Tangent and normal lines

In summary, the person is asking for help with two calculus problems involving finding the tangent and normal lines for given paths. They are unsure if they just need to take the derivative or if there is more to it. The expert suggests differentiating the components of the vectors and finding the tangent and normal vectors to solve the problem. They also remind the person about the relationship between tangents and derivatives.
  • #1
Feynmanfan
129
0
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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  • #2
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.
 
  • #3
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
 

What is a tangent line?

A tangent line is a straight line that touches a curve at only one point. It is perpendicular to the radius of the curve at that point, and therefore has the same slope as the curve at that point.

How do you find the equation of a tangent line?

To find the equation of a tangent line, you need to know the point of tangency and the slope of the tangent line. The slope can be found by taking the derivative of the curve at the point of tangency. Then, you can use the point-slope formula to find the equation of the tangent line.

What is a normal line?

A normal line is a straight line that is perpendicular to the tangent line at the point of tangency. It is also perpendicular to the curve at that point, and therefore has a slope that is the negative reciprocal of the slope of the tangent line.

How are tangent and normal lines related?

Tangent and normal lines are related because they both intersect the curve at the point of tangency. The tangent line is perpendicular to the radius of the curve at that point, while the normal line is perpendicular to the tangent line at that point.

Why are tangent and normal lines important in calculus?

Tangent and normal lines are important in calculus because they allow us to find the instantaneous rate of change of a curve at a specific point. This is done by finding the slope of the tangent line, which is the same as the slope of the curve at that point. Normal lines also help us understand the behavior of a curve and its relationship to the tangent line.

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