Part I
Winter Semester, 2002

(Styles, structure: 5 points)

Prepare nice notebook with styles. Do clear sectioning.

(Investigtion of functions: 10 points)

Where is the following function not defined:

(((x^6)-3 (x^5)+6 (x^3)-3 (x^2)-3 x+2)/((x^5)-4 (x^4)+(x^3)+14 (x^2)-20 x+8))

Transform it to a simpler form. Evaluate it at the values  x=1.5, -2.1.  (Do not use x=...)

(Series expansion: 10 points)

Consider the function

f(x)=((sin(x) log(x-1) )/((x^2)-x))

and the place (Subscript[x, 0])=2. Find the Taylor polynomials at x0 and visualize the approximation for n=1,...,5.

(Functions of two variables: 15 points)

Investigate the behavior the following function graphically. Then, find its extremal points.

f(x,y)=((x^2)/2)-0.2 x+(y^2)-0.1 y-(Sqrt[(x^2)+(y^2)])

(Study of a dataset: 10 points)

Investigate the behavior of the following dataset. Use special plots and/or transformations if needed. Fit a function to this set using the method of least squares.

data={{0.4,1.11111},{0.41,1.07498},{0.42,1.04058},{0.43,1.00781},{0.44,0.976563},{0.45,0.946746},{0.46,0.918274},{0.47,0.891067},{0.48,0.865052},{0.49,0.84016},{0.5,0.816327},{0.51,0.793493},{0.52,0.771605},{0.53,0.75061},{0.54,0.73046},{0.55,0.711111},{0.56,0.692521},{0.57,0.67465},{0.58,0.657462},{0.59,0.640923},{0.6,0.625},{0.61,0.609663},{0.62,0.594884},{0.63,0.580636},{0.64,0.566893},{0.65,0.553633},{0.66,0.540833},{0.67,0.528471},{0.68,0.516529},{0.69,0.504987},{0.7,0.493827},{0.71,0.483033},{0.72,0.47259},{0.73,0.462481},{0.74,0.452694},{0.75,0.443213},{0.76,0.434028},{0.77,0.425125},{0.78,0.416493},{0.79,0.408122},{0.8,0.4},{0.81,0.392118},{0.82,0.384468},{0.83,0.377038},{0.84,0.369822},{0.85,0.362812},{0.86,0.355999},{0.87,0.349375},{0.88,0.342936},{0.89,0.336672},{0.9,0.330579},{0.91,0.324649},{0.92,0.318878},{0.93,0.313259},{0.94,0.307787},{0.95,0.302457},{0.96,0.297265},{0.97,0.292205},{0.98,0.287274},{0.99,0.282466},{1.,0.277778},{1.01,0.273205},{1.02,0.268745},{1.03,0.264393},{1.04,0.260146},{1.05,0.256},{1.06,0.251953},{1.07,0.248},{1.08,0.244141},{1.09,0.24037},{1.1,0.236686},{1.11,0.233087},{1.12,0.229568},{1.13,0.226129},{1.14,0.222767},{1.15,0.219479},{1.16,0.216263},{1.17,0.213117},{1.18,0.21004},{1.19,0.207029},{1.2,0.204082},{1.21,0.201197},{1.22,0.198373},{1.23,0.195609},{1.24,0.192901},{1.25,0.19025},{1.26,0.187652},{1.27,0.185108},{1.28,0.182615},{1.29,0.180172},{1.3,0.177778},{1.31,0.175431},{1.32,0.17313},{1.33,0.170874},{1.34,0.168663},{1.35,0.166493},{1.36,0.164366},{1.37,0.162278},{1.38,0.160231},{1.39,0.158222},{1.4,0.15625},{1.41,0.154315},{1.42,0.152416},{1.43,0.150551},{1.44,0.148721},{1.45,0.146924},{1.46,0.145159},{1.47,0.143426},{1.48,0.141723},{1.49,0.140051},{1.5