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

[Feynmanfan]

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?

 

[Gza]

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.

 

[e(ho0n3]

"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