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