Issuing Function Calls within a Plot command
in [Mathematica]
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From: Andrew DeYoung on 3 Aug 2010 06:34 Hi, It seems like Mathematica has difficulty plotting calls to other functions. Is this true? For example, can you help me understand why the following does not plot? If I write myFunction[x_] := x^2; Plot[D[myFunction[x], x], {x, 0, 5}] nothing plots. If I write myFunction[x_] := x^2; Plot[D[myFunction[x][x], x], {x, 0, 5}] again, nothing plots. However, if I compute the derivative of the function outside of the Plot command, myFunction[x_] := x^2; myDerivative = D[myFunction[x], x] Plot[myDerivative, {x, 0, 5}] the derivative of x^2 (i.e., 2x) plots correctly. Can anyone please help me understand why my first two tries do not work, but my third try does? Many thanks in advance. Andrew DeYoung Carnegie Mellon University adeyoung(a)andrew.cmu.edu
From: David Park on 4 Aug 2010 05:48 The reason you obtain nothing is that Plot has the Attribute HoldFirst, which means that the plot expression is not immediately evaluated. Instead, Mathematica substitutes x values into it and only then evaluates. So say that Mathematica evaluates at x = 0.1. Then the plot expression becomes D[myFunction[0.1],0.1], which makes no sense and will not return a value. The solution is to tell Mathematica to evaluate the derivative first. myFunction[x_] := x^2; Plot[D[myFunction[x], x] // Evaluate, {x, 0, 5}] In this case, there is an even simpler construction. Plot[myFunction'[x], {x, 0, 5}] If you had a function with parameters, defined as f[a_,b_][x_]:= expression in a, b and x, then you could use f[2,3]`[x] say as a plot function of the derivative with specific parameters. If you have a plot function that involves complicated processing, such as Integral, then it is worthwhile to define the plot function outside of any Plot statement and make certain you know what you are dealing with - by looking at the resulting expression, or evaluating at some points, or looking at it with Table. This untangles the plot function definition from the plotting algorithm and simplifies any debugging. David Park djmpark(a)comcast.net http://home.comcast.net/~djmpark/ From: Andrew DeYoung [mailto:adeyoung(a)andrew.cmu.edu] Hi, It seems like Mathematica has difficulty plotting calls to other functions. Is this true? For example, can you help me understand why the following does not plot? If I write myFunction[x_] := x^2; Plot[D[myFunction[x], x], {x, 0, 5}] nothing plots. If I write myFunction[x_] := x^2; Plot[D[myFunction[x][x], x], {x, 0, 5}] again, nothing plots. However, if I compute the derivative of the function outside of the Plot command, myFunction[x_] := x^2; myDerivative = D[myFunction[x], x] Plot[myDerivative, {x, 0, 5}] the derivative of x^2 (i.e., 2x) plots correctly. Can anyone please help me understand why my first two tries do not work, but my third try does? Many thanks in advance. Andrew DeYoung Carnegie Mellon University adeyoung(a)andrew.cmu.edu
From: Curtis Osterhoudt on 4 Aug 2010 05:49 myFunction[x_] := x^2; Plot[Evaluate[ D[myFunction[x], x] ], {x, 0, 5}] The function "Plot" has the attribute "HoldAll", which means that it holds the argument unevaluated for as long as possible. I think it does this to keep things in an analytic state, and allows things such as symbolic manipulation (as your differentiation operator, "D", uses). In this case (as in many plotting cases), the use of Evaluation makes sure that things resolve to numbers when they should. At least in earlier versions of Mathematica, it was emphasized that one should Evaluate arguments to Plot whenever possible; in the latest documentation, I don't see that emphasis. On Tuesday, August 03, 2010 04:34:19 Andrew DeYoung wrote: > Hi, > > It seems like Mathematica has difficulty plotting calls to other > functions. Is this true? For example, can you help me understand why > the following does not plot? > > If I write > > myFunction[x_] := x^2; > Plot[D[myFunction[x], x], {x, 0, 5}] > > nothing plots. > > If I write > > myFunction[x_] := x^2; > Plot[D[myFunction[x][x], x], {x, 0, 5}] > > again, nothing plots. > > However, if I compute the derivative of the function outside of the > Plot command, > > myFunction[x_] := x^2; > myDerivative = D[myFunction[x], x] > Plot[myDerivative, {x, 0, 5}] > > the derivative of x^2 (i.e., 2x) plots correctly. > > Can anyone please help me understand why my first two tries do not > work, but my third try does? > > Many thanks in advance. > > Andrew DeYoung > Carnegie Mellon University > adeyoung(a)andrew.cmu.edu > > -- ================================== Curtis Osterhoudt cfo(a)remove_this.lanl.and_this.gov ==================================
From: David Reiss on 4 Aug 2010 05:50 Because you are differenciating with respect to a number. The value of x in the argument of the plot function is substituted first and then the argument is evaluated, effectively resulting in attempting to differentiate your function evaluated at numerical values with respect to those numerical values. Try the alternative Plot[D[myFunction[z], z] /. z -> x, {x, 0, 5}] or Plot[Evaluate[D[myFunction[x], x] ], {x, 0, 5}] to force the differentiation to take place first (since Plot has the HoldAll attribute). which will work. However you may want to think through why you are including the unevaluated differentiation within the Plot command anyway: it will take place at each plot point and thus be quite inefficient... Hope this helps, David On Aug 3, 6:34 am, Andrew DeYoung <adeyo...(a)andrew.cmu.edu> wrote: > Hi, > > It seems like Mathematica has difficulty plotting calls to other > functions. Is this true? For example, can you help me understand why > the following does not plot? > > If I write > > myFunction[x_] := x^2; > Plot[D[myFunction[x], x], {x, 0, 5}] > > nothing plots. > > If I write > > myFunction[x_] := x^2; > Plot[D[myFunction[x][x], x], {x, 0, 5}] > > again, nothing plots. > > However, if I compute the derivative of the function outside of the > Plot command, > > myFunction[x_] := x^2; > myDerivative = D[myFunction[x], x] > Plot[myDerivative, {x, 0, 5}] > > the derivative of x^2 (i.e., 2x) plots correctly. > > Can anyone please help me understand why my first two tries do not > work, but my third try does? > > Many thanks in advance. > > Andrew DeYoung > Carnegie Mellon University > adeyo...(a)andrew.cmu.edu
From: Themis Matsoukas on 4 Aug 2010 05:51
The error messages give a hint of what is going on: myFunction[x_] := x^2; Plot[D[myFunction[x], x], {x, 0, 5}] General::ivar: 0.00010214285714285715` is not a valid variable. >> Plot works by evaluating myFunction at fixed x but then D[] fails b/c it expects x to be a variable, not a number. You will get the same error is you try to execute the following: N[D[myFunction[x], x = 0.000102]] You can avoid this conflict by using separate variables in D[] and in Plot[]: myFunction[x_] := x^2; Plot[D[myFunction[y], y] /. y -> x, {x, 0, 5}] Themis |