Summer Semester, 2014

Version A

(10 points)

Define the following function (5 points)

f(x)=({{{, ({{sin(((x))^2), for 0≤x<(π/2)}, {(1/x), for≥(π/2)}, {-(x^2), x<0}})}})

(10 points)

Find all the zero points of the function: -0.8 x+1.023 (1.3^x)-sin(x)+0.472 sin(2.7 x).
What is the maximum of the derivative at these points?

(5+15 points)

The matrix

A=(({{1, 2, 1}, {0, -2, -2}, {2, 1, -1}}))

is the matrix of a linear transformation ((R^3)->(R^3)). 
Show that the transformation is singular. What is the image (the subspace of vectors A.{x,y,z} and the null-space (it is known that theay are either planes of lines passing through the origin) (5 point)

Visualize both the image-space and the null-space (vectors, planes, normal vector of the planes, what are necessary; image of unit vectors, etc...(10 point)

(15 points)

Create a function or (rule)  that recognizes any square matrix and calculates the maximum of the absolute values of the  elements at the co-diagonal. Give an example.

(25 points)

Write functions that calculates the so called Abel's sums (Subscript[R, n]) for a given series (∑_(i=0)^∞)(Subscript[a, i]). The n'th Abel's sum  is (Subscript[R, n])(r)=(∑_(i=0)^n)(r^i) (Subscript[a, i]) for r∈ [0,1]. Note that (Subscript[R, n])(1)=(Subscript[S, n])=(∑_(i=0)^n)(Subscript[a, i]). (10 points)

Use (Subscript[a, n])=1/(n+1) and visualize the approximation by animating (Subscript[R, n])(r)  (n=1,2, NN). (1 points)

(25 points)

Consider the following random motion. Let a particle move randomly from any point P only in either x-direction of y-direction (only one direction!) such that the maximum length of the step is equal to (1/(1+||P((||)^2))), where ||P|| is the distance of the point from the origin. It means that the maximal step size is smaller and smaller as the particle is getting far from the origin.
Write a function for the step of the motion for any space! Iterate the process. Animate the motion {(Subscript[P, 0]),(Subscript[P, 1]),...,(Subscript[P, n])} of the particle using different ways (points, path, coloring).

(20 + 5 points)

Consider the planar linear differential equation

(({{x'}, {y'}}))=(({{1, 1}, {-1, k}})).(({{x}, {y}}))

Find the eigenvalues of its matrix, the fundamental solutions (with initial conditions x(0)=1, y(0)=0 and x(0)=0, y(0)=1.  Visualize interactively (by changing the parameter k) its vectorfield (stream) and the trajectories of the fundamental solutions in the 2D phasespace. (20 points)

Give a criterion for the solutions to tend to zero as the time tends to infinity (5 points)

Version B

(10 points)

Define the following function (5 points)

f(x)=({{{, ({{(x^2), for 0<x<1}, {(1/x), for>1}, {sin(x), otherwise}})}})

(10 points)

Find the inflection points of the function

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

on the interval [0,6]. Find the point where growth rate of h(x) is the greatest.

(5+15 points)

The matrix

A=(({{1, 2, 0}, {2, -2, 1}, {2, 1, -1}}))

is the matrix of a linear transformation ((R^3)->(R^3)). 
Show that the transformation is nonsingular. What are the eigenvalues and eigenvectors (5 point)

Visualize the image of the unit sphere, the images of unit vectors, the eigenvectors. (10 point)

(15 points)

Give a substitution rule that replaces a positive number by its reciprocal. Apply it to the points of the graph of the function

f(x)=(x^3)-3(x^2)-x+3.

on the interval [-2,4], with the step-size 0.2. Plot the original and the transformed points.

(25 points)

Write functions that calculates the averaged partial sums for a given series (∑_(i=0)^∞)(Subscript[a, i]).  The partial sums are (Subscript[S, n])=(∑_(i=0)^n)(Subscript[a, i]). Calculate the modified sums  (Subscript[A, n])=(1/n)(∑_(i=0)^n)(n-i+1) (Subscript[a, i])=(1/n)(n (Subscript[a, 1])+(n-1)(Subscript[a, 2])+...+2(Subscript[a, n-1])+ (Subscript[a, n])). (10points)

Take  (Subscript[a, n])=1/(n+1), NN=20, and visualize the approximation by plotting (Subscript[S, n]) and (Subscript[A, n])  (n=1,2, NN) . (10 points)

(25 points)

Write programme for the following iteration:

(Subscript[X, n+1])=(Subscript[Y, n])+1+a (Subscript[X, n]^2)
(Subscript[Y, n+1])=b (Subscript[X, n]),

where (Subscript[X, 0])=(Subscript[Y, 0])=0, and a=1.4, b=0.3. Plot the result on the X-Y plane (10 points)

Prepare dynamic demonstration with Manipulate, in which the parameters and the initial values can be changed (15 points)

(20 + 5 points)

Consider the following nonlinear oscillation:

x''+0.2x' + x=Cos[k t]

Find the general solution and the particular solutions with initial conditions x(0)=1, y(0)=0 and x(0)=0, y(0)=1. Visualize interactively (by changing the parameter k) its vectorfield in the 3D extended phase space with the solution (t,x(t),y(t)), the trajectories (x(t),y(t)) in the 2D phasespace, as well as the coorinates of the particular solutions.  (20 points)

Give a criterion for the resonance, i.e. the Period of the own oscillations is equal to the forcing term.  (5 points)