ratk6.nb

•Tehtävä 1

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•Tehtävä 2

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${Sec[x]^2, 2 Sec[x]^2 Tan[x], 2 Sec[x]^4 + 4 Sec[x]^2 Tan[x]^2, 16 Sec[x]^4 Tan[x] + 8 Sec[x]^ ... 112949304754176 Sec[x]^6 Tan[x]^15 + 137433710592 Sec[x]^4 Tan[x]^17 + 524288 Sec[x]^2 Tan[x]^19}$

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•Tehtävä 3

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${{y^'[0] -> 1/2}}$

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$ParametricPlot[{y^3 + y^2 + y + 1, y}, {y, -2, 2}, PlotRange -> {{-2, 2}, {-2, 2}}, AspectRatio -> Automatic]$

[Graphics:HTMLFiles/ratk6_16.gif]

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•Tehtävä 4

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Kuitenkaan funktio ei ole vakio!

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[Graphics:HTMLFiles/ratk6_23.gif]

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•Tehtävä 5

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$f[x_] = x^4 - 7 x^3 + 11 x^2 + 7 x - 10 ; f1[x_] = D[f[x], x] ; f2[x_] = D[f[x], {x, 2}] ;$

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[Graphics:HTMLFiles/ratk6_27.gif]

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Nollakohdat:

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Ääriarvopisteet:

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${{x -> 7/4 + (-189 + 8 i 9069^(1/2))^(1/3)/(4 3^(2/3)) + 59/(4 (3 (-189 + 8 i 9069^(1/2)))^ ... 8 i 9069^(1/2))^(1/3))/(8 3^(2/3)) - (59 (1 + i 3^(1/2)))/(8 (3 (-189 + 8 i 9069^(1/2)))^(1/3))}}$

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${{3.5742137985635267` - 3.3306690738754696`*^-16 i, -0.8788980209096451` + 0.` i}, {1.9295354 ... 374924586228165` - 1.1102230246251565`*^-16 i, -10.949453288874253` - 1.1570987105878513`*^-30 i}}$

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Käännepisteet:

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${{x -> 1/12 (21 - 177^(1/2))}, {x -> 1/12 (21 + 177^(1/2))}}$

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•Tehtävä 6

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$f[x_] = 2 * x^4 - 12 * x^3 + 7 * x^2 + 41 * x - 3 ; Plot[f[x], {x, -2, 4.5}]$

[Graphics:HTMLFiles/ratk6_42.gif]

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•Tehtävä 7

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Tilavuus:

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$Integrate[Pi * f[x]^2, {x, 0, Pi}]$

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Pinta-ala

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$Integrate[2 * Pi * f[x] Sqrt[1 + f '[x]^2], {x, 0, Pi}]$

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$ParametricPlot3D[{x, Sin[x]^2 * Sin[t], Sin[x]^2 * Cos[t]}, {x, 0, Pi}, {t, 0, 2 * Pi}] ;$

[Graphics:HTMLFiles/ratk6_58.gif]

•Tehtävä 8

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$f[x_] = ArcSin[2 x * Sqrt[1 - x^2]] ; Plot[f[x], {x, -1, 1}, AspectRatio -> Automatic]$

[Graphics:HTMLFiles/ratk6_60.gif]

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Suljetulla välillä jatkuva funktio: ääriarvot saadaan joko päätepisteissä, derivaatan nollakohdissa tai derivaatan epäjatkuvuuskohdissa.

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${{x -> -1}, {x -> 1}, {x -> -1/2^(1/2)}, {x -> 1/2^(1/2)}}$

Pienin arvo ja suurin arvo:

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•Tehtävä 9

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$ImplicitPlot[{y^2 == x, x - y == 3}, {x, 0, 6}]$

[Graphics:HTMLFiles/ratk6_76.gif]

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${{y -> 1/2 (1 - 13^(1/2))}, {y -> 1/2 (1 + 13^(1/2))}}$

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•Tehtävä 10

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[Graphics:HTMLFiles/ratk6_86.gif]

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Vaihtoehto 1

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${{x -> 1/2 (10 - 3^(1/2))}, {x -> 1/2 (10 + 3^(1/2))}}$

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Vaihtoehto 2

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${{x -> 1/2 (10 - (3 - y)^(1/2))}, {x -> 1/2 (10 + (3 - y)^(1/2))}}$

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$Integrate[Pi * (x /. rat2[[2]])^2 - Pi * (x /. rat2[[1]])^2, {y, 0, 3}]$

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Converted by Mathematica (August 26, 2003)