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1

a

f[x_] := Log[2  x + E]  + (2  x + 2)/(x + 2)

Plot[f[x], {x, 0, 2}];

ArcTan[(f[2] - f[0])/(2 - 0)]

b1 = (f[2] - f[0])/(2 - 0) // N

a1 = ArcTan[b1] // N

a1 / Degree

ArcTan[(f'[x] /. x -> 1)]

b2 = (f'[x] /. x -> 1) // N

a2 = ArcTan[b2] // N

a2/Degree

(b1 - b2)/b1 100

(a1 - a2)/a1 100

g[x_] := 
      f[1] + Evaluate[(f'[v] /. v -> 1)] (x - 1)

s[x_] := f[0] + (f[2] - f[0])/(2 - 0) x

Plot[{f[x], g[x], s[x]}, {x, 0, 2}];

Plot[{g[x], s[x]}, {x, 0, 6}];

Solve[g[x] == s[x], {x}]

Solve[g[x] == s[x], {x}] // N

g[4.238482789359839`]

b

h[x_] := -Sqrt[(x^3)^2] + 8

Plot[h[x], {x, -2, 2}, PlotRange -> {0, 8.1}];

h[2]

Integrate[h[x], {x, -2, 2}]

rechteck[x_] :=  2  h[x] x;
rechteck[x]

rechteck'[x]

Solve[rechteck'[x] == 0, {x}]

Solve[rechteck'[x] == 0, {x}] // N

rechteck[1.2599210498948732`]

rechteck[1.2599210498948732`]/
      Integrate[h[x], {x, -2, 2}]

2

f[x_] := Cos[1/(2 - x^2)];  f[x]

Plot[f[x], {x, -1, 1}, PlotRange -> {0, 1}];

f'[x]

NIntegrate[
      Sqrt[1 + (Evaluate[f'[x]])^2], {x, -1, 1}]

NIntegrate[
      2  f[x] Pi Sqrt[1 + (Evaluate[f'[x]])^2], {x, -1, 
        1}]

NIntegrate[f[x]^2 Pi , {x, -1, 1}]

Solve[
      Evaluate[NIntegrate[f[x]^2 Pi , {x, -1, 1}]] == 
        r^2 Pi 2, {r}]

(0.8096308466898374`)/f[0] 100

3

Remove["Global`*"]

x0Vec = {44.36, 36.85}

x0 = x0Vec[[1]]

y0 = x0Vec[[2]]

A[λ_, φ_] := λ {{Cos[φ], -Sin[φ]}, {Sin[φ], Cos[φ]}};  
    A[λ, φ] // MatrixForm

uv[x_, y_, x0_, y0_, λ_, φ_] := 
      A[λ, φ] . {x - x0, y - y0}

uv[x, y, x0, y0, λ, φ]

uv[x0, y0, x0, y0, λ, φ]

uv[x0, y0, x0, y0, λ, φ] // Chop

{x1, y1}={28.96, 43.92}

{u1, v1}={7.32, 12.88}

Solve[
      uv[x1, y1, x0, y0, λ, φ] == {u1, 
          v1}, {λ, φ}]

uv[x_, y_] := 
      A[0.8742663363204899, -1.6572100129016138] . {x - x0, 
          y - y0}

uv[x, y] // Simplify

uv[44.36, 36.85]

uv[x1, y1]

uv[10, 20]

uv[18.57, 24.24]

Solve[uv[x, y] == {28, 15}, {x, y}]

4

Remove["Global`*"]

a1 = {1, 0, 2};   a2 = {2, 1, 1};  
    a3 = {1, 2, 3};

    mD = {{7, 0, 0}, {0, 14, 0}, {0, 0, -7}}; 
    B = Transpose[{a1, a2, a3}] . mD . 
        Inverse[Transpose[{a1, a2, a3}]];

    B // MatrixForm

B . a1

B . a2

B . a3

Det[{a1, a2, a3}] == 0

Eigensystem[B]

Eigensystem[Inverse[B]]

N[%]

B . (v1 a1 + v2 a2 +  v3 a3) // Simplify

a1 = {1, 0, 2};   a2 = {2, 1, 1};  
    a3 = {1, 2, 3};

    pD = {{1, 0, 0}, {0, 1, 0}, {0, 0, 0}}; 
    mC = Transpose[{a1, a2, a3}] . pD . 
        Inverse[Transpose[{a1, a2, a3}]];

    mC // MatrixForm

N[%] // MatrixForm

p1 = {1, 1, 1}; p2 = {4, 4, 4}; p3 = {5, 12, 20};

mC . p1

N[%]

mC . p2

N[%]

mC . p3

N[%]

Norm[Cross[p1 - p2, p3 - p2]]/Norm[p3 - p2]

u1 = N[%]

Norm[Cross[mC . p1 - mC . p2, mC . p3 - mC . p2]]/
      Norm[mC . p3 - mC . p2]

u2 = N[%]

u2/u1

5

Remove["Global`*"]

q[x_] := 10 - (E^(3  x/10) + E^(-3 x/10))/2

Plot[q[x], {x, -10, 10}];

Plot[q[x], {x, 9.9, 10.1}];

fr = FindRoot[q[x] == 0, {x, 10}]

ns = x /. fr[[1]]

FT = Integrate[q[x], {x, -ns, ns}]

VT = 2 * FT *40000

VP = 230.3^2  146.6/3

Solve[VT == s  VP, {s}]

solv = 
      Solve[VT == (500^3 - r^3) Pi 4/3, {r}] // 
        Flatten

500 - r /. solv[[3]]

f[x_] := Normal[Series[q[x], {x, 0, 6}]]

f[x]

N[%]

f[x] - q[x] /. x -> 10.

Plot[Evaluate[{f[x] - q[x]}], {x, -10, 10}];

f[x] - q[x] /. x -> ns

Plot[
        Evaluate[{f[x], q[x]}], {x, -10, 10}];

Plot[
        Evaluate[{f[x], q[x]}], {x, 9.5, 10.5}];

6

Remove["Global`*"]

a

DSolve[{y'[x] == x^2/2 y[x], y[0] == 1}, {y}, 
      x]

(E^(((x^3)/6))) /. x -> 1

N[%]

b

p[x_, a_, b_, c_, d_] := a x^3 + b x^2 + c x + d

D[p[x, a, b, c, d], {x}] /. x -> 0

D[p[x, a, b, c, d], {x}] /. x -> -1

{p[-2, a, b, c, d] == 0, 
      p[0, a, b, c, d] == 
        1, (D[p[x, a, b, c, d], {x}] /. x -> 0) == 
        0, 
(D[p[x, a, b, c, d], {x}] /. 
            x -> -1) == 0}

Solve[{-8 a + 4 b == 0, 
        3 a - 2 b == 0
}, {a, b}]

solv = 
      Solve[{p[-2, a, b, c, d] == 0, 
            p[0, a, b, c, d] == 
              1, (D[p[x, a, b, c, d], {x}] /. x -> 0) == 
              0, 
(D[p[x, a, b, c, d], {x}] /. 
                  x -> -1) == 0}, {a, b, c, d}] // 
        Flatten

N[%]

p[x_] := p[x, a, b, c, d] /. solv

Plot[p[x], {x, -2.5, 1}, 
        PlotRange -> {-1, 2}];

p'[x]

N[%]

p'[x] /. x -> -2

ArcTan[(p'[x] /. x -> -2)]/Degree  // N