Proving that $xf'(x) = fracxx+1(x+1)f(x+1) - xf(x) - frac12xf''(ξ)$
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Exercise :
Let $f:(0,infty) to mathbb R$ be a function such that $f in C^2$, $lim_x to infty xf(x) = 2$ and $lim_x to infty xf''(x) = 0$. If $x>0$, prove that for some $ξ in (x,x+1)$ it is
$$xf'(x) = fracxx+1(x+1)f(x+1) - xf(x) - frac12xf''(ξ)$$
and then calculate the limit $lim_x to infty xf'(x)$.
Question/Request : I need some help, hints or tips on how to prove the formula in the center of the question. The final limit asked is simple as you just use the formula proven, so no help needed on that.
calculus differential-equations derivatives
asked Sep 10 at 21:25
Rebellos
10.2k21039
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up vote
0
down vote
favorite
Exercise :
Let $f:(0,infty) to mathbb R$ be a function such that $f in C^2$, $lim_x to infty xf(x) = 2$ and $lim_x to infty xf''(x) = 0$. If $x>0$, prove that for some $ξ in (x,x+1)$ it is
$$xf'(x) = fracxx+1(x+1)f(x+1) - xf(x) - frac12xf''(ξ)$$
and then calculate the limit $lim_x to infty xf'(x)$.
Question/Request : I need some help, hints or tips on how to prove the formula in the center of the question. The final limit asked is simple as you just use the formula proven, so no help needed on that.
calculus differential-equations derivatives
asked Sep 10 at 21:25
Rebellos
10.2k21039
Is the first term in the right-hand side what you meant to write? Because then $x+1$ would cancel out.
– Îµ-δ
Sep 10 at 21:29
@ε-δ Was weird enough on when I was copying it from my textbook, but yes, according to it, this is the form of the expression.
– Rebellos
Sep 10 at 21:30
why do you write $fracxx+1(x+1)f(x+1)$ ?
– Ahmad Bazzi
Sep 10 at 21:33
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Exercise :
Let $f:(0,infty) to mathbb R$ be a function such that $f in C^2$, $lim_x to infty xf(x) = 2$ and $lim_x to infty xf''(x) = 0$. If $x>0$, prove that for some $ξ in (x,x+1)$ it is
$$xf'(x) = fracxx+1(x+1)f(x+1) - xf(x) - frac12xf''(ξ)$$
and then calculate the limit $lim_x to infty xf'(x)$.
Question/Request : I need some help, hints or tips on how to prove the formula in the center of the question. The final limit asked is simple as you just use the formula proven, so no help needed on that.
calculus differential-equations derivatives
asked Sep 10 at 21:25
Rebellos
10.2k21039
Exercise :
Let $f:(0,infty) to mathbb R$ be a function such that $f in C^2$, $lim_x to infty xf(x) = 2$ and $lim_x to infty xf''(x) = 0$. If $x>0$, prove that for some $ξ in (x,x+1)$ it is
$$xf'(x) = fracxx+1(x+1)f(x+1) - xf(x) - frac12xf''(ξ)$$
and then calculate the limit $lim_x to infty xf'(x)$.
Question/Request : I need some help, hints or tips on how to prove the formula in the center of the question. The final limit asked is simple as you just use the formula proven, so no help needed on that.
calculus differential-equations derivatives
calculus differential-equations derivatives
asked Sep 10 at 21:25
Rebellos
10.2k21039
asked Sep 10 at 21:25
Rebellos
10.2k21039
asked Sep 10 at 21:25
Rebellos
10.2k21039
asked Sep 10 at 21:25
Rebellos
10.2k21039
asked Sep 10 at 21:25
Rebellos
10.2k21039
10.2k21039
Is the first term in the right-hand side what you meant to write? Because then $x+1$ would cancel out.
– Îµ-δ
Sep 10 at 21:29
@ε-δ Was weird enough on when I was copying it from my textbook, but yes, according to it, this is the form of the expression.
– Rebellos
Sep 10 at 21:30
why do you write $fracxx+1(x+1)f(x+1)$ ?
