R3 Tangent Line at a Point (1,1,1) for x=t^4, y=t^4, z=t^3

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SUMMARY

The discussion focuses on finding the parametric equations for the tangent line to the curve defined by x=t^4, y=t^4, z=t^3 at the point (1,1,1). The key challenge is determining the value of t that corresponds to this point. It is established that t=1 satisfies the equations x=1, y=1, and z=1, as substituting t=1 yields the correct coordinates. Understanding this substitution is crucial for solving similar problems involving parametric equations.

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  • Understanding of parametric equations
  • Knowledge of derivatives and tangent lines
  • Familiarity with the concept of curves in three-dimensional space
  • Basic algebra for solving equations
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  • Study how to derive parametric equations from given functions
  • Learn about calculating derivatives of parametric equations
  • Explore the concept of tangent lines in three-dimensional geometry
  • Practice solving similar problems involving parametric curves
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Students studying calculus, particularly those focusing on parametric equations and tangent lines, as well as educators looking for examples to illustrate these concepts.

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Homework Statement



Find parametric equations for the tangent line to the curve x= t^4, y= t^4, z=t^3 at the point (1,1,1)

Homework Equations





The Attempt at a Solution



I understand everything about solving this problem with the exception of how to find what t =? to plug in. ie: This equation it equals 1, other problems I see it equals 0 or 2Pi, yet not a single place in 3 books do I see any mention as to how to determine what t is.

What magic are they doing to figure this out?

Thanks,
 
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Why don't you understand why t=1 in this problem? Now you are confusing me.
 
x= t^4= 1, y= t^4= 1, and z= t^3= 1. Is there one value of t that satisfies those equations?
 

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