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Limits pre- & post-Solve[]
*To*: mathgroup at smc.vnet.net
*Subject*: [mg14486] Limits pre- & post-Solve[]
*From*: "Richard W. Klopp" <rwklopp at unix.sri.com>
*Date*: Fri, 23 Oct 1998 20:59:06 -0400
*Organization*: SRI International
*Sender*: owner-wri-mathgroup at wolfram.com
I've run into difficulties while solving a practical problem. I can
illustrate with a simpler example. Consider two linear functions
f[x]:=m x + b, and g[x]:= n x + c. The question that I want to answer
is how far from the origin do I have to integrate each function so that
their definite integrals are equal. I ask for the answer as left and
right offset h from some target ordinate t, as shown below. Visualize
it as equating the areas under two straight lines bounded on the left
by the origin.
In[154]:=
Solve[Integrate[m x + b,{x,0,t+h}]==Integrate[n x + c,{x,0,t-h}],h]
Out[154]=
{{h -> (-b - c - m*t - n*t -
Sqrt[b^2 + 2*c*b + 4*n*t*b + c^2 + 4*m*n*t^2 + 4*c*m*t])/
(m - n)}, {h ->
(-b - c - m*t - n*t + Sqrt[b^2 + 2*c*b + 4*n*t*b + c^2 +
4*m*n*t^2 + 4*c*m*t])/(m - n)}}
I think that you can anticipate the problem if the slopes of the lines,
m and n are the same:
In[155]:=
Solve[Integrate[m x + b,{x,0,t+h}]==Integrate[n x +
c,{x,0,t-h}],h]/.n->m
\!\(Power::"infy" \( : \ \) "Infinite expression \!\(1\/0\)
encountered."\)
\!\(Power::"infy" \( : \ \) "Infinite expression \!\(1\/0\)
encountered."\)
Out[155]=
{{h -> ComplexInfinity}, {h -> ComplexInfinity}}
Taking the limit as one slope approaches the other m->n doesn't help.
In[157]:=
Limit[h/.Solve[Integrate[m x + b,{x,0,t+h}]==Integrate[n x +
c,{x,0,t-h}],h],
n->m]
Out[157]=
{DirectedInfinity[b + c + 2*m*t +
Sqrt[b^2 + 2*c*b + 4*m*t*b + c^2 + 4*m^2*t^2 + 4*c*m*t]],
DirectedInfinity[b + c + 2*m*t -
Sqrt[b^2 + 2*c*b + 4*m*t*b + c^2 + 4*m^2*t^2 + 4*c*m*t]]}
However, if I makes the slopes equal before I Solve[], everything's
fine.
In[158]:=
h/.Solve[((Integrate[m x + b,{x,0,t+h}]-Integrate[n x +
c,{x,0,t-h}])/.n->m)==
0,h]
Out[158]=
{((-b + c)*t)/(b + c + 2*m*t)}
What's going on and how do I get In[154] to behave as I desire, that is,
come out with something like Out[158]?
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