Find an equation of a parabola that has curvature 6 at the origin.


To find the equation of a parabola with a curvature of 6 at the origin, we proceed step by step.

1. **Equation of a Parabola**:  
   Start with the general form of a parabola:
   \[
   y = ax^2
   \]
   This is a simple parabola opening upwards with its vertex at the origin \((0, 0)\).

2. **Curvature Formula**:  
   The curvature \( \kappa \) of a function \( y = f(x) \) is given by the formula:
   \[
   \kappa(x) = \frac{|f''(x)|}{\left(1 + (f'(x))^2\right)^{3/2}}
   \]
   For a parabola \( y = ax^2 \), we first need to compute the first and second derivatives.

3. **First and Second Derivatives**:
   - First derivative: \( f'(x) = \frac{d}{dx}(ax^2) = 2ax \)
   - Second derivative: \( f''(x) = \frac{d^2}{dx^2}(ax^2) = 2a \)

4. **Curvature at the Origin**:  
   At the origin \( x = 0 \), the curvature formula simplifies. Since \( f'(0) = 0 \), the denominator becomes 1, and the curvature at the origin is:
   \[
   \kappa(0) = |f''(0)| = |2a|
   \]

   We are given that the curvature at the origin is 6, so:
   \[
   |2a| = 6
   \]
   Solving for \( a \):
   \[
   2a = 6 \quad \text{or} \quad 2a = -6
   \]
   \[
   a = 3 \quad \text{or} \quad a = -3
   \]

5. **Equation of the Parabola**:  
   Therefore, the equation of the parabola is either:
   \[
   y = 3x^2 \quad \text{or} \quad y = -3x^2
   \]
   These are the two possible equations of parabolas with curvature 6 at the origin, one opening upwards and the other downwards.

Would you like more details or further clarifications on this solution?

Here are 5 related questions to explore this topic further:
1. How do you compute the curvature of a general curve?
2. What is the significance of curvature in geometry?
3. How would the curvature change if we modify the value of \( a \) in the parabola equation?
4. Can curvature be negative? If so, what does that represent?
5. How would you find the curvature at a point other than the origin for a general curve?

**Tip**: The curvature gives an idea of how "bent" or "sharp" a curve is at a specific point. The higher the curvature, the sharper the bend.

Find an equation of a parabola that has curvature 6 at the origin.

Learn how to find the equation of a parabola with a curvature of 6 at the origin. This step-by-step solution covers key calculus concepts like curvature, derivatives, and parabolas. Suitable for advanced high school and pre-college students.

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