Part II
Summer Semester, 2004

(10 points)

Give a substitution rule that replaces a negative number by its reciprocal. Apply it to the the function

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

Plot the original and the transformed functions.

(15 points)

Consider the matrix  ((Subscript[x, i,k])),    i=1,...,n, k=1,...,m,. Create a general Mathematica function that calculates the value  (max_i)((∑_k)|(Subscript[x, i,k])|),  i.e., the maximum of the sum of the elements' absolute value in rows. Give an example.

(15 points)

Draw a hexagon with its all diagonal (see the enclosed figure).

(30 points)

Find (Sqrt[2]) iteratively using the iteration

(Subscript[x, n+1])=(1/(Subscript[x, n]))+((Subscript[x, n])/2).

Draw the cobweb diagram to illustrate the approximation.

(30 points)

Visualize the random motion on the plane. Let a particle move randomly on the square (0≤x,y≤1), i.e. the body can jump from any position to any point of the square. Let the points of the motion be (Subscript[P, 1]), (Subscript[P, 2]), ..., (Subscript[P, n])   (n ∈ N) in the consecutive seconds.  Visualize the motion  by animation:

show the points (Subscript[P, 1]), (Subscript[P, 2]), ..., (Subscript[P, n]) (use different colors),

draw the vectors ((Subscript[P, i]),(Subscript[P, i+1])) at every steps.

(10 points)

Give a definition of the following function:

f(x)={({{cos(x) for |x|≤(π/2) }, {0 otherwise}})

Plot it.

(15 points)

Consider a function f((Subscript[x, 1]),...,(Subscript[x, n])). The variables are given in the list var={x1,x2,...,xn}. The second derivative matrix is ((∂_((Subscript[x, i]),(Subscript[x, j])))f((Subscript[x, 1]),...,(Subscript[x, n]))). Create a general Mathematica function that calculates the trace (spur)  (∑_(i=1)^n)(∂_((Subscript[x, i]),(Subscript[x, i])))f. Give an example.

(15 points)

Take an octagon. Draw all the lines that connect the midpoints of the sides (see the figure).

(30 points)

Consider the difference equation

(Subscript[x, n+1])=(Subscript[x, n]) + (Subscript[x, n])(1-(Subscript[x, n])).

Plot its solution with the initial value (Subscript[x, 1])=0.5 for n=1,...,100 so that the coloring of the points be done by the subscript n.

(30 points)

Consider the graphics object (curve) created by the following stement:

p0=ParametricPlot[{Cos[3t],Sin[2t]},{t,0,2Pi}];

Consider the vector field

{y-x,-sin(x)-0.5y}.

Plot the elements of the given vector field at every explicitely available point of the object (Subscript[p, 0]).

(10 points)

Give a definition of the following function:

f(x)={({{cos(x) for |x|≤(π/2) }, {0 otherwise}})

Plot it.

(15 points)

Consider a function f((Subscript[x, 1]),...,(Subscript[x, n])). The variables are given in the list var={x1,x2,...,xn}. The second derivative matrix is ((∂_((Subscript[x, i]),(Subscript[x, j])))f((Subscript[x, 1]),...,(Subscript[x, n]))). Create a general Mathematica function that calculates the trace (spur)  (∑_(i=1)^n)(∂_((Subscript[x, i]),(Subscript[x, i])))f. Give an example.

(15 points)

Take an octagon. Draw all the lines that connect the midpoints of the sides (see the figure).

(30 points)

Consider the difference equation

(Subscript[x, n+1])=(Subscript[x, n]) + (Subscript[x, n])(1-(Subscript[x, n]))

Plot its solution with the initial value (Subscript[x, 1])=0.5 for n=1,...,100 so that the coloring of the points be done by the subscript n.

(30 points)

Consider the graphics object (curve) created by the following stement:

p0=ParametricPlot[{Cos[3t],Sin[2t]},{t,0,2Pi}];

Consider the vector field

{y-x,-sin(x)-0.5y}.

Plot the elements of the given vector field at every explicitely available points of the object (Subscript[p, 0]).

(10 points)

Give a substitution rule that replaces a negative number by its reciprocal. Apply it to the the function

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

Plot the original and the transformed functions.

(15 points)

Consider the matrix  ((Subscript[x, i,k])),    i=1,...,n, k=1,...,m,. Create a general Mathematica function that calculates the value (max_i)((∑_i)|(Subscript[x, i,k])|),  i.e., the maximum of the sum of the elements' absolute value in rows. Give an example.

(15 points)

Draw a hexagon with its all diagonal (see the enclosed figure).

(30 points)

Find (Sqrt[2]) iteratively using the iteration

(Subscript[x, n+1])=(a/(2 (Subscript[x, n])))+((Subscript[x, n])/2).

Draw the cobweb diagram to illustrate the approximation.

(30 points)

Visualize the random motion on the plane. Let a particle move randomly on the square (0≤x,y≤1), i.e. the body can jump from any position to any point of the square. Let the points of the motion be (Subscript[P, 1]), (Subscript[P, 2]), ..., (Subscript[P, n])   (n ∈ N) in the consecutive seconds. Visualize the motion  by animation:

show the points (Subscript[P, 1]), (Subscript[P, 2]), ..., (Subscript[P, n]) (use different colors),

draw the vectors ((Subscript[P, i]),(Subscript[P, i+1])) at every steps.