Write a script that implements the two-dimensional Newton method for the function f: R^2→ R^2 defined by f(x1+ x2)= (x1^2+ x2^2 -1, cos(πx1)). You may compute the explicit expression of the Jacobian matrix and its inverse to implement it directly. Store your solution in a column vector named . Store the final values of f(x) in a column vector named .

)a grad student comes up with the following algorithm to sort an array a[1..n] that works by first sorting the first 2/3rds of the array, then sorting the last 2/3rds of the (resulting) array, and finally sorting the first 2/3rds of the new array. 1: function g-sort(a, n) . takes as input an array of n numbers, a[1..n] 2: g-sort-recurse(a, 1, n) 3: end function 4: function g-sort-recurse(a, `, u) 5: if u ⒠` ≤ 0 then 6: return . 1 or fewer elements already sorted 7: else if u ⒠` = 1 then . 2 elements 8: if a[u] < a[`] then . swap values 9: temp ↠a[u] 10: a[u] ↠a[`] 11: a[`] ↠temp 12: end if 13: else . 3 or more elements 14: size ↠u ⒠` + 1 15: twothirds ↠d(2 ◠size)/3e 16: g-sort-recurse(a, `, ` + twothirds ⒠1) 17: g-sort-recurse(a, u ⒠twothirds + 1, u) 18: g-sort-recurse(a, `, ` + twothirds ⒠1) 19: end if 20: end function first (5 pts), prove that the algorithm correctly sorts the numbers in the array (in increasing order). after showing that it correctly sorts 1 and 2 element intervals, you may make the (incorrect) assumption that the number of elements being passed to g-sort-recurse is always a multiple of 3 to simplify the notation (and drop the floors/ceilings).

The “rule of 72” is used to approximate the time required for prices to double due to inflation. if the inflation rate is r%, then the rule of 72 estimates that prices will double in 72/r years. for instance, at an inflation rate of 6%, prices double in about 72/6 or 12 years. write a program to test the accuracy of this rule. for each interest rate from 1% to 20%, the program should display the rounded value of 72/r and the actual number of years required for prices to double at an r% inflation rate. (assume prices increase at the end of each year.)

Which statistical function in a spreadsheet you to see how far each number varies, on average, from the average value of the list?