Unit Tangent for a Curve: Finding T(t) using the Product Rule

In summary, a unit tangent for a curve is a vector with a magnitude of 1 that represents the direction of the curve at a specific point. It is calculated by finding the derivative of the curve at the point and dividing it by its magnitude. The unit tangent is significant because it provides information about the curve's direction and rate of change at that point. It can change at different points due to the changing slope of the curve. It is used in real-world applications, such as physics, engineering, and computer graphics, to analyze motion and rates of change.
  • #1
withthemotive
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Homework Statement



r(t) = 〈sin(t) − t cos(t), cos(t) +t sin(t), 5 t2 + 7 〉

Find the unit tangent. T(t)=


The Attempt at a Solution



r'(t) = <cos(t) + tsin(t), -sin(t) + tcos(t), 10t>

T(t) = r'(t)/ |r'(t)|

|r(t)| = sqrt( 1 + 101t^2)

And so on.
Supposably according to the homework I'm wrong, completely wrong.
 
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  • #2
Your derivative is wrong, completely wrong. You aren't using the product rule.
 

1. What is a unit tangent for a curve?

A unit tangent for a curve is a vector that is tangent to the curve at a specific point and has a magnitude of 1. It represents the direction of the curve at that point.

2. How is the unit tangent for a curve calculated?

The unit tangent for a curve can be calculated by finding the derivative of the curve at the specific point, and then dividing the derivative vector by its magnitude.

3. What is the significance of the unit tangent for a curve?

The unit tangent for a curve is significant because it provides information about the direction of the curve at a specific point. It can also be used to find the rate of change of the curve at that point.

4. Can the unit tangent for a curve change at different points?

Yes, the unit tangent for a curve can change at different points because the direction of the curve can vary at different points. This is due to the changing slope of the curve.

5. How is the unit tangent for a curve used in real-world applications?

The unit tangent for a curve is used in real-world applications, such as physics and engineering, to analyze the motion of objects or the rate of change in a system. It can also be used in computer graphics to create smooth and realistic curves.

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