Part II
Summer Semester, 2005

(5 points)

Write program that replaces the exponent of an exponential expression by its square.

(15 points)

Give a definition of a logical function MyCondition that gives True for every vector {u,v}, where  u and v are positive real numbers (type Real) and False otherwise! (10 points)

Use this function in defining the substitution rule that  changes the coordinates of a given planar vector {x,y} between each other and the function MyCondition  gives True for the original vector. (5 points)

(20 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). (10 points)

(30 points)

Write a program for the gradient method to approximate a local minimum of a function F(x,y). It means that the direction of the steapest decrease of F(x,y) on the plane is the -grad F/||grad F||=.

Let h>0 and {(Subscript[x, 0]),(Subscript[y, 0])} be given. Then define {(Subscript[x, 1]),(Subscript[y, 1])}={(Subscript[x, 0]),(Subscript[y, 0])}-h grad F((Subscript[x, 0]),(Subscript[y, 0])). Iterate the process and then visualize it in the 2D and 3D spaces.

(30 points)

Write program for the following process. Take the points (Subscript[P, 1]),(Subscript[P, 2]),...,(Subscript[P, k]) in the 2D space. Rotate them left to (Subscript[R, 1])=(Subscript[P, 2]),(Subscript[R, 2])=(Subscript[P, 3]),...,(Subscript[R, k-1])=(Subscript[P, k]),(Subscript[R, k])=(Subscript[P, 1]), and then map the points (Subscript[R, i])  into the new points (Subscript[Q, 1]),(Subscript[Q, 2]),...,(Subscript[Q, k]), where (Subscript[Q, i]) is the weight point (average) of (Subscript[R, i]),(Subscript[R, i+1]),...,(Subscript[R, k]) ((Subscript[Q, k])=(Subscript[R, k]), (Subscript[Q, k-1])=((Subscript[R, k])+(Subscript[R, k-1]))/2, ... (Subscript[Q, 1])=(∑_(i=1)^k)(Subscript[R, i]). Iterate and visualize this process. Use the iteration steps for coloring.

(5 points)

Write program that replaces the numerator of a fraction by its Logarithm.

(15 points)

Give a definition of a logical function MyCondition that gives True for every integer number greater than 1 (remember type Integer) number and  False otherwise (rmember the Head types). (10 points)

Use this function in defining the substitution rule that divides the second coordinate of a given planar vector {x,y} by 2 and the function MyCondition gives True for the original  first cordinate. (5 points)

(20 points)

Write functions that calculates the averaged 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)

(30 points)

Write functions that visualizes the Taylor's series expansions, and apply it to a function f(x)=Sin[x] at (Subscript[x, 0])=0.í

Find the Taylor's polynomial (Subscript[T, n])(x)=(∑_(i=0)^n)(Subscript[a, i]) (((x-(Subscript[x, 0])))^i)  (Series command). (10 points)

Find the coefficients and animate the spectrum {(Subscript[a, 1]),(Subscript[a, 2]),...,(Subscript[a, n])} of the expansion (CoefficientList) such that the element (Subscript[a, i]) of the spectrum is interpeted by a vertical line of length  (Subscript[a, i]) at the x-coordinate i in the Cartesian system. (20 points)

(30 points)

Write program for the following process. Take the points (Subscript[P, 1]),(Subscript[P, 2]),...,(Subscript[P, 4]) in the 3D space. Take now (Subscript[Q, 0]). Define (Subscript[Q, 1])as follows. Choose randomly one of the points  (Subscript[P, 1]),(Subscript[P, 2]),...,(Subscript[P, 4]), for example (Subscript[P, j]) and let (Subscript[Q, 1])be the midpoint on the line ((Subscript[P, j]),(Subscript[Q, 0])). Iterate and visualize the process. It you have enough time, use coloring, such as the colors are associated to the points (Subscript[P, 1]),(Subscript[P, 2]),...,(Subscript[P, 4]).