•Tentti 1

•Tehtävä 1

In[1]:=

Remove["Global`*"]

In[2]:=

yht1 = 5 x^2 + 6 x y + 5 y^2 - 16 x - 16 y - 16 == 0

Out[2]=

-16 - 16 x + 5 x^2 - 16 y + 6 x y + 5 y^2 == 0

In[3]:=

yht2 = x^2 + x y - 2 y^2 + x + 2    y == 0

Out[3]=

x + x^2 + 2 y + x y - 2 y^2 == 0

In[4]:=

<< Graphics`ImplicitPlot`

In[5]:=

ImplicitPlot[{yht1, yht2}, {x, -5, 5}, {y, -5, 35}, PlotPoints -> {200, 200}]

[Graphics:HTMLFiles/ratk1_8.gif]

Out[5]=

-Graphics -

In[6]:=

Solve[{yht1, yht2}, {x, y}] // N

Out[6]=

{{x -> -1.3065265388416374`, y -> 0.6532632694208187`}, {x -> 3.768065000380099`, y - ... 75054`, y -> 0.10805890929249462`}, {x -> 1.8919410907075054`, y -> 2.8919410907075056`}}

•Tehtävä 2

In[7]:=

Remove["Global`*"]

In[8]:=

a = {1, 2} ; b = {7, -1} ; c = {15, 7} ;

Sivujen keskipisteet:

In[9]:=

ab2 = (a + b)/2

Out[9]=

{4, 1/2}

In[10]:=

bc2 = (b + c)/2

Out[10]=

{11, 3}

In[11]:=

ca2 = (c + a)/2

Out[11]=

{8, 9/2}

Keskinormaalien leikkauspiste:

In[12]:=

p = {x, y}

Out[12]=

{x, y}

Kohtisuoruusehdot:

In[13]:=

yhtalot = {(p - ab2) . (b - a) == 0, (p - bc2) . (c - b) == 0, (p - ca2) . (a - c) == 0}

Out[13]=

{6 (-4 + x) - 3 (-1/2 + y) == 0, 8 (-11 + x) + 8 (-3 + y) == 0, -14 (-8 + x) - 5 (-9/2 + y) == 0}

In[14]:=

ratk = Solve[yhtalot, p]

Out[14]=

{{x -> 43/6, y -> 41/6}}

Koska ratkaisu löytyi, keskinormaalit leikkaavat samassa pisteessä, joka on

In[15]:=

p /. ratk[[1]]

Out[15]=

{43/6, 41/6}

•Tehtävä 3

In[16]:=

Remove["Global`*"]

In[17]:=

p = x^5 + x^4 + x^3 + x^2 + x + 1

Out[17]=

1 + x + x^2 + x^3 + x^4 + x^5

In[18]:=

q = x^3 - x^2 + x - 1

Out[18]=

-1 + x - x^2 + x^3

In[19]:=

om = PolynomialQuotient[p, q, x]

Out[19]=

2 + 2 x + x^2

In[20]:=

jj = PolynomialRemainder[p, q, x]

Out[20]=

3 + x + 2 x^2

In[21]:=

p == q * om + jj // Simplify

Out[21]=

True

•Tehtävä 4

In[22]:=

Remove["Global`*"]

In[23]:=

f = 1/(2 + Sin[x])

Out[23]=

1/(2 + Sin[x])

In[24]:=

intf = Integrate[f, x]

Out[24]=

(2 ArcTan[(Sec[x/2] (Cos[x/2] + 2 Sin[x/2]))/3^(1/2)])/3^(1/2)

In[25]:=

Plot[intf, {x, -2 Pi, 4 Pi}]

[Graphics:HTMLFiles/ratk1_45.gif]

Out[25]=

-Graphics -

Ei ole jatkuva, vaikka integraalifunktion pitäisi olla.

In[26]:=

Integrate[f, {x, 0, 2 Pi}]

Out[26]=

(2 π)/3^(1/2)

In[27]:=

% // N

Out[27]=

3.6275987284684352`

In[28]:=

NIntegrate[f, {x, 0, 2 Pi}]

Out[28]=

3.6275987284707627`

In[29]:=

(intf /. x -> 2 Pi) - (intf /. x -> 0)

Out[29]=

0

Integraalifunktioon sijoittaminen antaa väärän tuloksen, koska funktio f on kaikkialla positiivinen. Syynä on saadun integraalifunktion epäjatkuvuus. Muut kaksi tulosta ovat ilmeisesti oikein. Funktio f:

In[30]:=

Plot[f, {x, -2 Pi, 4 Pi}, AxesOrigin -> {0, 0}]

