Re: DSolving(?) for a given tangent

*To*: mathgroup at smc.vnet.net*Subject*: [mg81390] Re: DSolving(?) for a given tangent*From*: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>*Date*: Fri, 21 Sep 2007 03:13:28 -0400 (EDT)*Organization*: The Open University, Milton Keynes, UK*References*: <fct9c4$r6e$1@smc.vnet.net>

AngleWyrm wrote: > Don't know for sure if this is the right function, so here's the scenario: > > f[x_] := E^(0.22 x); > Plot[f[x], {x, 6, 36}] > > Which plots a nice escalating curve. > > What I would like to know is: Where is the point {x,f[x]} that has a > 45-degree tangent line; ie where is this curve's balance point before it > really starts taking off? So what you are looking for is the value of x for which f'[x] == Pi/4 (i.e. the slope of the tangent at x is 45 degrees but it must be expressed in radians rather than in degrees). The solution can be found by solving the equation f'[x] == Pi/4 for x; to do so one can use Solve or Reduce for an analytic solution (which implies exact coefficients such as 22/100 rather than 0.22) or NSolve or FindRoot for an numerical solution. For instance, In[1]:= f[x_] := E^(0.22 x); Plot[f[x], {x, 6, 36}] sol = NSolve[f'[x] == Pi/4, x] x /. sol[[1]] Out[3]= {{x -> 5.78438}} Out[4]= 5.78438 HTH, -- Jean-Marc