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

```

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