Part I
Winter Semester, 2003

(Styles, structure: 10 points)

Prepare nice notebook with styles. Do clear sectioning.

(Equations: 5 points)

Plot the functions f(x) and g(x) and find the places where f(x)=g(x).

f(x_):=(E^(x^2));   g(x_):=10 x sin(x)+0.9

(Study of a dataset: 10 points)

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

data={{2.`,0.04387655940762929`},{2.02`,0.04546515912946302`},{2.04`,0.04078766069113874`},{2.06`,0.03979159371765726`},{2.08`,0.045821880138058595`},{2.1`,0.04550627946390336`},{2.12`,0.03780485713146543`},{2.14`,0.0449009784114398`},{2.16`,0.040258189491989255`},{2.18`,0.04056021428684235`},{2.2`,0.03737722595964534`},{2.22`,0.03771053170601624`},{2.24`,0.03925000731229104`},{2.26`,0.039650839080869665`},{2.28`,0.03845476062999551`},{2.3`,0.03846892495320178`},{2.32`,0.038101707518971684`},{2.34`,0.03873855336877822`},{2.36`,0.03765038680559707`},{2.38`,0.033672895495508344`},{2.4`,0.03840499508969624`},{2.42`,0.03751621781181646`},{2.44`,0.03573761203938113`},{2.46`,0.03233148320001201`},{2.48`,0.034200283600388896`},{2.5`,0.03249128332250458`},{2.52`,0.03319001301697982`},{2.54`,0.03487352100930534`},{2.56`,0.033211975277525264`},{2.58`,0.031590741763211264`},{2.6`,0.03171043400624305`},{2.62`,0.03332794927730387`},{2.64`,0.030187479461944007`},{2.66`,0.031837958195647074`},{2.68`,0.029879484419666757`},{2.7`,0.03084401435949166`},{2.72`,0.03032165351550277`},{2.74`,0.028578390824274102`},{2.76`,0.02983007860578133`},{2.78`,0.02797344396017512`},{2.8`,0.02757528717524004`},{2.82`,0.027936588325721427`},{2.84`,0.027109212266110676`},{2.86`,0.02684330326451315`},{2.88`,0.02664470057733295`},{2.9`,0.026948816473863327`},{2.92`,0.026595042789029195`},{2.94`,0.025827698410470754`},{2.96`,0.02620337396096793`},{2.98`,0.025581404546644714`},{3.`,0.02592403451881346`},{3.02`,0.024966588662506735`},{3.04`,0.02522385757174387`},{3.06`,0.024874641288403266`},{3.08`,0.02440310772320574`},{3.1`,0.024028008209084937`},{3.12`,0.02386367480837555`},{3.14`,0.02355219677712993`},{3.16`,0.023304502187020106`},{3.18`,0.023020684643791728`},{3.2`,0.02285475367684569`},{3.22`,0.022660203405527497`},{3.24`,0.02219306121944968`},{3.26`,0.022035743292844235`},{3.28`,0.021968240302389374`},{3.3`,0.02127257298655768`},{3.32`,0.021194445400700854`},{3.34`,0.021108785592541274`},{3.36`,0.02117810348891925`},{3.38`,0.02070948008373825`},{3.4`,0.020708685838087`},{3.42`,0.020211999003046124`},{3.44`,0.019796756531459223`},{3.46`,0.02019552658147617`},{3.48`,0.019862707628540425`},{3.5`,0.019528471027759634`},{3.52`,0.018855478385710712`},{3.54`,0.019352004916573344`},{3.56`,0.0183153007293996`},{3.58`,0.019056142122758645`},{3.6`,0.017649220566498786`},{3.62`,0.017091784244884552`},{3.64`,0.01806825784758385`},{3.66`,0.017302342124163157`},{3.68`,0.01715669071757546`},{3.7`,0.016415210566404728`},{3.72`,0.016453492548443284`},{3.74`,0.017427972105782532`},{3.76`,0.016743121673093057`},{3.78`,0.017365096695814826`},{3.8`,0.01695248335351365`},{3.82`,0.015320079457741385`},{3.84`,0.01627308926921998`},{3.86`,0.015108871360540342`},{3.88`,0.016683605894461957`},{3.9`,0.015502302070232109`},{3.92`,0.014459737755202422`},{3.94`,0.014628847420790755`},{3.96`,0.014003908618190525`},{3.98`,0.015551596457700309`},{4.