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f1 @ x D •• Expand -1 + 2 x + 3 x 2 + 4 x 3 + x 4

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1

Remove @ "Global`*" D

f1 @ x_ D := H x - 1 L H x + 1 L H x ^ 2 + 1 L + 4 x ^ 3 + 3 x ^ 2 + 2 x;

f1 @ x D •• Expand -1 + 2 x + 3 x 2 + 4 x 3 + x 4

f1 ' @ x D

2 + 6 x + 12 x 2 + 2 H -1 + x L x H 1 + x L + H -1 + x L H 1 + x 2 L + H 1 + x L H 1 + x 2 L

f1 ' @ x D •• Expand 2 + 6 x + 12 x 2 + 4 x 3

2

Remove @ "Global`*" D

f2 @ x_ D := Sin @ x D E ^ x Cosh @ x D ; f2 ' @ x D

ã x Cos @ x D Cosh @ x D + ã x Cosh @ x D Sin @ x D + ã x Sin @ x D Sinh @ x D

% • . 8 Cosh @ x D ® H E ^ x + E ^-x L • 2, Sinh @ x D ® H E ^ x - E ^-x L • 2 <

€€€€ 1

2 ã x Hã

-x

+ ã x L Cos @ x D + 1

€€€€ 2 ã x H-ã

-x

+ ã x L Sin @ x D + 1

€€€€ 2 ã x Hã

-x

+ ã x L Sin @ x D

% •• Simplify

€€€€ 1

2 H Cos @ x D + ã 2 x Cos @ x D + 2 ã 2 x Sin @ x DL

3

Remove @ "Global`*" D

f3 @ x_ D := H x - 1 L H x + 2 L • H x ^ 2 - 1 L ; f3 @ x D •• Simplify

2 + x

€€€€€€€€€€€€

1 + x

(2)

D @ %, x D

€€€€€€€€€€€€ 1

1 + x - 2 + x

€€€€€€€€€€€€€€€€€€€€

H 1 + x L 2

% •• Together

- 1

€€€€€€€€€€€€€€€€€€€€

H 1 + x L 2

4

Remove @ "Global`*" D

f4 @ x_ D := Cos @ Cos @ x DD ; f4 ' @ x D

Sin @ x D Sin @ Cos @ x DD

5

Remove @ "Global`*" D

f5 @ x_ D := Cos @ Cos @ Cos @ x DDD ; f5 ' @ x D

-Sin @ x D Sin @ Cos @ x DD Sin @ Cos @ Cos @ x DDD

6

Remove @ "Global`*" D

f6 @ x_ D := H 2 x L ^ H 3 x L ; f6 ' @ x D

3 2 3 x x 3 x Log @ 2 D + 2 3 x x 3 x H 3 + 3 Log @ x DL

% •• Simplify

3 8 x x 3 x H 1 + Log @ 2 D + Log @ x DL

7

Remove @ "Global`*" D Sign @ x D

Sign @ x D

(3)

f7 @ x_ D := x ^ 3 Log @ Abs @ x ^ 3 DD ; f7 ' @ x D • . Abs

¢

@ x D ® 0

3 x 2 Log @ Abs @ x D 3 D

f7 @ x_ D := x ^ 3 Log @ x ^ 3 Sign @ x ^ 3 DD ; f7 ' @ x D • . Sign

¢

@ x D ® 0 3 x 2 + 3 x 2 Log @ x 3 Sign @ x D 3 D

% •• Simplify

3 x 2 H 1 + Log @ x 3 Sign @ x D 3 DL

8

Remove @ "Global`*" D

f8 @ x_ D := 2 Sin @ 2 x D ; f8 ' @ x D

4 Cos @ 2 x D

Solve @ 4 Cos @ 2 x D Š Tan @ 30 Degree D , 8 x <D 99 x ® - 1

€€€€ 2 ArcCos A 1

€€€€€€€€€€€€€€

4 •!!!! 3 E= , 9 x ® 1

€€€€ 2 ArcCos A 1

€€€€€€€€€€€€€€

4 •!!!! 3 E==

% •• N

88 x ® -0.712976 < , 8 x ® 0.712976 <<

9

Remove @ "Global`*" D

f9 @ x_ D := E ^-x - 2 x ^ 3 + 4; f9 ' @ x D

-ã

-x

- 6 x 2

f91 @ x_ D := Evaluate @ f9 ' @ x DD w @ 0 D = 1;

w @ n_ D := w @ n - 1 D - f9 @ w @ n - 1 DD • f91 @ w @ n - 1 DD •• N

(4)

Table @8 n, w @ n D< , 8 n, 0, 9 <D •• TableForm

0 1

1 1.37185 2 1.29304 3 1.28825 4 1.28823 5 1.28823 6 1.28823 7 1.28823 8 1.28823 9 1.28823

Plot @ f9 @ x D , 8 x, 0, 2 <D ;

0.5 1 1.5 2

-10 -7.5 -5 -2.5 2.5 5

10

Remove @ "Global`*" D f10 @ x_ D := Sqrt @ 4 - x ^ 2 D ;

Dreieck10 @ x_ D := x f10 @ x D • 2;

Plot @ f10 @ x D , 8 x, 0, 2 < , AspectRatio ® Automatic D ;

0.5 1 1.5 2

0.5

1

1.5

2

(5)

