Part I
Winter Semester, 2000

(Investigtion of Functions)

Do total investigation of one of the functions given below.

Graphical views (local and global).

Zero positions.

Limits where necessary.

Monotonity and extremal points.

Concavity and inflection points.

f[x_]:=((Log[x]-((Cos[x])^2))/(2 (x^3)-2 x))

f[x_]:=((((Sin[x])^2)-((Cos[x])^2))/(2 (x^3)-2 x))

f[x_]:=(((E^(-x))-Cos[x])/(2 (x^3)-2x))

f [x_]:=(((x^11)+3 (x^10)-(x^5)-2 (x^3)-1)/((x^2)-x))

f[x_]:=(((x^6)-(x^4)-2 (x^5)-4)/(2 (x^6)- (x^3)))

f[x_]:=((Log[(x^7)+2(x^6)+(x^3)-2(x^3)-1])/((x^3)-3x))

f[x_]:=(((x^2)-(x^4)-1)/(2 (x^3)-2 x))

f [x_]:=((2(x^6)+3(x^3)-(x^2)-1)/(Sin[x]))

f[x_]:=((E^((Cos[x])^2))/(2 (x^3)-2 x))

f[x_]:=(((Sin[x])^2)/(2 (x^6)+3 (x^3)))

(Study of a dataset)

Investigate the behavior of one of the following datasets. Use Special plot if needed. Fit a function to this set using the method of least squares.

Case 1

data={{-1.5, -14.07018575923918}, 
   {-1.4, -5.708755863528245}, 
   {-1.3, -3.598730644966069}, 
   {-1.2, -2.557015741511812}, 
   {-1.1, -1.874613916979771}, 
   {-1., -1.550171547194927}, 
   {-0.9, -1.25901401877474}, 
   {-0.7999999999999998, -0.980720628269496}, 
   {-0.6999999999999998, -0.863940874943487}, 
   {-0.5999999999999998, -0.6311929958548359}, 
   {-0.4999999999999999, -0.6098307280759957}, 
   {-0.3999999999999998, -0.5172337051481137}, 
   {-0.2999999999999998, -0.3750162074150962}, 
   {-0.1999999999999998, -0.1835456740704539}, 
   {-0.0999999999999998, -0.1149694591807097}, 
   {1.942890293094024*10^-16, -0.05120651361073937}, 
   {0.1000000000000002, 0.1737762133016124}, 
   {0.2000000000000002, 0.1484522904236703}, 
   {0.3000000000000002, 0.3359704489002492}, 
   {0.4000000000000003, 0.4134651492855475}, 
   {0.5000000000000002, 0.6247725039099829}, 
   {0.6000000000000002, 0.7075007806331626}, 
   {0.7000000000000002, 0.804364681290384}, 
   {0.8000000000000001, 1.081819117255669}, 
   {0.9, 1.20739404368399}, {1., 1.591643844991731}, 
   {1.1, 2.023464155074051}, 
   {1.2, 2.509196301717121}, 
   {1.3, 3.559192533832756}, 
   {1.400000000000001, 5.724883658359765}, 
   {1.500000000000001, 14.05898024622165}};

