Part I
Winter Semester, 2006

(Elements: 5 points)

Plot the function

h(x)=(e^(-(x^4)+2.3(x^3)-x))

on reasonable domains and nice parameters (plotpoints, plotrange, aspectratio...).

(Graphics options: 10 points)

Create a table of points of the graph of the function f(x)=sin((x^2)) on the interval [-2,2] at uniform distance on the x-axis. (5 points)

Plot these points with different color and pointsize depending on x. (5 points)

(Investigtion of functions: 15 points)

Consider the following functions:

f(x):=((2(x^4) (e^(x^2)))/(((x^8)+(x^4)+1) (1+(e^(x^2)))));
g(x):=(0.6/((x^2)+1))

Use plots to compare the local and global behavior of the functions f(x) and g(x). (5 points)

Find the places where f(x)=g(x). Use initial points as good as possible by using the plots from the previous part. (5 points)

Find the maxima and minima of f(x)  and calculate the values of g(x) at these points. (5 points)

(Linear Algebra: 5 points)

Calculate the eigenvalues and  eigenvectors of the matrix:

(({{1, 1, 1}, {1, 0, 0}, {1, 0, 2}}))

(Functions of two variables: 15 points)

Consider the following function:

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

Find the extremal points. Use graphical study to find good initial values.

(Differential equations: 15 points)

Consider the predator-prey system of differential equations

x'=2x-x y, 
y'=-y+ x y,     
x,y ≥0.

Find the equilibrium point. Solve the system for t ϵ [0,10] with initial values x(0)=1, y(0)=1. Plot the trajectory of the solution as well as the tangent and normal vectors of this trajectory at t=0,0.5,1,...10.