,0.138408},{1.51,0.136794},{1.52,0.135208},{1.53,0.13365},{1.54,0.132118},{1.55,0.130612},{1.56,0.129132},{1.57,0.127677},{1.58,0.126247},{1.59,0.12484},{1.6,0.123457},{1.61,0.122096},{1.62,0.120758},{1.63,0.119442},{1.64,0.118147},{1.65,0.116874},{1.66,0.11562},{1.67,0.114387},{1.68,0.113173},{1.69,0.111979},{1.7,0.110803},{1.71,0.109646},{1.72,0.108507},{1.73,0.107385},{1.74,0.106281},{1.75,0.105194},{1.76,0.104123},{1.77,0.103069},{1.78,0.10203},{1.79,0.101008},{1.8,0.1},{1.81,0.0990075},{1.82,0.0980296},{1.83,0.0970662},{1.84,0.0961169},{1.85,0.0951814},{1.86,0.0942596},{1.87,0.0933511},{1.88,0.0924556},{1.89,0.091573},{1.9,0.0907029},{1.91,0.0898452},{1.92,0.0889996},{1.93,0.0881659},{1.94,0.0873439},{1.95,0.0865333},{1.96,0.0857339},{1.97,0.0849455},{1.98,0.084168},{1.99,0.0834011},{2.,0.0826446},{2.01,0.0818984},{2.02,0.0811622},{2.03,0.080436},{2.04,0.0797194},{2.05,0.0790123},{2.06,0.0783147},{2.07,0.0776262},{2.08,0.0769468},{2.09,0.0762762},{2.1,0.0756144},{2.11,0.0749611},{2.12,0.0743163},{2.13,0.0736798},{2.14,0.0730514},{2.15,0.072431},{2.16,0.0718184},{2.17,0.0712137},{2.18,0.0706165},{2.19,0.0700268},{2.2,0.0694444},{2.21,0.0688693},{2.22,0.0683013},{2.23,0.0677404},{2.24,0.0671862},{2.25,0.0666389},{2.26,0.0660982},{2.27,0.0655641},{2.28,0.0650364},{2.29,0.0645151},{2.3,0.064},{2.31,0.0634911},{2.32,0.0629882},{2.33,0.0624912},{2.34,0.0620001},{2.35,0.0615148},{2.36,0.0610352},{2.37,0.0605611},{2.38,0.0600925},{2.39,0.0596294},{2.4,0.0591716},{2.41,0.058719},{2.42,0.0582717},{2.43,0.0578294},{2.44,0.0573921},{2.45,0.0569598},{2.46,0.0565323},{2.47,0.0561096},{2.48,0.0556917},{2.49,0.0552784},{2.5,0.0548697},{2.51,0.0544655},{2.52,0.0540657},{2.53,0.0536704},{2.54,0.0532793},{2.55,0.0528926},{2.56,0.05251},{2.57,0.0521315},{2.58,0.0517572},{2.59,0.0513868},{2.6,0.0510204},{2.61,0.0506579},{2.62,0.0502993},{2.63,0.0499444},{2.64,0.0495933},{2.65,0.0492459},{2.66,0.0489021},{2.67,0.048562},{2.68,0.0482253},{2.69,0.0478921},{2.7,0.0475624},{2.71,0.0472361},{2.72,0.0469131},{2.73,0.0465934},{2.74,0.046277},{2.75,0.0459638},{2.76,0.0456538},{2.77,0.0453468},{2.78,0.045043},{2.79,0.0447422},{2.8,0.0444444},{2.81,0.0441496},{2.82,0.0438577},{2.83,0.0435687},{2.84,0.0432825},{2.85,0.0429992},{2.86,0.0427186},{2.87,0.0424408},{2.88,0.0421656},{2.89,0.0418932},{2.9,0.0416233},{2.91,0.0413561},{2.92,0.0410914},{2.93,0.0408292},{2.94,0.0405696},{2.95,0.0403124},{2.96,0.0400577},{2.97,0.0398054},{2.98,0.0395554},{2.99,0.0393078},{3.,0.0390625}};

(Differential equations: extra 15 points)

Consider the differential equation of population growth

x'=(((sin(x)+1))^2)x(x-2).

Plot its direction field and the solution with the initial value x(0)=0.2,0.4,0.6.