# Marking points of intersection between two curves

I'm trying to illustrate the solutions numerically and graphically for an equation such as Tan[x] == x. I think I did everything ok except I wanted to mark each intersection between Tan[x] and x.

Does anyone know how such a thing can be done?

• Comment by the OP (migrated from the question) -> ** I'm sorry if I'm not 'commenting' properly. I'll figure it out when I get more time here. I wanted to say THANKS to EVERYONE that posted. This is exactly what I was looking for. I will use all the input and make sure I learn from what was given. What a goldmine this site is for learning something like this. Thanks again, it's greatly appreciated!! Commented Sep 12, 2012 at 19:23
• See also the solution using RootsInRange. This solution is more general as it will work when either the exact intersections are not known, or NSolve fails. Commented Mar 26, 2014 at 1:49

Edited to make it a function. For the strange Exclusions specification I use below, see my answer here. Thanks to @Oleksandr and @JM for their great comments.

plInters[{f1_, f2_}, {min_, max_}] :=
Module[{sol, x},
sol = x /. NSolve[f1[x] == f2[x] && min < x < max, x];
Framed@Show[
ListPlot[{#, f1[#]} & /@ sol, PlotStyle -> PointSize[Large]],
Plot[{f1[x], f2[x]}, {x, min, max}, Exclusions -> {True, f2[x] == 10, f1[x] == 10}]
]
]

GraphicsRow[plInters[#, {-10, 10}] & /@ {{# &, Tan}, {Tan, Coth}, {Sin, 1/# &}}]


• Damn, yours is better.
– kale
Commented Sep 12, 2012 at 1:00
• @kale That is one of the good things in this site. You can always find other ways. Commented Sep 12, 2012 at 1:01
• Anyway definite +1 from me.
– kale
Commented Sep 12, 2012 at 1:06
• I don't really know why Plot can sometimes determine exclusions properly and sometimes not. Still, this'll look a bit better if you set Exclusions -> Pi/2 Range[-5, 5, 2] for Plot (otherwise, it may be harder for students to see that these aren't really solutions). Commented Sep 12, 2012 at 1:45
• @Alexey I tried to explain why here mathematica.stackexchange.com/a/10502/193. It gets the plot refinement routine to work to your advantage. Commented Sep 13, 2012 at 1:52

Update 3: Using GraphicsMeshFindIntersections to get the intersection points (see also):

showIntersections =
Block[{Annotation = # &},
Show[#, Graphics@{Red, PointSize[Large],
Point@GraphicsMeshFindIntersections@#}]] &;


Using the two examples in the original answer:

Row[showIntersections /@ {Plot[{Cos[x], x Sin[x]}, {x, -3 Pi, 3 Pi},
ImageSize -> 400],
Plot[{Tan[x], x Sin[x]}, {x, -3 Pi, 3 Pi}, ImageSize -> 400,
Exclusions -> Range[-5 Pi/2, 5 Pi/2, Pi]]}]


You can also use MeshFunctions:

  Plot[{Cos[x], x Sin[x]}, {x, -3 Pi, 3 Pi},
MeshFunctions -> {(Cos[#] - # Sin[#]) &}, Mesh -> {{0}},
MeshStyle -> Directive[Red, PointSize[Large]]]


Update: Dealing with Tan[x] using Exclusions

Plot[{Tan[x], x Sin[x]}, {x, -3 Pi, 3 Pi},
MeshFunctions -> {(Tan[#] - # Sin[#]) &}, Mesh -> {{0}},
MeshStyle -> Directive[Red, PointSize[Large]],
Exclusions -> Range[-5 Pi/2, 5 Pi/2, Pi]]
(* or Exclusions -> (Cos[x] == 0) *)


Update 2: Using just Mesh and MeshStyle:

points = NSolve[Tan[x] == x Sin[x] && -3 Pi < x < 3 Pi, x][[All, 1, 2]];
Plot[{Tan[x], x Sin[x]}, {x, -3 Pi, 3 Pi},
Mesh -> {points},
MeshStyle -> {Directive[Red, PointSize[Large]]},
Exclusions -> Range[-5 Pi/2, 5 Pi/2, Pi]]
(* same picture as above *)

• very nice! :) 
– rm -rf
Commented Sep 12, 2012 at 1:45
• This has some problems with the exclusions :( Commented Sep 12, 2012 at 5:03
• @R.M thank you. Belisarius, right... without excluding the vertical segments of the Tan function (using Exclusions or RegionFunction) MeshFunctions do not work.
– kglr
Commented Sep 12, 2012 at 5:43
• But that is not always easy > mathematica.stackexchange.com/q/10501/193 Commented Sep 12, 2012 at 13:28