If $xmapsto F(x)=int^x_af(s)ds$, then $F$ is $C^1$ and $F'(x)=f(x),;;forall;xin[a,b]$
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
Assuming that beginalignf:[a,b]toBbbR^nendalign
is continuous, and
beginalignF:[a,b]toBbbR^nendalign
beginalignxmapsto F(x)=int^x_af(s)ds.endalign
I want to prove that $F$ is $C^1$ and $F'(x)=f(x),;;forall;xin[a,b].$
MY WORK
To prove differentiablity, it suffices to prove that beginalignVert F(x_0+h)- F(x_0)-hf(x_0)Vert leqVert h Vertepsilon(h)endalign
Since, $f:[a,b]toBbbR^n$, then it is uniformly continuous.
beginalignVert F(x_0+h)- F(x_0)-hf(x_0) Vertendalign
beginalign=Vert int^x_0+h_af(s)ds- int^x_0_af(s)ds-hf(x_0) Vertendalign
beginalign=Vert int^x_0+h_x_0f(s)ds-hf(x_0) Vertendalign
I'm stuck at this point, can anyone help me out?
calculus integration differential-equations multivariable-calculus derivatives
 |Â
show 2 more comments
up vote
1
down vote
favorite
Assuming that beginalignf:[a,b]toBbbR^nendalign
is continuous, and
beginalignF:[a,b]toBbbR^nendalign
beginalignxmapsto F(x)=int^x_af(s)ds.endalign
I want to prove that $F$ is $C^1$ and $F'(x)=f(x),;;forall;xin[a,b].$
MY WORK
To prove differentiablity, it suffices to prove that beginalignVert F(x_0+h)- F(x_0)-hf(x_0)Vert leqVert h Vertepsilon(h)endalign
Since, $f:[a,b]toBbbR^n$, then it is uniformly continuous.
beginalignVert F(x_0+h)- F(x_0)-hf(x_0) Vertendalign
beginalign=Vert int^x_0+h_af(s)ds- int^x_0_af(s)ds-hf(x_0) Vertendalign
beginalign=Vert int^x_0+h_x_0f(s)ds-hf(x_0) Vertendalign
I'm stuck at this point, can anyone help me out?
calculus integration differential-equations multivariable-calculus derivatives
1
Mean value theorem for integrals...
â amsmath
Aug 28 at 15:27
2
@amsmath Doesn't the Mean value theorem fail for vector-valued functions?
â Sobi
Aug 28 at 15:29
1
@Sobi Sure it does. I did not read carefully enough. My bad.
â amsmath
Aug 28 at 15:30
@amsmath Happened to me a million times! :)
â Sobi
Aug 28 at 15:31
1
@Sobi That happens because we wanna be the first to help, right? ;-)
â amsmath
Aug 28 at 18:02
 |Â
show 2 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Assuming that beginalignf:[a,b]toBbbR^nendalign
is continuous, and
beginalignF:[a,b]toBbbR^nendalign
beginalignxmapsto F(x)=int^x_af(s)ds.endalign
I want to prove that $F$ is $C^1$ and $F'(x)=f(x),;;forall;xin[a,b].$
MY WORK
To prove differentiablity, it suffices to prove that beginalignVert F(x_0+h)- F(x_0)-hf(x_0)Vert leqVert h Vertepsilon(h)endalign
Since, $f:[a,b]toBbbR^n$, then it is uniformly continuous.
beginalignVert F(x_0+h)- F(x_0)-hf(x_0) Vertendalign
beginalign=Vert int^x_0+h_af(s)ds- int^x_0_af(s)ds-hf(x_0) Vertendalign
beginalign=Vert int^x_0+h_x_0f(s)ds-hf(x_0) Vertendalign
I'm stuck at this point, can anyone help me out?
calculus integration differential-equations multivariable-calculus derivatives
Assuming that beginalignf:[a,b]toBbbR^nendalign
is continuous, and
beginalignF:[a,b]toBbbR^nendalign
beginalignxmapsto F(x)=int^x_af(s)ds.endalign
I want to prove that $F$ is $C^1$ and $F'(x)=f(x),;;forall;xin[a,b].$
MY WORK
To prove differentiablity, it suffices to prove that beginalignVert F(x_0+h)- F(x_0)-hf(x_0)Vert leqVert h Vertepsilon(h)endalign
Since, $f:[a,b]toBbbR^n$, then it is uniformly continuous.
beginalignVert F(x_0+h)- F(x_0)-hf(x_0) Vertendalign
beginalign=Vert int^x_0+h_af(s)ds- int^x_0_af(s)ds-hf(x_0) Vertendalign
beginalign=Vert int^x_0+h_x_0f(s)ds-hf(x_0) Vertendalign
I'm stuck at this point, can anyone help me out?
calculus integration differential-equations multivariable-calculus derivatives
asked Aug 28 at 15:20
Micheal
25710
25710
1
Mean value theorem for integrals...
â amsmath
Aug 28 at 15:27
2
@amsmath Doesn't the Mean value theorem fail for vector-valued functions?
â Sobi
Aug 28 at 15:29
1
@Sobi Sure it does. I did not read carefully enough. My bad.
â amsmath
Aug 28 at 15:30
@amsmath Happened to me a million times! :)
â Sobi
Aug 28 at 15:31
1
@Sobi That happens because we wanna be the first to help, right? ;-)
â amsmath
Aug 28 at 18:02
 |Â
show 2 more comments
1
Mean value theorem for integrals...
â amsmath
Aug 28 at 15:27
2
@amsmath Doesn't the Mean value theorem fail for vector-valued functions?
â Sobi
Aug 28 at 15:29
1
@Sobi Sure it does. I did not read carefully enough. My bad.
