Correction to "Fundamental Theorem of Calculus and Mathematica"
Len
I define a function (using f[x_]:=) as the definite integral (from 0
to x) of sin(t^2). When I differentiate using Mathematica I get the
correct answer of sin(x^2).
But when I define a function (using g[x_]:=) as the definite integral
(from 0 to x) of e^(-t^2) and differentiate, I get the incorrect
answer of 0. (The correct answer is e^(-x^2).)
Why the inconsistency?
Oddly, if I define the function g above using "=" instead of ":=", all
works well.
Can someone explain the odd behavior?
Thanks,
Len
Simon
Running both 6.0.3 and 7.0.1, I don't seem to get that problem:
In[1]:= f[x_]:=Integrate[Sin[t^2],{t,0,x}]
In[2]:= D[f[x],x]
Out[2]= Sin[x^2]
In[3]:= g[x_]:=Integrate[Exp[-t^2],{t,0,x}]
In[4]:= D[g[x],x]
Out[4]= E^-x^2
Simon
Murray Eisenberg
f'[x]
Sin[x^2]
g'[x]
0
I believe that's the discrepancy to which the original poster refers.
And this originates from what you'll see if you use the FullForm, the
FullForm of h'[x] being Derivative[1][h][x]:
Derivative[1][f]
(FresnelS^\[Prime])[Sqrt[2/\[Pi]] #1]&
Derivative[1][g]
0&
The latter arises from:
g[x]
1/2 Sqrt[\[Pi]] Erf[x]
So I don't understand why the derivative of g is the constantly 0
function. After all, Mathematica DOES know:
Derivative[1][Erf]
(2 E^-#1^2)/Sqrt[\[Pi]]&
And that surely is not the zero function!
--
Murray Eisenberg mur...@math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower phone 413 549-1020 (H)
University of Massachusetts 413 545-2859 (W)
710 North Pleasant Street fax 413 545-1801
Amherst, MA 01003-9305
David Bailey
that, and as a result, g had some sort of definition. If the problem
happens again, try typing
??g
If you get a strange result, it is often useful to kill the kernel so
that you start with a clean slate.
David Bailey
http://www.dbaileyconsultancy.co.uk
Curtis Osterhoudt
In[4]:= f[x_] := Integrate[Sin[t^2], {t, 0, x}]
In[5]:= D[f[x], x]
Out[5]= Sin[x^2]
In[6]:= g[x_] := Integrate[Exp[-t^2], {t, 0, x}]
In[7]:= D[g[x], x]
Out[7]= E^-x^2
On Thursday 11 June 2009 07:44:34 pm Len wrote:
> Greetings:
>
> I define a function (using f[x_]:=) as the definite integral (from 0
> to x) of sin(t^2). When I differentiate using Mathematica I get the
> correct answer of sin(x^2).
>
> But when I define a function (using g[x_]:=) as the definite integral
> (from 0 to x) of e^(-t^2) and differentiate, I get the incorrect
> answer of 0. (The correct answer is e^(-x^2).)
>
> Why the inconsistency?
>
> Oddly, if I define the function g above using "=" instead of ":=", all
> works well.
>
> Can someone explain the odd behavior?
>
> Thanks,
>
> Len
>
>
--
==================================
Curtis Osterhoudt
cfo@remove_this.lanl.and_this.gov
PGP Key ID: 0x4DCA2A10
==================================
Alexei Boulbitch
I define a function (using f[x_]:=) as the definite integral (from 0
to x) of sin(t^2). When I differentiate using Mathematica I get the
correct answer of sin(x^2).
But when I define a function (using g[x_]:=) as the definite integral
(from 0 to x) of e^(-t^2) and differentiate, I get the incorrect
answer of 0. (The correct answer is e^(-x^2).)
Why the inconsistency?
Oddly, if I define the function g above using "=" instead of ":=", all
works well.
Can someone explain the odd behavior?
Thanks,
Len
Hi, Len,
I tried and this is the result:
In[3]:= g[x_] := Integrate[Exp[-t^2], {t, 0, x}]
In[4]:= D[g[x], x]
Out[4]= \[ExponentialE]^-x^2
Sorry, Alexei
--
Alexei Boulbitch, Dr., habil.
Senior Scientist
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Len
Interesting! Like you, I get correct results when differentiating
using "D". But when I differentiate using the prime sign (g'[x] ) I
get the incorrect answer of 0.
Len
Jens-Peer Kuska
this is nonsens.
f[x_] := Integrate[Sin[t^2], {t, 0, x}]
g[x_] := Integrate[Exp[-t^2], {t, 0, x}]
D[#, x] & /@ {g[x], f[x]}
gives
{E^(-x^2), Sin[x^2]}
That is why it is useful to post your full input and not a
verbal description.
Regards
Jens
Albert Retey
> Interesting! Like you, I get correct results when differentiating
> using "D". But when I differentiate using the prime sign (g'[x] ) I
> get the incorrect answer of 0.
I think this is a case where Trace is useful, and when looking at the
output of Trace it seems clear that this is why it goes wrong for g':
In[14]:= Integrate[Exp[-t^2], {t, 0, #}]
Out[14]= 1/2 Sqrt[\[Pi]] Erf[3]
In[15]:= Integrate[Sin[t], {t, 0, #}]
Out[15]= 1 - Cos[#1]
I think Out[14] clearly is a bug, because whatever # may be, it is not 3
in almost all cases :-)
hth,
albert
peter
In[2]:= D[f[x],x]
Out[2]= Sin[x^2]
In[3]:= g[x_]:=Integrate[Exp[-t^2],{t,0,x}]
In[4]:= D[g[x],x]
Out[4]= E^-x^2
Out[9]= 7.0 for Mac OS X x86 (32-bit) (February 18, 2009)
2009/6/12 Len <lwap...@gmail.com>:
> Greetings:
>
> I define a function (using f[x_]:=) as the definite integral (from 0
> to x) of sin(t^2). When I differentiate using Mathematica I get the
> correct answer of sin(x^2).
>
> But when I define a function (using g[x_]:=) as the definite integral
> (from 0 to x) of e^(-t^2) and differentiate, I get the incorrect
> answer of 0. (The correct answer is e^(-x^2).)
>
> Why the inconsistency?
>
> Oddly, if I define the function g above using "=" instead of ":=", al=
l
> works well.
>
> Can someone explain the odd behavior?
>
> Thanks,
>
> Len
>
>
--
Peter Lindsay