`,0.015203237860933007`},{4.02`,0.014327838378966262`},{4.04`,0.014939038484316458`},{4.06`,0.015218009504028043`},{4.08`,0.013955241335312034`},{4.1`,0.013572172600674575`},{4.12`,0.012602918646845169`},{4.14`,0.013573361485944329`},{4.16`,0.012317458944598682`},{4.18`,0.012611281757193027`},{4.2`,0.01393188699052675`},{4.22`,0.013539725348746251`},{4.24`,0.012825309773851483`},{4.26`,0.012260506335739553`},{4.28`,0.013909756181610982`},{4.3`,0.012505454628653276`},{4.32`,0.013180027460409654`},{4.34`,0.011123926236613896`},{4.36`,0.010859903198934892`},{4.38`,0.0131954535047791`},{4.4`,0.012132594115044557`},{4.42`,0.012314303051716165`},{4.44`,0.012556735517865694`},{4.46`,0.01055419635813635`},{4.48`,0.012313052314220412`},{4.5`,0.010633902688495544`},{4.52`,0.01015428783226052`},{4.54`,0.010302620701038366`},{4.56`,0.010554117083262782`},{4.58`,0.011668733569102733`},{4.6`,0.009564060627109293`},{4.62`,0.011032346261589068`},{4.64`,0.009951181818930128`},{4.66`,0.010576289735459168`},{4.68`,0.009602856955170658`},{4.7`,0.011329316407793095`},{4.72`,0.010740286780434661`},{4.74`,0.009428625551092442`},{4.76`,0.009272541041435812`},{4.78`,0.00966416052883997`},{4.8`,0.010775832688093992`},{4.82`,0.008932326052949583`},{4.84`,0.00891226967769663`},{4.86`,0.009406158235861852`},{4.88`,0.009021487421715527`},{4.9`,0.00916697627215297`},{4.92`,0.008880237142112245`},{4.94`,0.008828845885394043`},{4.96`,0.008995198243015916`},{4.98`,0.008378570247003588`},{5.`,0.008417375389337197`},{5.02`,0.008175379016245383`},{5.04`,0.008062190484472487`},{5.06`,0.008419792057329895`},{5.08`,0.008150309202517754`},{5.1`,0.008509150516922338`},{5.12`,0.009147050150814035`},{5.14`,0.008935251794911796`},{5.16`,0.007605000036614381`},{5.18`,0.008334047306907788`},{5.2`,0.0075781561575297635`},{5.22`,0.008247884167888535`},{5.24`,0.008681681091644697`},{5.26`,0.007442267587669049`},{5.28`,0.00747514505013423`},{5.3`,0.007765857715784807`},{5.32`,0.008110615581458454`},{5.34`,0.008023027743654104`},{5.36`,0.00826071063890203`},{5.38`,0.008378775829281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(Series expansion: 10 points)

Consider the function

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

Let (Subscript[x, 0])=2. Determine the Taylor polinomials (Subscript[T, n])(x)  for n=1,...,10. Create a table that contains the difference between f(x)  and the Taylor polinomials (Subscript[T, n])(x)  at x=2.1,2.2,2.3,2.4,2.5.

(Investigtion of functions: 15 points)

Use Mathematica (plots and analitic methods) to investigate the following function:

f(x_):=((sin(log(x)))/((x^2)-x))

find the domain of definition;

find the limit at the boundaries of the domain (left- and right-hand-side if necessary);

find the zero points on the interval [-100,100]. How many zeros are there on this interval?

(Functions of two variables: 15 points)

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

f(x_,y_):=-(((x^4)+(y^4))^(1/4))+(y^4)+((x^2)/2)-(y^2)-0.25 x

(Differential equations: 10 points)

Consider the following differential equation

x'=t (e^(-0.1 t))x(1-x),

of a students learning activity, where x(t) is the learned portion of the whole material (x(t)=1 is the total amount of knowledge, x'(t) is the learning speed). Investigate the model by solving the equation for different initial conditions 0≤x(0)≤1. Animate the dependence of the solutions with respect to the initial values.