1

a

f[x_] := 6 x^3 + 5 x^2 - 6 x^1 + x^0 - 2/x^2

i

D[f[x], x]

ii

D[f[x], x] /. x -> 1

iii

ArcTan[D[f[x], x]] /. x -> 1

N[%]

%/Degree

iv

Plot[
        6 x^3 + 5 x^2 - 6 x^1 + x^0 - 
          2/x^2, {x, -2, -0.5}];

Plot[-6 + (4/(x^3)) + 10 x + 18 (x^2), {x, 0.5, 2}];

Plot[-6 + (4/(x^3)) + 10 x + 18 (x^2), {x, -2, 
          2}];

NSolve[-6 + (4/(x^3)) + 10 x + 18 (x^2) == 
        Tan[Pi/6], {x}]

v

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

b

f[x_] := (E^x + Cos[x])^
          x + ((Sin[x/3 - 3])^2) /(Log[x] - x^2)

D[f[x], x]

D[f[x], x] // Simplify

D[(E^x + Cos[x])^x, x]

D[f[x], x] /. x -> (1. )

2

Remove["Global`*"]

p[1] = {0, 0}; p[2] = {1, 1}; p[3] = {2, 0}; 
    p[4] = {3, 1};
p[5] = {4, -1}; p[6] = {6, 0}; p[7] = {7, 3};

f1[x_] := 
        a1 x^3 + b1 x^2  + c1 x + d1;
f2[x_] := 
        a2 x^3 + b2 x^2  + c2 x + d2;
f3[x_] := a3 x^3 + b3 x^2  + c3 x + d3;

s1 = 
      Flatten[Solve[
          Evaluate[{(D[f1[x], x] == 0 /. x -> 0), 
              f1[0] == 0, f1[1] == 1, f1[2] == 0}], {a1, 
            b1, c1, d1}]]

f1[x_] := (a1 x^3 + b1 x^2  + c1 x + d1 /. s1)

pl1 = Plot[Evaluate[f1[x]], {x, 0, 2}];

s2 = 
      Flatten[Solve[
          Evaluate[{(D[f1[x], x] == D[f2[x], x] /. x -> 2), 
              f2[2] == 0, f2[3] == 1, 
              f2[4] == -1}], {a2, b2, c2, d2}]]

N[%]

f2[x_] := (a2 x^3 + b2 x^2  + c2 x + d2 /. s2)

pl2 = Plot[Evaluate[f2[x]], {x, 2, 4}];

s3 = 
      Flatten[Solve[
          Evaluate[{(D[f2[x], x] == D[f3[x], x] /. x -> 4), 
              f3[4] == -1, f3[6] == 0, 
              f3[7] == 3}], {a3, b3, c3, d3}]]

N[%]

f3[x_] := (a3 x^3 + b3 x^2  + c3 x + d3 /. s3)

pl3 = Plot[Evaluate[f3[x]], {x, 4, 7}];

Show[pl1, pl2, pl3];

D[f1[x], x] /. x -> 2

D[f2[x], x] /. x -> 4

D[f3[x], x] /. x -> 7

N[%]

3

f[x_] := E^x;
Plot[{f[x], -(x - 0.1614393539403694)  2/(E - 1/E) + 
            E^0.1614393539403694, x (E - 1/E)/2 + 0.985477463224239, 
          E^x, x (E - 1/E)/2, x (E - 1/E)/2 + 0.985477463224239, 
          x (E - 1/E)/2, 
          x (E - 1/E)/2 + E -  (E - 1/E)/2}, {x, -1, 1}, 
        AspectRatio -> Automatic];

-(x - 
                  0.1614393539403694)  2/(E - 1/E) + 
          E^0.1614393539403694 // Expand // N

x (E - 1/E)/2 + 0.985477463224239 // Expand // N

fl1 = 
      Solve[1.3125728575551499` - 
              0.8509181282393216` x == 0, {x}] // Flatten

x1 = x /. fl1

fl2 = 
      Solve[0.985477463224239` + 
              1.1752011936438014` x == 0, {x}] // Flatten

x2 = x /. fl2

basis = x1 - x2

(1.5425371889432675 - (-0.8385606384288044`)) 
      E^0.1614393539403694 /2

Plot[{x (E - 1/E)/
                2 + ((-1 + (E^2) + Log[(1/2) (-(1/E) + E)] - (E^2) Log[(1/2) (-(1/E) + E)])/(2 E)), -(x - 0.1614393539403694)  2/(E - 1/E) + 
            E^0.1614393539403694} , {x, -1, 2}, 
        AspectRatio -> Automatic];

Plot[{E^x, x (E - 1/E)/2, 
          x (E - 1/E)/2 + 0.985477463224239, x (E - 1/E)/2, 
          x (E - 1/E)/
                2 + ((-1 + (E^2) + Log[(1/2) (-(1/E) + E)] - (E^2) Log[(1/2) (-(1/E) + E)])/(2 E)), -(x - 0.1614393539403694)  2/(E - 1/E) + E^0.1614393539403694, 
          x (E - 1/E)/2 + E -  (E - 1/E)/2} , {x, -1, 1}, 
        AspectRatio -> Automatic];

fr = FindRoot[E^x ==  (E - 1/E)/2, {x, 1}]

x0 = x /. fr

Log[(E - 1/E)/2] // N

h = E^x0

flaeche = basis h /2

Solve[
      0.1614393539403694 (E - 1/E)/2 + b == 
        E^0.1614393539403694, {b}]

Solve[
      Log[(E - 1/E)/2] (E - 1/E)/2 + b == 
        E^Log[(E - 1/E)/2], {b}]

N[%]

4

a

Remove["Global`*"]

f[a_, x_, b_] := -a x^2 + b;
bSolv = 
      Flatten[Solve[{f[a, 0, b] == 20, f[a, 20, b] == 0}, {a, 
            b}]]

f[x_] := -1/20  x^2 + 20;

Plot[f[x], {x, -20, 20}];

b

A[x1_] := 2 (x1 f[x1]  + (20 - x1) f[x1]/2) //  Simplify;  
    A[x1]

Plot[A[x1], {x1, 0, 20}, PlotRange -> {0, 500}];

D[A[x1], x1] //  Simplify

Solve[Evaluate[D[A[x1], x1] == 0], {x1}]

N[%]

(Integrate[f[x], {x, -20, 20}] - A[x1]) 20

% // Expand

N[%]

(Integrate[f[x], {x, -20, 20}] - A[x1]) 20 /. {x1 -> 
          (20/3)}

N[%]

4

Remove["Global`*"]

a

a1 = 1; a2 = -1; a3 = 2; a4 = 12;
a[
          n_] := α n^3 + β n^2 + γ n + δ;
solv = 
      Solve[{a[1] == a1, a[2] == a2, a[3] == a3, 
            a[4] == a4}, {α, β, γ, δ}] // 
        Flatten

a[n_] := α n^3 + β n^2 + γ n + δ /. 
        solv;  a[n]

a[5]

s[n_] := Sum[a[k], {k, 1, n}];  s[n]

s[100]

s[1000]/s[999]

N[%]

b

b[1] = 1; b[n_] := (n + 1) (b[n - 1] - 1)

Table[b[n], {n, 1, 50}]

Table[b[n], {n, 1, 50}] // N

c

Limit[(E^2 - E)/
          Sum[1/k!, {k, 0, n}] (Cos[n^2] + n^2 - 2  n + 
              2)/(3  n^3 + 4  n^2 + 2  n - 1) (n + 1), 
      n -> Infinity]

Limit[Sum[1/k!, {k, 0, n}], n -> Infinity]