Case 1

data={{0,1.0686793079491528`},{0.1`,1.1895644947815431`},{0.2`,1.2431059876330366`},{0.30000000000000004`,1.3178636912926247`},{0.4`,1.4234591711404598`},{0.5`,1.4328961836341594`},{0.6000000000000001`,1.4891456578545317`},{0.7000000000000001`,1.5850144381586675`},{0.8`,1.62837390687407`},{0.9`,1.6839710365390088`},{1.`,1.705254426430055`},{1.1`,1.719592766614182`},{1.2000000000000002`,1.7120063546259088`},{1.3`,1.7501966990227922`},{1.4000000000000001`,1.8125495952430477`},{1.5`,1.8247390197077615`},{1.6`,1.8161183505524252`},{1.7000000000000002`,1.9155913215803548`},{1.8`,1.867363748895313`},{1.9000000000000001`,1.950197849750699`},{2.`,1.9335513580904284`},{2.1`,1.9383275390954837`},{2.2`,1.9205294632398713`},{2.3000000000000003`,1.9396645541293327`},{2.4000000000000004`,1.909489380088476`},{2.5`,1.984297055907064`},{2.6`,1.9952223026768532`},{2.7`,2.014035973138044`},{2.8000000000000003`,1.9456180535765712`},{2.9000000000000004`,1.971931991127763`},{3.`,1.981751518574765`},{3.1`,2.0545925415341593`},{3.2`,1.9879630412321316`},{3.3000000000000003`,1.999531347503322`},{3.4000000000000004`,2.0250314493808053`},{3.5`,2.016980510193137`},{3.6`,1.9882009562250944`},{3.7`,1.988962496377418`},{3.8000000000000003`,2.0768873883003116`},{3.9000000000000004`,2.0790668023134597`},{4.`,2.079194171236572`},{4.1000000000000005`,1.9988385018129067`},{4.2`,2.0515999462191`},{4.3`,2.085973685773729`},{4.4`,2.0163458288951968`},{4.5`,2.04351821332796`},{4.6000000000000005`,2.025211065692739`},{4.7`,2.0505235698320976`},{4.800000000000001`,2.0201860883713563`},{4.9`,2.080798572264315`},{5.`,2.0590290735471`},{5.1000000000000005`,2.072280614490484`},{5.2`,2.0164711547978262`},{5.300000000000001`,2.056298350244159`},{5.4`,2.0297118526609443`},{5.5`,2.0746488456898184`},{5.6000000000000005`,2.089564610291607`},{5.7`,2.021529463789036`},{5.800000000000001`,2.0727961595170497`},{5.9`,2.0288182786940587`},{6.`,2.0752590431590363`},{6.1000000000000005`,2.008946538679264`},{6.2`,2.0745361234696524`},{6.300000000000001`,2.0304127049765337`},{6.4`,2.0785664279372225`},{6.5`,2.0942747899911045`},{6.6000000000000005`,2.008609663028823`},{6.7`,2.0314758530649595`},{6.800000000000001`,2.0504910412642863`},{6.9`,2.0401432338888315`},{7.`,2.073795247663443`},{7.1000000000000005`,2.0722628131105734`},{7.2`,2.0224423951833783`},{7.300000000000001`,2.052230325207447`},{7.4`,2.0083288563216826`},{7.5`,2.094157472607692`},{7.6000000000000005`,2.000700810339727`},{7.7`,2.0911630926486846`},{7.800000000000001`,2.0743019405002756`},{7.9`,2.0156041963090985`},{8.`,2.0076033251371754`}}

(Series expansion)

Solve on of the following problems :

Do animation for the discontinuity of the function below at x=0 and x=1.

f(x_):=((e^((cos^2)(x)))/(2 (x^3)-2 x))

Animate the error of the approximation of the Taylor polynomials for log(x-1) at (Subscript[x, 0])=0.

Animate the error of the approximation of the Taylor polynomials for sin((x^2)) at (Subscript[x, 0])=0.

(Animation, Linear Algebra)

Do an animation showing the affect of the linear transformation with matrix

(({{1, c}, {-c, -1}}))

on the unit circle.

Do an animation showing the affect of the linear transformation with matrix

(({{1, c}, {-1, (c^2)}}))

on the unit circle.

Do an animation showing the affect of the linear transformation with matrix

(({{c, 1}, {0, -(c^2)}}))

on the unit circle.

(Functions of two variables)

Investigate the behavior graphically of one of the follwing functions. Then, try to find its extremal points, and plot the tangent plane at another point.

f[x_,y_]:=(12 (x^2)-6 y) (E^(-x y))

f[x_,y_]:=(12 (x^2)-6 y) (E^(-x- y))

f[x_,y_]:=((x^2)/5)+x y Log[(x^2)+(y^2)]

f[x_,y_]:=((x^2)+(y^2)) (E^(x-(y^2)))

f[x_,y_]:=(x/5)+x y Log[(x^2)+(y^2)]

f[x_,y_]:=(12 (x^2)+y) (E^(-(x^2)-(y^2)))

f[x_,y_]:=((x^2)+(y^4)) (E^(x-y))

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

(Differential equations)

Solve one of the following prblems.

Consider the differential equation of population growth

x'=x(x-2).

Find its general solution, plot its direction field, and do an animation showing the dependence of the solutions on the initial values. Animate the change of the population in the time.  For these, solve the equation numerically.

Consider the differential equation of population growth

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

Find its general solution, plot its direction field, and do an animation showing the dependence of the solutions on the initial values. Animate the change of the population in the time.  For these, solve the equation numerically.

Consider the oscillator equation

x'=y, 
y'=-0.1y -sin(x).

Plot its vectorfield some trajectories in the phaseplane. Do an animation showing the dependence of the trajectories on the initial velocity. Animate the motion in the phase space.

Consider the equation

x'=2x-x y, 
y'=-y+x y.

Plot its vectorfield some trajectories in the phaseplane. Do an animation showing the dependence of the trajectories on the initial values. Animate the motion in the phase space.