We study the vertical position of an engine cab after an obstacle has been driven over. One model for the motion is that the position at time t is given by y $($t $)=$ ec t cos $\backslash$omega t $,$ where c and $\backslash$omega are real constants. Suppose we have the following measurement data.

t$1=0\mathrm{.}2,$ y$1=0\mathrm{.}5,$ t$2=0\mathrm{.}4,$ and y$2=0\mathrm{.}05.$

i$.$ Set up the problem of determining the unknown constants c and $\backslash$omega as a system of the form f $($x$)=0,$ clearly state what the components of x correspond to

as well as calculate the Jacobian matrix of f $.[4$p$]$

ii$.$ Suppose we want to solve this system of nonlinear equations, f $($x$)=0,$ using Newton's method. Given a starting guess x$(0),$ what will be the next updated updated guess x $(1)?$ The Jacobian J to f is available.