Homework 3

Due Date:  Friday, September 19

1. Use the formula \[ \vec{T}\hspace{0.08333em}'(t) \,=\, s'(t)\,\kappa_g(t)\,\vec{U}(t) \] to compute the curvature \(\kappa_g(t)\) of the tractrix \(\vec{x}(t) = (t - \tanh t,\operatorname{sech} t)\) for \(t>0\).
2. 1. Use equation 1.12 in the textbook to compute the curvature of the ellipse \[ \left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 \;=\; 1 \] at each of its four vertices.
   2. Find the equation of the osculating circle for this ellipse at the point \((a,0)\).
3. 1. Find the center \(\vec{C}(h)\) of the circle that intersects the catenary \(y=\cosh x\) at the points \((-h,\cosh h)\), \((0,1)\), and \((h,\cosh h)\).
   2. Use L'Hôpital's rule to compute \(\displaystyle\lim_{h\to 0}\, \vec{C}(h)\).
   3. Based on your answer to part (b), what is the curvature of the catenary \(y = \cosh x\) at the point \((0,1)\)? Explain.
4. Let \(\vec{x}(s)\) be a unit-speed curve of class \(C^2\) (where \(s\geq 0\)), and suppose that:
   • \(\vec{x}(0) = (1,0)\),
   • \(\vec{x}\hspace{0.08333em}'(0) = (1,0)\), and
   • The curvature of \(\vec{x}(s)\) is given by the formula \(\kappa_g(s) = (2s)^{-1/2}\).
   1. Find a formula for \(\theta(s)\).
   2. Find a formula for \(\vec{x}\hspace{0.08333em}'(s)\).
   3. Find a formula for \(\vec{x}(s)\).
   4. Find a parametrization for this curve that does not involve any square roots.