Lösungen / Solutions

1

?Limit

a

Limit[Sin[5  x^2]/Sin[x]^2, x -> 0]

b

Limit[(E^(x^2) - 1)/x^2, x -> 0]

c

Limit[((E^x)^2 - 1)/x^2, x -> 0]

2

f1[x_] := Sin[x + b];      f1[x]

f2[x_] := Sin[x] Cos[b] + Cos[x] Sin[b];      f2[x]

f1[x] == f2[x]

D[f1[x], x] == D[f2[x], x]

Additionstheoreme / Théorèmes d'addition

3

a

f[x_] := x^5/5 - x^4/4 - x^3/3 + x^2/2;    f[x]

Plot[f[x], {x, -2, 2}];

D[f[x], x]

solv1a = Solve[D[f[x], x] == 0, {x}]

D[f[x], {x, 2}]

solv2a = Solve[D[f[x], {x, 2}] == 0, {x}]

solv2a // N

Keine Symmetrie / pas de symétrie

b

f[x_] := x^4/12 - x^3/9 - x^2 + 1;    f[x]

Plot[f[x], {x, -3.5, 4.5}];

D[f[x], x]

solv1a = Solve[D[f[x], x] == 0, {x}]

D[f[x], {x, 2}]

solv2a = Solve[D[f[x], {x, 2}] == 0, {x}]

solv2a // N

Keine Symmetrie / pas de symétrie

c

f[x_] := 1/50 x^2 (x - 5) (x - 9);    f[x]

Plot[f[x], {x, -3.5, 11}];

D[f[x], x]

solv1a = Solve[D[f[x], x] == 0, {x}]

D[f[x], {x, 2}]

solv2a = Solve[D[f[x], {x, 2}] == 0, {x}]

solv2a // N

Keine Symmetrie / pas de symétrie

4

f[x_] := a x (x - 5);  f[x]

f[2.5]

solv4 = Solve[f[2.5] == 25, {a}] // Flatten

a = a /. solv4

f[x]  //  Expand

aA[x_] :=  ((5 - x) - x) f[x];  aA[x]

aA[x] // Expand

Plot[aA[x], {x, -1.5, 6.5}];

D[aA[x], x] // Expand

solv1a = Solve[D[aA[x], x] == 0, {x}]

solv1a = 
      Solve[100 - 120  x + 24 (x^2) == 
          0, {x}]

D[aA[x], {x, 2}] // Expand

solv4a = Solve[D[aA[x], {x, 2}] == 0, {x}]

5

a

Limit[Log[x]/(x - 1), x -> 1]

b

Limit[Log[x]/(x - 1)^2, x -> 1]

c

Limit[(Cos[x] - 1)/x^2, x -> 0]

d

Limit[Sin[2 x]/Cos[x]^2, x -> Pi/2]

6

f1[x_] := Sum[x^n/n!, {n, 0, 25}];  f1[x]

D[f1[x], x]

f1[x] - D[f1[x], x]

f[x_] := Sum[x^n/n!, {n, 0, Infinity}];  f[x]

D[f[x], x]

f[x] - D[f[x], x]

7

a

f[x_] := 1/6 (x^3 - 3 x^2 - 9 x + 17);    f[x]

Plot[f[x], {x, -6, 6}];

D[f[x], x]

solv1a = Solve[D[f[x], x] == 0, {x}]

D[f[x], {x, 2}]

solv2a = Solve[D[f[x], {x, 2}] == 0, {x}]

Symmetrie zum Wendepunkt/ Symétrie par rapport au point d'inflexion

b

f[x_] := 0.1 (x^4 - 2 x^3 - 12 x^2 + 8 x + 20);    f[
      x]

Plot[f[x], {x, -3.2, 4.5}];

D[f[x], x]

solv1a = Solve[D[f[x], x] == 0, {x}]

D[f[x], {x, 2}]

solv2a = Solve[D[f[x], {x, 2}] == 0, {x}]

Keine Symmetrie / pas de symétrie

c

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

Plot[f[x], {x, -2, 3}];

D[f[x], x]

solv1a = Solve[D[f[x], x] == 0, {x}]

Keine reelles Extremum / pas d'extremum réel

D[f[x], {x, 2}]

D[f[x], {x, 2}] // Together

solv2a = Solve[D[f[x], {x, 2}] == 0, {x}]

Symmetrie zu (0/0) /   Symétrie à (0/0)

d

f[x_] := E^(Sin[x]);    f[x]

Plot[f[x], {x, -7, 7}];

D[f[x], x]

solv1a = Solve[D[f[x], x] == 0, {x}]

Lösungen von cos(x)=0 periodisch mit π / Solutions de cos(x)=0, périodique avec π

D[f[x], {x, 2}]

solv2a = Solve[D[f[x], {x, 2}] == 0, {x}]

Lösungen periodisch mit 2π /    Solutions périodique avec 2 π

Symmetrie / pas de symétrie

8

V[x_] := x (16 - 2  x) (6 - 2  x) // Expand;  V[x]

Plot[V[x], {x, -2, 16}];

D[V[x], x]

solv1a = Solve[D[V[x], x] == 0, {x}]

solv1a // N

D[V[x], {x, 2}]

solv2a = Solve[D[V[x], {x, 2}] == 0, {x}]

solv2a // N