Exercise 12

Consider the single variable function defined by $y = 4x^2 - x^3.$

1. Find a parameterization of the form $\vr(t) = \langle x(t), y(t) \rangle$ that traces the curve $y = 4x^2 - x^3$ on the interval from $x = -3$ to $x = 3$.

2. Write a definite integral which, if evaluated, gives the exact length of the given curve from $x = -3$ to $x = 3$. Why is the integral difficult to evaluate exactly?

3. Determine the curvature, $\kappa(t)$, of the parameterized curve. (Exercise might be useful here.)

4. Use appropriate technology to approximate the absolute maximum and minimum of $\kappa(t)$ on the parameter interval for your parameterization. Compare your results with the graph of $y = 4x^2 - x^3$. How do the absolute maximum and absolute minimum of $\kappa(t)$ align with the original curve?