Re: Point of intersection (brain dead example)
- From: David W. Cantrell <DWCantrell@xxxxxxxxxxx>
- Date: 07 Aug 2007 12:00:28 GMT
quasi <quasi@xxxxxxxx> wrote:
On Tue, 7 Aug 2007 11:25:05 +0200, "Konrad Viltersten"
<tmp1@xxxxxxxxxxxxxx> wrote:
quasi wrote/skrev/kaita/popisal/schreibt :
On Tue, 7 Aug 2007 10:44:34 +0200, "Konrad Viltersten"
<tmp1@xxxxxxxxxxxxxx> wrote:
Awkwardly enough, i'm trying to solve the following.
Find the points of intersection between exp(x) and 2x.
So, i do:
e^x = 2x
x = log(2) + log(x)
...and then i get stuck. Boy, do i feel stupid! How do we
find x there?!
Graph the functions with a graphing calculator or CAS.
You will discover something.
To make it more evident, let h(x) = e^x - 2x, and graph h.
To prove what the graph of h implies about the solutions, find the
minimum value of h (using Calculus).
Perhaps i wasn't clear. I want to compute it algebraically.
You can switch 2x for ax. That was just an example. I
don't care if the intersection(s) is/are real or complex (as
was the case with 2x)...
Graphically, it's clear that the equation e^x = a*x has
2 real solutions if a>e
one real solution x=1 if a=e
no real solutions if 0<=a<e
one real solution if a<0
As far as an algebraic solution, Maple gives the symbolic solution
x = -LambertW(-1/a)
which is real valued if a<0 or a>=e.
For a>e, the above solution only gives the smaller of the 2 real
solutions.
As Konrad will find if he reads about the Lambert W function, it's
multivalued. Maple's solution uses just the principal branch. But using
all branches, we get all the complex solutions. If you want only real
solutions, then you need consider only the principal branch and the -1
branch.
David
.
- References:
- Point of intersection (brain dead example)
- From: Konrad Viltersten
- Re: Point of intersection (brain dead example)
- From: quasi
- Re: Point of intersection (brain dead example)
- From: Konrad Viltersten
- Re: Point of intersection (brain dead example)
- From: quasi
- Point of intersection (brain dead example)
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