1. Zobrazte graf funkcie f(x) a jej prvých štyroch derivácií, krivky odlíšte farebne.
V bode [x0, f(x0)], v ktorom platí f(x0) = f´´(x0), nájdite rovnicu dotyčnice grafu funkcie f(x)
a farebne zobrazte grafy oboch funkcií aj dotyčnicu.

f[x_] = Exp[Sin[x]]

g[x_] = D[f[x], x]

h[x_] = D[f[x], x, x]//Simplify

k[x_] = D[f[x], x, x, x]//Simplify

l[x_] = D[f[x], x, x, x, x]//Simplify

Plot[{f[x], g[x], h[x], k[x], l[x]}, {x, -7, 5}, PlotStyle {RGBColor[.9, .1, .1], RGBColor[.1, .9, .1], RGBColor[.1, .1, .9], RGBColor[.9, .9, .1], RGBColor[.1, .9, .9]}]

FindRoot[f[x0] h[x0], {x0, 2.5}]

t[x_] = f[x0] + g[x0] * (x - x0)/.%

Plot[{f[x], h[x], t[x]}, {x, -7, 5}, PlotStyle {RGBColor[.9, .1, .1], RGBColor[.1, .9, .1], RGBColor[.1, .1, .9],}]

Sin [ x ]

Sin [ x ] Cos [ x ]

Sin [ x ] ( Cos [ x ] 2 - Sin [ x ] )

Sin [ x ] Cos [ x ] ( - 1 + Cos [ x ] 2 - 3 Sin [ x ] )

Sin [ x ] ( Cos [ x ] 4 + Sin [ x ] ( 1 + 3 Sin [ x ] ) - 2 Cos [ x ] 2 ( 2 + 3 Sin [ x ] ) )

[Graphics:HTMLFiles/cvicenie1L_10.gif]

Graphics

{ x0 3.141592653589793 }

1.0000000000000002 - 1.0000000000000002 ( - 3.141592653589793 + x )

[Graphics:HTMLFiles/cvicenie1L_11.gif]

Graphics

2. Zobrazte graf funkcie f(x) a jej prvých štyroch derivácií,krivky odlíšte farebne.
V bode [x0, f(x0)], v ktorom platí f(x0) = f´´(x0), nájdite rovnicu dotyčnice grafu funkcie f(x)
a farebne zobrazte grafy oboch funkcií aj dotyčnicu.

Clear All

f[x_] = Log[Cos[x] + 2]

g[x_] = D[f[x], x]//Simplify

h[x_] = D[f[x], x, x]//Simplify

k[x_] = D[f[x], x, x, x]//Simplify

l[x_] = D[f[x], x, x, x, x]//Simplify

Plot[{f[x], g[x], h[x], k[x], l[x]}, {x, -6.5, 6.5}, PlotStyle {RGBColor[.9, .1, .1], RGBColor[.1, .9, .1], RGBColor[.1, .1, .9], RGBColor[.9, .9, .1], RGBColor[.1, .9, .9]}]

FindRoot[f[x0] h[x0], {x0, 2.5}]

t[x_] = f[x0] + g[x0] * (x - x0)/.%

Plot[{f[x], h[x], t[x]}, {x, -5, 5}, PlotStyle {RGBColor[.9, .1, .1], RGBColor[.1, .9, .1], RGBColor[.1, .1, .9],}]

All Clear

Log [ 2 + Cos [ x ] ]

- Sin [ x ] 2 + Cos [ x ]

- 1 + 2 Cos [ x ] ( 2 + Cos [ x ] ) 2

- 2 ( - 1 + Cos [ x ] ) Sin [ x ] ( 2 + Cos [ x ] ) 3

- 2 ( - 9 - 10 Cos [ x ] + Cos [ 2 x ] ) Sin [ x 2 ] 2 ( 2 + Cos [ x ] ) 4

[Graphics:HTMLFiles/cvicenie1L_22.gif]

Graphics

{ x0 2.3616452473153435 }

0.2539051055992167 - 0.5455508437318334 ( - 2.3616452473153435 + x )

[Graphics:HTMLFiles/cvicenie1L_23.gif]

Graphics

3. Vypočítajte obsah oblasti ohraničenej grafmi funkcií f(x) a g(x).

Clear All

All Clear

f [ x_ ] = 4 * x 3 - 7 * x + 4

4 - 7 x + 4 x 3

g [ x_ ] = 5 * x * x + 2

2 + 5 x 2

Solve [ f [ x ] g [ x ] , x ]

{ { x - 1 } , { x 1 4 } , { x 2 } }

Plot [ { f [ x ] , g [ x ] } , { x , - 5 , 5 } , PlotStyle { RGBColor [ .9 , .1 , .1 ] , RGBColor [ .1 , .9 , .1 ] } ]

[Graphics:HTMLFiles/cvicenie1L_24.gif]

Graphics

- 1 1 4 ( f [ x ] - g [ x ] ) x

2375 768

1 4 2 ( g [ x ] - f [ x ] ) x

5831 768

1 4 2 ( g [ x ] - f [ x ] ) x + - 1 1 4 ( f [ x ] - g [ x ] ) x // N

- 2.665291226812883

4. Vypočítajte obsah oblasti ohraničenej grafmi funkcií k(x) a l(x).

Clear All

All Clear

k[x_] = Sin[2 * x] * Exp[x]

l[x_] = Log[x^2]

Plot[{k[x], l[x]}, {x, -2, 4}, PlotStyle {RGBColor[.9, .1, .1], RGBColor[.1, .9, .1]}]

x Sin [ 2 x ]

Log [ x 2 ]

[Graphics:HTMLFiles/cvicenie1L_28.gif]

Graphics

r1 = FindRoot[k[x] l[x], {x, -1}]

r2 = FindRoot[k[x] l[x], {x, 2}]

r3 = FindRoot[k[x] l[x], {x, 3}]

{ x - 0.798594356775926 }

{ x 1.4810061416183924 }

{ x 3.1894452405860383 }

x0=x/.r1[[1]]

- 0.798594356775926

x1=x/.r2[[1]]

1.4810061416183924

x2=x/.r3[[1]]

3.1894452405860383

x0 0 ( k [ x ] - l [ x ] ) x + 0 x1 ( k [ x ] - l [ x ] ) x + x1 x2 ( l [ x ] - k [ x ] ) x

19.635897358090173


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