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Re: Correction to "Fundamental Theorem of Calculus and

  • To: mathgroup at smc.vnet.net
  • Subject: [mg100770] Re: [mg100727] Correction to "Fundamental Theorem of Calculus and
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Sat, 13 Jun 2009 06:04:43 -0400 (EDT)
  • Reply-to: hanlonr at cox.net

g[x_] := Integrate[E^(-t^2), {t, 0, x}]

D[g[x], x] === g'[x]

False

Trace[D[g[x], x]]

{{HoldForm[g[x]], HoldForm[Integrate[E^(-t^2), 
         {t, 0, x}]], HoldForm[(1/2)*Sqrt[Pi]*
         Erf[x]]}, HoldForm[
     D[(1/2)*Sqrt[Pi]*Erf[x], x]], 
   HoldForm[E^(-x^2)]}

Trace[g'[x]]

{{HoldForm[Derivative[1][g]], 
     {HoldForm[g[#1]], HoldForm[
         Integrate[E^(-t^2), {t, 0, #1}]], 
       HoldForm[(1/2)*Sqrt[Pi]*Erf[3]]}, 
     HoldForm[0 & ]}, HoldForm[(0 & )[x]], 
   HoldForm[0]}

This appears to be the problem

Integrate[Exp[-t^2], {t, 0, #1}]

(1/2)*Sqrt[Pi]*Erf[3]

g[x_] := Evaluate[Integrate[E^(-t^2), {t, 0, x}]]

D[g[x], x] === g'[x]

True

g[x_] = Integrate[E^(-t^2), {t, 0, x}];

D[g[x], x] === g'[x]

True


Bob Hanlon

---- L Wapner <lwapner2 at gmail.com> wrote: 

=============
Hi Bob:

Mine works as well using the "D" notation for derivative.  But why will it
not work using the "prime" notation?  See below.

In[1]:= g[x_] := Integrate[E^(-t^2), {t, 0, x}]

In[2]:= g'[x]

Out[2]= 0

In[3]:= D[g[x], x]

Out[3]= E^(-x^2)

Thanks,

Len


On Thu, Jun 11, 2009 at 8:18 PM, Bob Hanlon <hanlonr at cox.net> wrote:

> Works in my version.
>
> $Version
>
> 7.0 for Mac OS X x86 (64-bit) (February 19, 2009)
>
> f[x_] := Integrate[Sin[t^2], {t, 0, x}]
>
> D[f[x], x]
>
> Sin[x^2]
>
> g[x_] := Integrate[Exp[-t^2], {t, 0, x}]
>
> D[g[x], x]
>
> E^(-x^2)
>
>
> Bob Hanlon
>
> ---- Len <lwapner2 at gmail.com> 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
>


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