RECENT ASSIGNMENT

You can write the submission in word or using latex and export it as a file. You can take a screenshot of the codes you modify and enter in the file and everything should be a single file document.
Task 1
Explain what the program below performs line by line:

Task 2
Given system of differential equations:

Set up the system in matrix form (x´=Ax there x= ).
Sketch the direction vector for the solution curve at the point (-2, 1).
Determine eigenvalues and eigenvectors of A
Solve the system given x(0)=0,y(0)=10 (the solution curve passes the point
(0,10) at t = 0).
Solve the differential equation system:

Write briefly what the following program performs and adapt the program by writing appropriate syntax / mathematical expressions or numbers to sketch the solution curve in part c)

Given the 2nd order differential equation
mx’’+ ax’ + kx = cos x
there m = 1[kg],a = 0.4 [kg/s],k = 0.5 [kg/s^2 ],x(0) = 2,x’ (0) = 1.5.

Write about the given differential equation of a differential equation system. (Hint: Let ’= y). Then customize the code below to this system as must outline the solution curve of the system over the interval [0,6].
Task 3
Let f (x,y)= x y^2 be a function with the definition set the (x,y) so that
2x^2+y^2=6
Explain how we can know that f has the definition set of maximum point and minimum point. Explain why f does not have stationary points.
Determine the largest and smallest value of f using the Lagranges method.
What is the answer to parts a) and b) if the definition set is 2x^2 + y^2= 6?
Let h: R^2 ? R be a function given by h(x,y)=4-x^2-y^2
Find the level curve of f in the point P(1,-1). Determine ?h(1 ,-1).
Draw the point P, as well as the level curve and the gradient to h at this point.
Calculate how fast h changes at point P in the direction
In which direction from point P is this change greatest? (Hint: the direction can be given as a vector).