Finding the Closest Point on a Parabola: How Can We Minimize Distance?

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Discussion Overview

The discussion revolves around finding the point on the parabola defined by the equation y = x^2 that is closest to the point (1,0). Participants are exploring methods to minimize the distance between a point on the parabola and the given point, focusing on the use of derivatives and the distance formula.

Discussion Character

  • Exploratory, Mathematical reasoning, Homework-related

Main Points Raised

  • One participant describes using the distance formula and taking the derivative to find the closest point, but encounters a complex fourth power equation.
  • Another participant suggests deriving the solution in the thread.
  • A different participant emphasizes the importance of minimizing the distance squared instead of the distance itself as a hint for simplification.
  • One participant shares their current progress, detailing the distance formula they derived and the steps taken to find the derivative, but expresses difficulty in solving the resulting equations.
  • A later reply recommends taking the derivative of the squared distance instead, as previously suggested by another participant.

Areas of Agreement / Disagreement

Participants appear to agree on the approach of minimizing the distance, but there is no consensus on the specific method or solution, as some participants face challenges in their calculations and interpretations.

Contextual Notes

Participants have not resolved the complexities of the equations involved, and there are indications of missing assumptions or steps in the mathematical reasoning presented.

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Find the point P on the parabola y = x^2 closest to the point (1,0).

To solve for P...

I used the distance formula, and then took the derivate. Using points (x, x^2) and (1,0)...

however, while taking the derivative I get a really nasty 4th power equation, which I am not sure I can solve without a calculator.
 
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Can you derive your solution here?
 
I don't know how you did it, but the big hint here is of course to minimize the distance *squared*, rather than the distance.
 
ok this is what I have so far...

D = sqrt. ((x-2)^2 + (x^2)^2)

then I took the derivative...dD/dx = (1/2)((x-1)^2 + x^4)^-1/2)*(2(x-1) + 4x^3)

this simplifies to..

dD/dx = ((x-1) +2x^3)/sqrt.((x-1)^2 + x^4)

setting that equal to zero...

this is where I encounter issues..

2x^3 + x - 1 =0

and

sqrt. ((x-1)^2 + x^4) = 0
 
As Dead Wolfe suggested, try taking the derivative of D2 instead.
 

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