(Solved): A formula for the curvature of a parametrized plane curve a. Show that the curvature of a smooth cu ...

A formula for the curvature of a parametrized plane curve a. Show that the curvature of a smooth curve $\mathbf{r}(t)=f(t)\mathbf{i}+$ $g(t)\mathbf{j}$ defined by twice-differentiable functions $x=f(t)$ and $y=g(t)$ is given by the formula $?=\frac{?\ddot{x}\ddot{y}?\ddot{y}\ddot{x}?}{{\left({\stackrel{?}{x}}^2+{\stackrel{?}{y}}^2\right)}^{3/2}}.$ The dots in the formula denote differentiation with respect to $t$, one derivative for each dot. Apply the formula to find the curvatures of the following curves. b. $\mathbf{r}(t)=t\mathbf{i}+(\ln \sin t)\mathbf{j},0<t<?$ c. $\mathbf{r}(t)=\left[{\tan }^{?1}(\sinh t)\right]\mathbf{i}+(\ln \cosh t)\mathbf{j}$