Tricky use of MeshFunction

Question

MeshFunctions helps to get an approximation of the roots

Plot[Sin[x], {x, -5, 5}, MeshFunctions -> {#2 &}, Mesh -> {{0}}, MeshStyle -> {PointSize[Large]}]

of the plotfunction, wish is accessed as a pure function #2&

How can I detect points of a given slope of (#2&) in the same way? Thanks!

Answer 1

If you're only plotting one function at a time you could use the following:

f = Sin[x];

Plot[f, {x, -5, 5}, MeshFunctions -> {Function[x, Evaluate[D[f, x]]]},
  Mesh -> {{-0.5}}, MeshStyle -> {PointSize[Large]}]

The mesh function uses Evaluate since we need to make sure the derivative is taken with a symbolic value of $x$, before it starts receiving numeric values.

Answer 2

Here is an approach that follows @Szabolcs' suggestion in comments, i.e of plotting the derivative of the function to extract the values, then reporting those values back to a plot of the original function:

Clear[f]
f[x_?NumericQ] := Sin[x]

pts = Cases[
  Normal@
   Plot[
     D[f[t], t] /. t -> x, {x, -5, 5},
     MeshFunctions -> {#2 &}, Mesh -> {{-0.5}}
   ], p_Point :> First[p], All];

tangents = Function[{x}, f[#1] + #2 (x - #1) & @@@ pts];

Plot[
  {f[x], Style[tangents[x], GrayLevel[0.5], Dashed]},
  {x, -5, 5}, PlotRange -> 1.2,
  Epilog -> {
    PointSize[Large], Red,
    Point[pts /. {x_Real, y_Real} :> {x, f[x]}]
  },
  ImageSize -> Large
]

I have defined the function in question as something that only evaluates when given numeric input to show that this should work with reasonable functions more complex than the simple Sin[x] as well.

---

An alternative way to find those points is to use the NDSolve machinery and WhenEvent:

Reap[
  NDSolve[
    {D[y[x], x] == f'[x], y[0] == f[0], WhenEvent[y'[x] == -0.5, Sow[x]]}, 
    y, {x, -5, 5}
  ]
][[2, 1]]

(* Out: {-2.0944, -4.18879, 2.0944, 4.18879} *)


Plot[
  f[x], {x, -5, 5},
  Epilog -> {
   PointSize[Large], Red,
   Point[{#1, f[#1]} & /@ ptsNDSolve]
  }
]