Summer Semester, 2011

(4 points)

Plot the function

h(x)=(2^x) cos(x)

on the interval [-π,π], such that the plotted y-range is [-1,10], and the x-axis is twice longer than the y-axis. Color the graph by the value of the coordinate "y".

(6 points)

Give a definition of the following function:

f(x)={({{sin(x)                               for x≤0}, {f(x-1) -f(x-2) otherwise}}).

Plot it on the interval [-π,5].

(5 points)

Find the values of k, for which the matrix

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

is singular.

Solve[Det[{{1,0,1},{1,k,0},{1,0,k}}]==0,k]

(20 points)

Create a list of the points of the curve

c(t)={cos  t, sin(t), sin 5 t}

at uniform distance of parameter c(0), c(d),c(2d) ... Plot the points such that the size and color of them change according to the parameter value. (5 points)

Apply the transformation

{x,y,z}->0.5{(y-z),((y^2)-x),((x^2)-y)}

to each point. Iterate the transformation on the set of points (from 0 to 6)! Plot the iterated sets. (15 points)

c[t_]={Cos[t],Sin[t],Sin [5 t]}

rul={x_,y_,z_}->0.5{y-z,(y^2)-x,(x^2)-y}

ParametricPlot3D[{c[t],c[t]/.rul,c[t]/.rul/.rul},{t,0,2Pi},ColorFunction->(Hue[#4]&)]

(15 points)

Consider the following function:

f(x_,y_):=(((y^4)+2 (x^3)-y-0.3 y+(x^2)-(Sqrt[(y^4)+(x^4)]))/(e^((x^4)+(y^2)-1)))

Investigate the function graphically. Find the extremal and saddle points. Use graphical study to find good initial values.

Plot3D[(((y^4)+2 x^3-y-0.3 y+(x^2)-(Sqrt[(y^4)+(x^4)]))/(E^((x^4)+(y^2)-1))),{x,-1,1},{y,-2,2},PlotRange->All]

ContourPlot[(((y^4)+2 x^3-y-0.3 y+(x^2)-(Sqrt[(y^4)+(x^4)]))/(E^((x^4)+(y^2)-1))),{x,-.5,1},{y,1.7,2.},Contours->60,PlotRange->All]

(15 points)

Consider the nonlinear system of species competing for the same territory:

(Subscript[p, 1])' =(Subscript[p, 1])(1-(Subscript[p, 1])-(Subscript[p, 2]))-0.1(Subscript[p, 1]) - 0.7 (Subscript[p, 1])(Subscript[p, 2]),
(Subscript[p, 2])'= 0.5(Subscript[p, 2])(1-(Subscript[p, 2]))-0.1 (Subscript[p, 2])- (Subscript[p, 1])(Subscript[p, 2]), 
where 0≤(Subscript[p, 1]), (Subscript[p, 2]) and (Subscript[p, 1])+(Subscript[p, 2])≤1.

Find the equilibria. Solve the system for t ϵ [0,15] with several initial values  x(0), y(0)  on the line x(0)+ y(0)=0.9. Plot the trajectories of the solutions (colored by the time), the equilibria, and the vector field.