Summer Semester, 2012

Version A

(5 points)

Plot the function

h(x)=(((x^2)+log(x+1)sin(2x) )/(2^x))

on the interval [0,π]  such that the y-range be [-2,2], and the lengths of the x-axis and y-axis are equal. Color the graph according to the values of  Max(x,y).

(5 points)

Find the values of k, at which the eigenvalues of the matrix

A=(({{1, 0, 1}, {1, k, k}, {1, 0, k}}))

are real! 
Calculate the determinant of A at these cases.

(5 points)

Prepare a substitution rule that recognizes a 3 x 2 (3 rows, 2 columns) and replaces it by the crossproduct of the columns. Give an example!

(20 points)

Consider the following functions:

f(x_):=(((1.2 (x^2) (2^((x^2)-x))) x)/(((x^4)+1) ((3^((x^2)-x))+1)));        g(x_):=(((x^2) x (1.4^(-(x^2))))/((x^4)-x+1))

Investigate the behavior of the functions graphically (5 pts).

Find the extremal points of f(x) (5 pts).

Find the solutions of the equation f(x)=g(x), and draw the tangent lines of g(x) at these places (10 pts).

(15 points)

Consider the nonlinear system that is a model of a disease (SIRS model, s: susceptible; i: infected; r: recovered)
 s' =-0.5s i+0.1 r,
i'=0.5s i -0.2i
r'=0.2 i-0.1r
where 0≤s,i,r, and s+i+r = 1
Find the equilibria. Plot the vector field in the space (s,i,r). (5 pts)
Find the numeric solution if s(0)=0.95, i(0)=0.5, r(0)=0. Plot the solutions s(t), i(t) and r(t), and the trajectory in 3D (5 pts)
Find the tangent vector (s’,i’,r’) at t=0. (5 pts)

(20 points)

Consider the following difference equation (van der Pol map):

{(Subscript[x, n+1]),(Subscript[y, n+1])}:={0.1` (10.` (Subscript[x, n])+ (Subscript[y, n])),0.1` (-Sin[(Subscript[x, n])]+10 (Subscript[y, n])+1 (p- (Subscript[x, n]^2)) (Subscript[y, n]))}

Iterate it from an appropriate initial point with given 1.5>p>0 (by your choice). Plot the iterated points and the trajectory  (the points joined by a line). (10 pts)
Use coloring of the points by the time.
Change p and the initial point interactively in Manipulate. (10 pts)

Version B

(5 points)

Plot the function

h(x)=((x log((x^2))sin(x) )/((x^2)+1))

on the interval [-π,π]  such that the y-range be [-2,2], and the lengths of the x-axis and y-axis are equal. Color the graph according to the values of  |x| +|y|.

(5 points)

Find the values of k, at which the vectors

A=(({{1, 2, 2}})) , 
B=(({{1, 1, k}})) ,
C=(({{1, k, 1}})) ,

are linear dependent? What is the dimension of the space formed by the vectors in this case?

(5 points)

Prepare a substitution rule that recognizes a 3 x 3 (3 row, 3 column) and replaces it by the weight-point of the column vectors. Give an example.

(20 points)

Consider the following function:

f(x_,y_):=((-(Sqrt[(x^2)+(y^2)])+(x^2)+0.1 x+2 (y^3)+(y^2)-0.7 y)/(e^((x^6)+(y^4))))

Investigate the function graphically.  Use appropriate coloring and meshing. Plot the zero contours of the partial derivatives. (5 points).
Find the extremal and saddle points. Use graphical study to find good initial values. (5 points)
Draw the gradient vector at each extremal points (10 points)

(15 points)

Consider the nonlinear system that is a model of drug elimination:
 (Subscript[p, 1])' =-2 (Subscript[p, 1]),
(Subscript[p, 2])'=2(Subscript[p, 1])-(Subscript[p, 2]), 
where 0≤(Subscript[p, 1]), (Subscript[p, 2])
Find the equilibria. Plot the vector field in the space ((Subscript[p, 1]), (Subscript[p, 2])). (5 pts)
Find the symbolic solution if (Subscript[p, 1])(0)=p0>0, (Subscript[p, 2])(0)=0. Plot the solution and find its maximum. (5 pts)
Give its tangent line at t=0 (5 pts)

(20 points)

Consider the following difference equation (Mandelbrot's map):

{(Subscript[x, n+1]),(Subscript[y, n+1])}:={p+(Subscript[x, n]^2)-(Subscript[y, n]^2), q+2(Subscript[x, n]) (Subscript[y, n])} 

Iterate it from an appropriate initial point with given p,q (by your choice). Plot the iterated points and the trajectory  (the points joined by aline). (10 pts)
Use coloring by the time.
Change p,q interactively in Manipulate. (10 pts)