In this assignment we look again at how we can approximate the derivative of a discretisation of the sine function on the interval $(0,2\pi)$ via a difference formula. We divide the interval into 25 equidistant points. This assignment is specified in the Python language, but other programming languages like Matlab and R provide similar built-in functions. So, in other programming language you can also do this assignment.

Use the numpy and matplotlib package to calculate the derivatives in the points on the interval via the central 5-point difference formula and to make a scatterplot of the function and its derivatives in one diagram.

1. Do it via cross-correlation the first time (and use the 'same' option to get an array of the same length as the number of points.
   
   
2. Do it the second time using the numpy function roll which allows you to rotate an array.
   
   
3. Repeat part (1) with $101$ equidistant points, with $1001$ equidistant points, and with $1\,000\,001$ equidistant points. Do the approximations improve as you take more points on the interval?