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Re: Re: Correction to "Fundamental Theorem of Calculus and

  • To: mathgroup at smc.vnet.net
  • Subject: [mg100786] Re: [mg100754] Re: Correction to "Fundamental Theorem of Calculus and
  • From: "David Park" <djmpark at comcast.net>
  • Date: Sun, 14 Jun 2009 05:38:11 -0400 (EDT)
  • References: <h0t84r$r7k$1@smc.vnet.net> <25935701.1244890559823.JavaMail.root@n11>

As Bob Hanlon points out, there is a bug with:

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

g'[x] // Trace

I think someone left off the &!

But since this is a theorem about functions, we might use Function in the
first place:

g := Function[x, Integrate[E^-t^2, {t, 0, x}]]
g'[x] // Trace

which now works.

Also, Mathematica can handle the following purely symbolic expression.

Implies[F[x] == Integrate[f[t], {t, 0, x}], 
\!\(\*SuperscriptBox["F", "\[Prime]",
MultilineFunction->None]\)[x] == f[x]]
MapAt[D[#, x] &, %, 1]


David Park
djmpark at comcast.net
http://home.comcast.net/~djmpark/  


From: Len [mailto:lwapner2 at gmail.com] 


Hi Bob:

For some reason Mathematica doesn't like the "prime notation".  (See
below).  The prime notation does work for the
sin (t^2) example.  Do you know why this is the case?

Thanks -

Len

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

On Jun 12, 2:46 am, Bob Hanlon <hanl... 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 <lwapn... 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 ":=", al=
l
> works well.
>
> Can someone explain the odd behavior?
>
> Thanks,
>
> Len





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