Part I
Winter Semester, 2005

(Elements: 5 points)

Plot the function

h(x)=(e^x) sin(x)

on the interval [-π,π], such that the plotted y-range is [-1,10], and the x-axis is twice longer than the y-axis.

(Transformations: 10 points)

Create a table of the 20 points at equal distance of the unit circle around the origin. Plot these points such that the color of them be red. (5 points)

Apply the transformation

{x,y}->{Sin(x)-Cos(y), ((x-y)^(1/3))}

to each point of the list and plot the transformed list. (5 points)

(Transformatios: 10 points)

Consider the linear transformation on the plane:

(({{u}, {v}}))=A[s] . (({{x}, {y}}))

that depends on the parameter s ϵ [0,1], where

A[s_]=(({{s, -s}, {s, 1}})).

Do an animation with respect to s that show the dependence on the parameter s. For example, animate the effect of the transformation on the unit circle c[t_]:={Cos[t],Sin[t]}.

(Investigtion of functions: 25 points)

Consider the following functions:

f(x):=((e^x)/(((x^2)+0.92) (1+(e^x))));
g(x):=(((e^(-(x^2))) (x^2))/((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 there tangent lines are parallel (f'(x)=g'(x)). Use initial points as good as possible by using the plots from the previous part. (10 points)

Plot the functions and the parallel tangent lines together. (10 points)

(Functions of two variables: 10 points)

Consider the following function:

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

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

(Differential equations: 20 points)

Consider the following potential field:

f(x_,y_):=(1/((x^2)+(y^2)+1))+(1/((((x-2))^2)+(y^2)+1))

Plot the contour lines for

f(x,y)=0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.

of the given potential without shading. You can also use PlotRange for the ContourPlot!

Calculate the negative gradient vector field and plot it.

Find and plot curves that fit to the given vector field with initial values from the circles with radius 0.1 and with center {0,0} and {2,0}, i.e., solve the system of differential equations

x'=-(∂_x) f(x,y), 
y'=-(∂_y) f(x,y),

with initial conditiosn from the given circles.