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explain The parameter  u takes the values 0, 0.125 and 0.25 at the angles 0, 45 and 90°

In this example, the parameter \( u \) represents a variable used to trace points along the circle. The angles (0°, 45°, and 90°) are mapped to different values of \( u \) because the parameterization of the circle involves using \( u \) to determine positions along the circumference.

1. **Circle Parametric Equation**: The parametric equations for a circle with radius \( R \) and centered at the origin are given by:
   \[
   x = R \cos(2 \pi u), \quad y = R \sin(2 \pi u), \quad 0 \leq u \leq 1
   \]
   This describes a complete rotation around the circle as \( u \) goes from 0 to 1.

2. **Mapping of Angles to Parameter \( u \)**:
   - At **0°**, which corresponds to the point \( (R, 0) \), \( u = 0 \).
   - At **45°**, the point lies on the first quadrant (i.e., at an angle halfway between 0° and 90°). This corresponds to \( u = 0.125 \), since:
     \[
     u = \frac{45°}{360°} = 0.125
     \]
   - At **90°**, which corresponds to the point \( (0, R) \), \( u = 0.25 \), since:
     \[
     u = \frac{90°}{360°} = 0.25
     \]

Therefore, the parameter \( u \) is a fraction of the full revolution of the circle, where \( u = 0 \) is 0°, \( u = 0.125 \) is 45°, and \( u = 0.25 \) is 90°.

The tangent vectors at these points are computed by differentiating the parametric equation with respect to \( u \), and then the slopes at these angles are calculated using the ratio \( \frac{y'}{x'} \), which gives:
- **0° (u = 0)**: Slope = \( \infty \) (vertical slope).
- **45° (u = 0.125)**: Slope = \( -1 \) (diagonal).
- **90° (u = 0.25)**: Slope = 0 (horizontal slope).

Let me know if you want further clarification or details on this, and here are some related questions:

1. How would the parametric equation change if the radius was different?
2. What happens to the tangent vector if \( u = 0.5 \) (180°)?
3. How does the slope change for angles greater than 90°?
4. Can we generalize the parametric equation for an ellipse?
5. How do the tangent vectors behave at other key points like \( u = 0.75 \) (270°)?

**Tip**: The parameter \( u \) typically represents the fraction of the full angle (out of 360°), so dividing an angle by 360° gives the corresponding \( u \)-value for parametric equations involving circles.

The parameter u takes the values 0, 0.125, and 0.25 at the angles 0, 45, and 90°. What does this mean?

Learn how to find the parametric equation of a circle and compute the tangent vectors and slopes at specific angles (0°, 45°, and 90°). This problem covers concepts in parametric geometry, derivatives, and circle properties.

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