– Ahmad Bazzi
Sep 10 at 21:33
add a comment |Â
Is the first term in the right-hand side what you meant to write? Because then $x+1$ would cancel out.
– Îµ-δ
Sep 10 at 21:29
@ε-δ Was weird enough on when I was copying it from my textbook, but yes, according to it, this is the form of the expression.
– Rebellos
Sep 10 at 21:30
why do you write $fracxx+1(x+1)f(x+1)$ ?
– Ahmad Bazzi
Sep 10 at 21:33
Is the first term in the right-hand side what you meant to write? Because then $x+1$ would cancel out.
– Îµ-δ
Sep 10 at 21:29
Is the first term in the right-hand side what you meant to write? Because then $x+1$ would cancel out.
– Îµ-δ
Sep 10 at 21:29
@ε-δ Was weird enough on when I was copying it from my textbook, but yes, according to it, this is the form of the expression.
– Rebellos
Sep 10 at 21:30
@ε-δ Was weird enough on when I was copying it from my textbook, but yes, according to it, this is the form of the expression.
– Rebellos
Sep 10 at 21:30
why do you write $fracxx+1(x+1)f(x+1)$ ?
– Ahmad Bazzi
Sep 10 at 21:33
why do you write $fracxx+1(x+1)f(x+1)$ ?
– Ahmad Bazzi
Sep 10 at 21:33
add a comment |Â
1 Answer
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I don't know why you would write the identity in this peculiar why, but this is just Taylor's theorem on the interval $[x,x+1]$. Since $fin C^2$, the theorem guarantees the existence of $xiin(x,x+1)$, so that $f(x+1)=f(x)+f'(x)+frac12f''(xi)$. Multiply by $x$ and you're done.
answered Sep 10 at 22:12
ε-δ
1285
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
I don't know why you would write the identity in this peculiar why, but this is just Taylor's theorem on the interval $[x,x+1]$. Since $fin C^2$, the theorem guarantees the existence of $xiin(x,x+1)$, so that $f(x+1)=f(x)+f'(x)+frac12f''(xi)$. Multiply by $x$ and you're done.
answered Sep 10 at 22:12
ε-δ
1285
add a comment |Â
up vote
2
down vote
I don't know why you would write the identity in this peculiar why, but this is just Taylor's theorem on the interval $[x,x+1]$. Since $fin C^2$, the theorem guarantees the existence of $xiin(x,x+1)$, so that $f(x+1)=f(x)+f'(x)+frac12f''(xi)$. Multiply by $x$ and you're done.
answered Sep 10 at 22:12
ε-δ
1285
add a comment |Â
up vote
2
down vote
up vote
2
down vote
I don't know why you would write the identity in this peculiar why, but this is just Taylor's theorem on the interval $[x,x+1]$. Since $fin C^2$, the theorem guarantees the existence of $xiin(x,x+1)$, so that $f(x+1)=f(x)+f'(x)+frac12f''(xi)$. Multiply by $x$ and you're done.
answered Sep 10 at 22:12
ε-δ
1285
I don't know why you would write the identity in this peculiar why, but this is just Taylor's theorem on the interval $[x,x+1]$. Since $fin C^2$, the theorem guarantees the existence of $xiin(x,x+1)$, so that $f(x+1)=f(x)+f'(x)+frac12f''(xi)$. Multiply by $x$ and you're done.
answered Sep 10 at 22:12
ε-δ
1285
answered Sep 10 at 22:12
ε-δ
1285
answered Sep 10 at 22:12
ε-δ
1285
answered Sep 10 at 22:12
ε-δ
1285
1285
add a comment |Â
add a comment |Â
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Is the first term in the right-hand side what you meant to write? Because then $x+1$ would cancel out.
– Îµ-δ
Sep 10 at 21:29
@ε-δ Was weird enough on when I was copying it from my textbook, but yes, according to it, this is the form of the expression.
– Rebellos
Sep 10 at 21:30
why do you write $fracxx+1(x+1)f(x+1)$ ?
– Ahmad Bazzi
Sep 10 at 21:33