[Graphics:HTMLFiles/ratk1_56.gif]

Out[30]=

-Graphics -

•Tehtävä 5

In[31]:=

Remove["Global`*"]

In[32]:=

g[x_, 1] := x^2 + c

In[33]:=

g[x_, n_] := g[x^2 + c, n - 1]

In[34]:=

g[0, 10]

Out[34]=

c + (c + (c + (c + (c + (c + (c + (c + (c + c^2)^2)^2)^2)^2)^2)^2)^2)^2

In[35]:=

Table[{c, N[g[0, 10]]}, {c, -5, 2, 0.25}] // TableForm

Out[35]//TableForm=
-5 2.309603905871268693326167505`15.9546*^332
-4.75` 2.216688084727524281196813046075`15.6536*^319
-4.5` 3.0480750038298485`*^305
-4.25` 4.507895089606707`*^290
-4.` 4.997069910920882`*^274
-3.75` 2.6042826298265917`*^257
-3.5` 3.4250583986028373`*^238
-3.25` 4.778757842639549`*^217
-3.` 1.96369242848753`*^194
-2.75` 2.992347978995658`*^167
-2.5` 3.498854814714579`*^135
-2.25` 2.0079413697564395`*^94
-2.` 2.`
-1.75` -1.7485504881729272`
-1.5` -0.07668938219279631`
-1.25` 0.28797008359130105`
-1.` 0.`
-0.75` -0.30771899377394707`
-0.5` -0.3576176642011428`
-0.25` -0.2070903849467524`
0.` 0.`
0.25` 0.430549106102856`
0.5` 2.0773872763941816`*^16
0.75` 5.79191080893208`*^53
1.` 3.7918623102659254`*^90
1.25` 3.443304123599781`*^123
1.5` 5.729605653575896`*^152
1.75` 6.524184988853353`*^178
2.` 1.781492681120714`*^202

In[36]:=

Table[{c, N[g[0, 20]]}, {c, -2.1, 0.5, 0.05}] // TableForm

Out[36]//TableForm=
-2.1` 4.06751216365183857739536292939`13.5463*^60043
-2.0500000000000003` 4.74208369290156842813236418`13.5463*^42103
-2.` 2.`
-1.9500000000000002` 1.6190753727338199`
-1.9000000000000001` -1.1700361412284164`
-1.85` -1.1899875761852914`
-1.8` 1.3933572761123025`
-1.75` 1.3057686660440533`
-1.7000000000000002` -0.19019996744184886`
-1.6500000000000001` -0.965038082564891`
-1.6` -1.1473398913429562`
-1.55` -1.4936134318653849`
-1.5` -0.4934387873578123`
-1.45` 0.16126827921070497`
-1.4000000000000001` -0.21593115971098586`
-1.35` -0.06441154019348216`
-1.3` 0.019428527429930842`
-1.25` 0.12770582381440843`
-1.2` 0.16149709683057178`
-1.1500000000000001` 0.13202741579245125`
-1.1` 0.09160285309769223`
-1.05` 0.04772255495975841`
-1.` 0.`
-0.9500000000000001` -0.05278640195850959`
-0.9` -0.11269660530142978`
-0.85` -0.18336339944075797`
-0.8` -0.2679035625064362`
-0.75` -0.3540177529021947`
-0.7` -0.4103706786166595`
-0.65` -0.42590710980181257`
-0.6` -0.4151690828576661`
-0.55` -0.39270363371975836`
-0.5` -0.36565623470002073`
-0.45` -0.3365951484477454`
-0.4` -0.3062168141057926`
-0.35` -0.27459575685220705`
-0.3` -0.24161978651690041`
-0.25` -0.20710677874887562`
-0.19999999999999998` -0.17082039320748751`
-0.14999999999999997` -0.1324555320334743`
-0.09999999999999998` -0.0916079783099615`
-0.049999999999999975` -0.04772255750516609`
2.7755575615628914`*^-17 2.7755575615628914`*^-17
0.05000000000000003` 0.05278640450004209`
0.10000000000000003` 0.11270166537925236`
0.15000000000000002` 0.18377223382226`
0.20000000000000004` 0.27639246946886226`
0.25000000000000006` 0.4599437228607875`
0.30000000000000004` 1.0159111725670561`*^91
0.35000000000000003` 1.684896094391456103966631888906189`15.0515*^1478
0.4` 4.247496437039256321030651582056`14.4494*^5001
0.45000000000000007` 1.08966010747907856607104638390942`14.1484*^10268
0.5` 1.373683041368883599327114`14.1484*^16709

Siis väli [-2, 0.25].

Converted by Mathematica  (August 27, 2003)