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 >