How Do I Calculate the Tangent Line and Normal Line of a Path in Calculus?

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SUMMARY

The discussion focuses on calculating the tangent and normal lines of paths defined by parametric equations in calculus. Specifically, it addresses the paths s(t) = (t+1, e^t) and s(t) = (2t, t^2, ln(t)). To find the tangent line, participants confirm that differentiating each component of the vector with respect to the parameter t yields the tangent vector. For the normal line, a second derivative is required to determine the normal vector at the specified points.

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  • Understanding of parametric equations in calculus
  • Knowledge of derivatives and their applications
  • Familiarity with vector calculus concepts
  • Basic understanding of exponential and logarithmic functions
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  • Study the process of finding derivatives of parametric equations
  • Learn how to calculate normal vectors from tangent vectors
  • Explore applications of tangent and normal lines in physics
  • Investigate the role of velocity vectors in motion along a path
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Students studying calculus, particularly those focusing on vector calculus and parametric equations, as well as educators teaching these concepts in a mathematical or physical context.

Feynmanfan
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Hello everybody!

I'm having trouble with this calculus problem, where I don't know if I can apply what I've learned in MECHANICS.

"given a path s(t)=(t+1,E^t) calculate it's tangent line and the normal line at
this point s(0)"

and this is another version of the problem in R3

"given a path s(t)=(2t,t^2,Lnt) calculate the velocity vector and the tangent line at (2,1,0)"

How do I solve this? Is it just the derivative and that's all?
 
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Just differentiate each component of the vectors with respect to the parameter to find the vector's derivative. For the first one, you have to differentiate twice to find the normal vector.
 
Feynmanfan said:
"given a path s(t)=(t+1,E^t) calculate it's tangent line and the normal line at
this point s(0)"
Just find the tangent and normal vectors and you can find the corresponding lines from there. Remember the relationship between tangents and derivatives?
"given a path s(t)=(2t,t^2,Lnt) calculate the velocity vector and the tangent line at (2,1,0)"

How do I solve this? Is it just the derivative and that's all?
Since s(t) is some position function, how would you find the velocity? What does the velocity have to do with the tangent? It should all be clear once you answer these questions.

e(ho0n3
 

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