Dreieck10 ' @ x D

- x 2

€€€€€€€€€€€€€€€€€€€€€€€€

2 •!!!!!!!!!!!!!! 4 - x 2 + •!!!!!!!!!!!!!! 4 - x 2

€€€€€€€€€€€€€€€€€€€€

2

Solve @ Evaluate @ Dreieck10 ' @ x D Š 0 D , 8 x <D 99 x ® - •!!!! 2 = , 9 x ® •!!!! 2 ==

% •• N

88 x ® -1.41421 < , 8 x ® 1.41421 <<

1.4142135623730951 • 2 0.707107

11

Remove @ "Global`*" D

f11 @ x_ D := Sqrt @ 4 - H x • 2 L ^ 2 D ; Dreieck11 @ x_ D := x f11 @ x D • 2;

Plot @ f11 @ x D , 8 x, 0, 4 < , AspectRatio ® Automatic D ;

1 2 3 4

0.5 1 1.5 2

Dreieck11 ' @ x D

- x 2

€€€€€€€€€€€€€€€€€€€€€€€€€€

8 "################ 4 - €€€€€ x 4

2

+ 1

€€€€ 2 $%%%%%%%%%%%%%%%%% 4 - x 2

€€€€€€€

4

Solve @ Evaluate @ Dreieck11 ' @ x D Š 0 D , 8 x <D 99 x ® -2 •!!!! 2 = , 9 x ® 2 •!!!! 2 ==

% •• N

88 x ® -2.82843 < , 8 x ® 2.82843 <<

2.8284271247461903 • 4

0.707107

(6)

12

Remove @ "Global`*" D f12 @ x_ D := x ^ 2;

t12 @ x_, x1_ D := Evaluate @ f12 @ y D + f12 ' @ y D H x - y L • . y ® x1 D ; a = 1;

Plot @8 f12 @ x D , t12 @ x, a D< , 8 x, 0, 1.4 < ,

AspectRatio ® Automatic, Epilog ® 8 Line @88 a, 0 < , 8 a, f12 @ a D<<D<D ;

0.20.40.60.8 1 1.21.4

-1 -0.5 0.5 1 1.5 2

13

Remove @ "Global`*" D f13 @ x_ D := Sin @ x D ;

h13 @ x_ D := H Pi - x - x L f13 @ x D h13 ' @ x D

H p - 2 x L Cos @ x D - 2 Sin @ x D

Solve @ Evaluate @ h13 ' @ x D Š 0 D , 8 x <D Solve @H p - 2 x L Cos @ x D - 2 Sin @ x D Š 0, 8 x <D

fr = FindRoot @ Evaluate @ h13 ' @ x D Š 0 D , 8 x, 0.5 <D 8 x ® 0.710463 <

x1 = x • . fr

0.710463

(7)

Plot @ f13 @ x D , 8 x, 0, Pi < , AspectRatio ® Automatic,

Epilog ® 8 Line @88 x1, 0 < , 8 x1, f13 @ x1 D< , 8 Pi - x1, f13 @ x1 D< , 8 Pi - x1, 0 <<D<D ;

0.5 1 1.5 2 2.5 3

0.2 0.4 0.6 0.8 1

14

Remove @ "Global`*" D

f14 @ x_ D := H x - 1 L H x - 2 L H x + 3 L H x + 5 L ; f14 @ x D •• Expand

30 - 29 x - 7 x 2 + 5 x 3 + x 4

f14 '' @ x D •• Simplify 2 H -7 + 15 x + 6 x 2 L

Solve @ Evaluate @ f14 ' @ x D Š 0 D , 8 x <D •• N •• Chop 88 x ® 1.54619 < , 8 x ® -1.1238 < , 8 x ® -4.17239 <<

sv = Solve @ Evaluate @ f14 '' @ x D Š 0 D , 8 x <D •• N •• Chop 88 x ® -2.90202 < , 8 x ® 0.402019 <<

8 x1 = x • . sv @@ 1 DD , x2 = x • . sv @@ 2 DD<

8 -2.90202, 0.402019 <

Plot @ f14 @ x D , 8 x, -6, 3 < ,

Epilog ® 8 PointSize @ 0.03 D , Point @8 x1, f14 @ x1 D<D , Point @8 x2, f14 @ x2 D<D<D ;

-6 -4 -2 2

50 100 150

15

Remove @ "Global`*" D

(8)

x : 2.3 =(x+3) : y; L^2=(x+3)^2+y^2 = (x^2+2.3^2)/x *(x+3)

f15 @ x_ D := Sqrt @ x ^ 2 + H 2.3 L ^ 2 D H x + 3 L • x; f15 @ x D H 3 + x L •!!!!!!!!!!!!!!!!!!!!!! 5.29 + x 2

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€

x

Plot[f15[x],{x,0,10}];

2 4 6 8 10

20 40 60 80

f15'[x]//Simplify -15.87 + 0. x 2 + x 3

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€

x 2 •!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 5.29 + 1. x 2

so1 = Solve[Evaluate[f15'[x]==0],{x}]

88 x ® -1.2565 - 2.17632 ä < , 8 x ® -1.2565 + 2.17632 ä < , 8 x ® 2.513 <<

xMax = x • . so1 @@ 3 DD 2.513

Laenge = Sqrt @H xMax ^ 2 + 2.3 ^ 2 LD • xMax * H xMax + 3 L 7.47345

WinkelInRad = ArcTan @ 2.3 • xMax D 0.741172

WinkelInGrad = ArcTan @ 2.3 • xMax D • Degree 42.466

Das ist fast 45 Grad, jedoch nicht exakt 45 Grad!

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