Summer Semester, 2014

Version A

(10+5+10 pont)

Consider the functions -cos(x)  and -0.4(x^2)+3.9 (1.3^x)-0.175 cos(2.7 x). 
Plot the funtions on an appropriate interval such that the are colored by their first derivative. (10 points)
Find the points where their changing speed are equal (derivative).  (5 points)
Draw the points and the tangent lines to the graphs. (10 points)

(5+10 points)

The matrix 
A=(({{k, 1, 1}, {1, k, k}, {1, k, 0}})) 
is the matirx of a linear transformation  ((R^3)->(R^3)). At which values of parameters is the transformation nonsingular? What is the determinant of the matrix in these cases? (5 pont)

Draw the eigenvectors depending on the parameter (Manipulate) (10 points)

(10+15 points)

Consider the potential function of  a planar “tripole” :

V(X)=V(x,y):=((Subscript[Q, 1])/(‖X-(Subscript[P, 1])‖))+((Subscript[Q, 2])/(‖X-(Subscript[P, 2])‖))+((Subscript[Q, 3])/(‖X-(Subscript[P, 3])‖))

where X={x,y} is a general point in the plane; (Subscript[Q, i]) is the charge and  (Subscript[P, i])={(Subscript[x, i]),(Subscript[y, i])} of the i-th pole. Here ‖{u,v}‖=(Sqrt[(u^2)+(v^2)]), is the Euclidean norm.

(10 points)

Prepare a substitution rule that recognizes a Line object of any number of points, and replaces by the weight point of the vertices. Give an example!

(10+15 points)

Consider the Ricker’s system:

(Subscript[x, n+1]):=(Subscript[x, n])(e^(r(1-(Subscript[x, n])/K)))
where 
	K: Carrying capacity
	r: Growth rate of the population

Define the mapping. Do the iteration from appropriate given values, plot the iterated values, find its fixed points at given parameter values (10 points).
Do the above tasks by dynamically changing the parameters and the initial value (Manipulate). 15 points.

Version B

(10+5+10 points)

Consider the following function:

h(x)=x(2(e^(-(((x-1))^2)))+(e^(-(((x-3))^2))))  .

(5+10 pont)

Find the values of  k at which the vectors  
A=(({{k, k, 1}})) , 
B=(({{k, 0, k}})) ,
C=(({{1, k, 0}})) ,

are linear dependent.  What is the dimension of the space formed by them in such cases? (5 points)

Prepare a Manipulate, that shows the determinant of the matrix formed by the vectors, such that the parameter k changes dynamically.  (10 points)

(10+5+10 pont)

Create a list of the points of the curve {cos  t, sin  t}/(t+1) at uniformly distributed t-values on the interval [0,2π]. Plot the points, such that the color and size of them change according to the distance from the origin. (10 points)

Plot the velocity and acceleration vectors as well as their cross products at the point belonging to the parameter value (Subscript[t, 0])=2. (5 points)

Prepare a Manipulate for these vectors such that (Subscript[t, 0]) changes dynamically. (10 points)

(10 points)

Prepare a substitution rule that recognizes a Polygon object of any points, and replaces by  its normal vector  drawn at the first vertex. Give an example!

(10+15 points)

Consider the delayed logistic system:

(Subscript[x, n+1]):=a (Subscript[x, n])(1-(Subscript[x, n-1]))where
a:Growth rate of the population

Define the mapping, to be iterated (2D iteration with   {(Subscript[x, n]),(Subscript[x, n+1])} and two initial values needed).
At given parameter values, do the iteration from appropriate given values, plot the iterated points in the 2D space, find the fixed points  (10 points).
Do the above tasks by dynamically changing the parameters and the initial values (Manipulate).15 points