Manipulate not accepting default values

Question

I have pasted my Mathematica code below. For some reason, the default value for h, or Δx, while set to 4.5 in the code, keeps displaying at -4 instead. I have been over and over this example, and I don't understand why this is happening. If someone would be willing to help me to understand what is going on here and how to fix it, I would really appreciate it.

Manipulate[
  f[x_] = 2.195^x - 1;
  Grid[{
    {Row[{Style[
       Text["  approximation of the derivative f'(x) :  " <> 
         ToString[(f[a + h] - f[a])/h]], Green, 15]}]},
    {Row[{Style[
       Text["  derivative: f'(x) \[TildeTilde] " <> 
         ToString[D[f[x], x] /. x -> a]], Red, 15]}]},
    {Plot[{f[x], 
        f[a] + (D[f[t], t] /. t -> a)*(x - a), 
        f[a] + ((f[a + h] - f[a])/h)*(x - a)}, 
     {x, -0.5, 6.5}, 
     PlotStyle -> {Blue, Red, Green},
     PlotRange -> {{-0.5, 6.25}, {-50, 130}}, Axes -> True, 
     AxesOrigin -> {0, -50}, 
     AxesLabel -> {"time (sec)", "distance (ft)"},
     ImageSize -> 450,
     Prolog -> {
       {Line[{{a + h, f[a + h]}, {a, f[a + h]}}]},
       {Line[{{a, f[a + h]}, {a, f[a]}}]}, 
       {PointSize[.02], Blue, Point[{a, f[a]}]},
       {Blue, Text["(x,f(x))", {a + .3, f[a] - 5}]}, 
       {Text["Δx", {a + h/2, f[a + h] - 5}]}, 
       {Text["f(x+Δx) - f(x)", {a + .6, f[a] - (f[a] - f[a + h])/2}]}, 
       {PointSize[.02], Green, Point[{a + h, f[a + h]}]}, 
       {Green, Text["(x+Δx,f(x+Δx))", {a + h - .4, f[a + h] + 7}]}}]}}], 
  {{a, 1, "x"}, 0, 6},
  {{h, 4.5, "Δx"}, -a, 6 - a},
  TrackedSymbols -> {a, a + h},
  SaveDefinitions -> True]

Answer 1

I think the problem is with SaveDefinitions->True. Change this to SaveDefinitions->False, save, and then run... it will probably be OK. Then add SaveDefinitions->True back at the end. This should always be the very last thing you do. (In other words, the problem is that an old definition is conflicting with the new one. Enabling SaveDefinitions at the end will fix this)

Also, you should probably move the function definition outside the manipulate (and add the colon to use SetDelayed instead of Set):

 f[x_] := 2.195^x - 1;
 Manipulate[ ... ]