Interesting problem i came acrosss

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Discussion Overview

The discussion revolves around finding the angle between two exponential curves, specifically y=a^x and y=b^x, at their point of intersection, given that a and b are not equal. Participants explore methods for determining the point of intersection and the implications of the curves' behavior at that point.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant describes a method for finding the angle between curves using the tangent of their slopes at the point of intersection, referencing a different example with polynomial curves.
  • Another participant clarifies that the angle of interest is actually between the tangents of the curves at the intersection point.
  • A participant derives the condition for intersection by equating the two curves, leading to the conclusion that x must be zero for a and b to yield the same y value.
  • Some participants express uncertainty about the problem's validity and the implications of the curves intersecting at x=0.
  • One participant acknowledges their initial confusion and expresses gratitude for the clarification provided by others.

Areas of Agreement / Disagreement

Participants generally agree that the curves intersect at x=0, but there is some uncertainty regarding the implications of this intersection and the validity of the problem as initially posed.

Contextual Notes

The discussion does not resolve the broader implications of the intersection point or the nature of the angle between the curves beyond the specific case of x=0.

oneomega
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okay,
there are two curves: curve1: y=a^x and curve2: b^x. what is the angle between them at their point of intersection. given that (a≠b).
My general method:
if i wanted angle b/w the curves y=x^2 and y=(x-2)^2, i would easily do it.
let A be angle b/w them,
then, tanA= [tan(m1)-tan(m2)]/1-tanm1.tanm2
can be used at their point of intersection.this is what my textbook says. but the problem with a^x and b^x is that I CANNOT FIND THE POINT OF INTERSECTION. which brings me to the very basic question, how can we find pt of intersection of curves like this?
 
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a small addendum, what i actually mean by angle b/w curves is the angle b/w their tangents at that point.
 
a^(x)=b^(x)-->(a/b)^x=1.
What does that tell you x must be, when "a" does not equal "b"?
 
you tell me, is there something wrong with this problem?
 
beacuse, at the same x point, a and b has to be equal to get you to the same y value as the equation suggests
 
oh wait, i got it, x mustt be zero, so that for diff values of a and b, they get you the same y value.
i got it now.
 
That's right!
:smile:
Both functions equal 1 when x=0
 
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i'm not this dumb usually, its just that i got stuck here. thanks a lot.
 

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