â amsmath
Aug 28 at 15:30
@amsmath Happened to me a million times! :)
â Sobi
Aug 28 at 15:31
1
@Sobi That happens because we wanna be the first to help, right? ;-)
â amsmath
Aug 28 at 18:02
1
1
Mean value theorem for integrals...
â amsmath
Aug 28 at 15:27
Mean value theorem for integrals...
â amsmath
Aug 28 at 15:27
2
2
@amsmath Doesn't the Mean value theorem fail for vector-valued functions?
â Sobi
Aug 28 at 15:29
@amsmath Doesn't the Mean value theorem fail for vector-valued functions?
â Sobi
Aug 28 at 15:29
1
1
@Sobi Sure it does. I did not read carefully enough. My bad.
â amsmath
Aug 28 at 15:30
@Sobi Sure it does. I did not read carefully enough. My bad.
â amsmath
Aug 28 at 15:30
@amsmath Happened to me a million times! :)
â Sobi
Aug 28 at 15:31
@amsmath Happened to me a million times! :)
â Sobi
Aug 28 at 15:31
1
1
@Sobi That happens because we wanna be the first to help, right? ;-)
â amsmath
Aug 28 at 18:02
@Sobi That happens because we wanna be the first to help, right? ;-)
â amsmath
Aug 28 at 18:02
 |Â
show 2 more comments
1 Answer
1
active
oldest
votes
up vote
4
down vote
accepted
Use that the integral over a constant is the constant times the interval length, in reverse
beginalign
leftVert int^x_0+h_x_0f(s),ds-hf(x_0) rightVert
&=leftVert int^x_0+h_x_0(f(s)-f(x_0)),ds rightVert\
&le int^x_0+h_x_0Vert f(s)-f(x_0) Vert,ds
endalign
Wow, I love this!
â Micheal
Aug 28 at 15:28
Now, use uniform continuity of $f$ on $[a,b]$.
â amsmath
Aug 28 at 15:29
@amsmath: Done!
â Micheal
Aug 28 at 18:55
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
Use that the integral over a constant is the constant times the interval length, in reverse
beginalign
leftVert int^x_0+h_x_0f(s),ds-hf(x_0) rightVert
&=leftVert int^x_0+h_x_0(f(s)-f(x_0)),ds rightVert\
&le int^x_0+h_x_0Vert f(s)-f(x_0) Vert,ds
endalign
Wow, I love this!
â Micheal
Aug 28 at 15:28
Now, use uniform continuity of $f$ on $[a,b]$.
â amsmath
Aug 28 at 15:29
@amsmath: Done!
â Micheal
Aug 28 at 18:55
add a comment |Â
up vote
4
down vote
accepted
Use that the integral over a constant is the constant times the interval length, in reverse
beginalign
leftVert int^x_0+h_x_0f(s),ds-hf(x_0) rightVert
&=leftVert int^x_0+h_x_0(f(s)-f(x_0)),ds rightVert\
&le int^x_0+h_x_0Vert f(s)-f(x_0) Vert,ds
endalign
Wow, I love this!
â Micheal
Aug 28 at 15:28
Now, use uniform continuity of $f$ on $[a,b]$.
â amsmath
Aug 28 at 15:29
@amsmath: Done!
â Micheal
Aug 28 at 18:55
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
Use that the integral over a constant is the constant times the interval length, in reverse
beginalign
leftVert int^x_0+h_x_0f(s),ds-hf(x_0) rightVert
&=leftVert int^x_0+h_x_0(f(s)-f(x_0)),ds rightVert\
&le int^x_0+h_x_0Vert f(s)-f(x_0) Vert,ds
endalign
Use that the integral over a constant is the constant times the interval length, in reverse
beginalign
leftVert int^x_0+h_x_0f(s),ds-hf(x_0) rightVert
&=leftVert int^x_0+h_x_0(f(s)-f(x_0)),ds rightVert\
&le int^x_0+h_x_0Vert f(s)-f(x_0) Vert,ds
endalign
answered Aug 28 at 15:27
LutzL
50.1k31849
50.1k31849
Wow, I love this!
â Micheal
Aug 28 at 15:28
Now, use uniform continuity of $f$ on $[a,b]$.
â amsmath
Aug 28 at 15:29
@amsmath: Done!
â Micheal
Aug 28 at 18:55
add a comment |Â
Wow, I love this!
â Micheal
Aug 28 at 15:28
Now, use uniform continuity of $f$ on $[a,b]$.
â amsmath
Aug 28 at 15:29
@amsmath: Done!
â Micheal
Aug 28 at 18:55
Wow, I love this!
â Micheal
Aug 28 at 15:28
Wow, I love this!
â Micheal
Aug 28 at 15:28
Now, use uniform continuity of $f$ on $[a,b]$.
â amsmath
Aug 28 at 15:29
Now, use uniform continuity of $f$ on $[a,b]$.
â amsmath
Aug 28 at 15:29
@amsmath: Done!
â Micheal
Aug 28 at 18:55
@amsmath: Done!
â Micheal
Aug 28 at 18:55
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%2f2897389%2fif-x-mapsto-fx-intx-afsds-then-f-is-c1-and-fx-fx-fo%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
1
Mean value theorem for integrals...
â amsmath
Aug 28 at 15:27
2
@amsmath Doesn't the Mean value theorem fail for vector-valued functions?
â Sobi
Aug 28 at 15:29
1
@Sobi Sure it does. I did not read carefully enough. My bad.
â amsmath
Aug 28 at 15:30
@amsmath Happened to me a million times! :)
â Sobi
Aug 28 at 15:31
1
@Sobi That happens because we wanna be the first to help, right? ;-)
â amsmath
Aug 28 at 18:02