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).

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).

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