Point on the curve closest to (18,1)

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SUMMARY

The discussion focuses on finding the point on the curve defined by the equation y = x² + 1 that is closest to the point (18, 1). Participants emphasize the importance of minimizing the squared distance rather than the distance itself, as this simplifies the problem. The first derivative must be calculated to determine the minimum point, and testing various values for x is recommended to find the solution. The conversation highlights the necessity of understanding derivatives and distance minimization techniques in calculus.

PREREQUISITES
  • Understanding of calculus concepts, specifically derivatives
  • Familiarity with distance formulas in a Cartesian coordinate system
  • Knowledge of minimizing functions and optimization techniques
  • Ability to work with quadratic equations
NEXT STEPS
  • Learn how to calculate derivatives and their applications in optimization
  • Study the concept of minimizing squared distances in calculus
  • Explore quadratic functions and their properties
  • Practice solving optimization problems involving distance in coordinate geometry
USEFUL FOR

Students studying calculus, mathematics educators, and anyone interested in optimization problems in geometry.

Faraz Ahmed
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Question:
Question: Find the point on the curve y=x^2 +1 that is closest to the point (18,1).

Please see the image and that’s where I’m stucked- after taking the first derivate. Please solve it further step by step completely. It’d mean a lot.
 

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We won't do your homework for you here - it is your homework.

Which value does the derivative have to reach at the minimum?

Afterwards you'll have to try different values for u to find the solution (there is a formula but that is more complicated than testing different values).
 
Faraz Ahmed said:
Question:
Question: Find the point on the curve y=x^2 +1 that is closest to the point (18,1).

Please see the image and that’s where I’m stucked- after taking the first derivate. Please solve it further step by step completely. It’d mean a lot.

It is against PF rules for us to do complete solutions; we are allowed to offer hints, but not more.

Here is a hint: minimizing the squared-distance will give the same solution as minimizing the distance itself----can you see why? --- and the squared-distance problem is easier.
 

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