supremum/infimum, we need only compute the limit as $n$ grows without bound.
$endgroup$
– Cameron Buie
Dec 16 '18 at 21:46
$begingroup$
Have you tried comparing the Riemann sums as $n$ increases? You should find that the greater sums are eventually decreasing, and that the lesser sums are eventually increasing. Thus, to find the supremum/infimum, we need only compute the limit as $n$ grows without bound.
$endgroup$
– Cameron Buie
Dec 16 '18 at 21:46
$begingroup$
what is the sup(1/(1-c/n)) as n increases?
$endgroup$
– man
Dec 16 '18 at 22:44
$begingroup$
what is the sup(1/(1-c/n)) as n increases?
$endgroup$
– man
Dec 16 '18 at 22:44
$begingroup$
That depends. If $c$ is a positive integer, then the supremum is $+infty$. However, if we concentrate only on those $n$ which are greater than $c,$ the supremum is $1.$
$endgroup$
– Cameron Buie
Dec 16 '18 at 23:56
$begingroup$
That depends. If $c$ is a positive integer, then the supremum is $+infty$. However, if we concentrate only on those $n$ which are greater than $c,$ the supremum is $1.$
$endgroup$
– Cameron Buie
Dec 16 '18 at 23:56
add a comment |
$begingroup$
I have solved the problem but I'm not sure if the solution is correct. If you check this and comment my solution I would appreciate it. Be aware that instead of c the constant h is used :
solution
answered Dec 17 '18 at 1:35
manman
33
$endgroup$
$begingroup$
I'm all for benefit of the doubt, but I'm seriously impressed that you've ostensibly gone from having no apparent idea what LaTeX is to being able to render a publishable document in a few hours. However, it seems that you've figured out my hints, so feel free to accept my answer, if so. Welcome to the site!
$endgroup$
– Cameron Buie
Dec 17 '18 at 3:16
add a comment |
$begingroup$
I have solved the problem but I'm not sure if the solution is correct. If you check this and comment my solution I would appreciate it. Be aware that instead of c the constant h is used :
solution
answered Dec 17 '18 at 1:35
manman
33
$endgroup$
$begingroup$
I'm all for benefit of the doubt, but I'm seriously impressed that you've ostensibly gone from having no apparent idea what LaTeX is to being able to render a publishable document in a few hours. However, it seems that you've figured out my hints, so feel free to accept my answer, if so. Welcome to the site!
$endgroup$
– Cameron Buie
Dec 17 '18 at 3:16
add a comment |
$begingroup$
I have solved the problem but I'm not sure if the solution is correct. If you check this and comment my solution I would appreciate it. Be aware that instead of c the constant h is used :
solution
answered Dec 17 '18 at 1:35
manman
33
$endgroup$
I have solved the problem but I'm not sure if the solution is correct. If you check this and comment my solution I would appreciate it. Be aware that instead of c the constant h is used :
solution
answered Dec 17 '18 at 1:35
manman
33
answered Dec 17 '18 at 1:35
manman
33
answered Dec 17 '18 at 1:35
manman
33
answered Dec 17 '18 at 1:35
manman
33
33
$begingroup$
I'm all for benefit of the doubt, but I'm seriously impressed that you've ostensibly gone from having no apparent idea what LaTeX is to being able to render a publishable document in a few hours. However, it seems that you've figured out my hints, so feel free to accept my answer, if so. Welcome to the site!
$endgroup$
– Cameron Buie
Dec 17 '18 at 3:16
add a comment |
$begingroup$
I'm all for benefit of the doubt, but I'm seriously impressed that you've ostensibly gone from having no apparent idea what LaTeX is to being able to render a publishable document in a few hours. However, it seems that you've figured out my hints, so feel free to accept my answer, if so. Welcome to the site!
$endgroup$
– Cameron Buie
Dec 17 '18 at 3:16
$begingroup$
I'm all for benefit of the doubt, but I'm seriously impressed that you've ostensibly gone from having no apparent idea what LaTeX is to being able to render a publishable document in a few hours. However, it seems that you've figured out my hints, so feel free to accept my answer, if so. Welcome to the site!
$endgroup$
– Cameron Buie
Dec 17 '18 at 3:16
$begingroup$
I'm all for benefit of the doubt, but I'm seriously impressed that you've ostensibly gone from having no apparent idea what LaTeX is to being able to render a publishable document in a few hours. However, it seems that you've figured out my hints, so feel free to accept my answer, if so. Welcome to the site!
$endgroup$
– Cameron Buie
Dec 17 '18 at 3:16
add a comment |
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I've cleaned up your layout to the best of my ability, though I think you may have made a few mistakes. Specifically, your left Riemann sum and the last two equations should probably have $e^c$ instead of $e^{frac{c}{n}}$ in at least one place each.
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– Cameron Buie
Dec 16 '18 at 21:44