(a) Rješenje sustava jednadžbi:
\(x = 6\)
\(y = 8\)
\(z = 2\)

(b) Izvod funkcije:
\(f'(x) = 2\cos(2x) + e^x\)

(c) Rješenje diferencijalne jednadžbe:
\(y(x) = x^3 + x^2 - x + 2\)

(d) Određeni integral:
\(\int_0^1 x^2 \cos(x) \,dx = \frac{1}{3}(\sin(1) + \frac{2}{3})\)

(e) Razlaganje na parcijalne razlomke:
\(f(x) = \frac{1}{3} \cdot \frac{1}{x-7} - \frac{1}{2} \cdot \frac{1}{x+8}\)

(f) Laplaceova transformacija:
\(F(s) = \frac{12}{(s-8)^2 + 16}\)

(g) Determinanta matrice:
\(-10\)

(h) Invertibilnost:
\(f(x) = x^3\) je invertibilna na \(\mathbb{R}\)

(i) Integralna jednadžba:
\(y(x) = e^{x^2}\)

(j) Kompleksni broj:
\(\frac{(1+2i)(3-4i)}{2-3i} = 1 + 3i\)

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Drugi dio

(a) Diferencijalna jednadžba:
\(y(x) = e^{-8x} \cdot (x^2 + 2x + 3)\)

(b) Kompleksni korijeni:
\(z_1 = \pm i, z_2 = \pm i\sqrt{3}\)

(c) Sustav diferencijalnih jednadžbi:
\(x(t) = e^{-5t} \cdot \cos(2t)\)
\(y(t) = e^{-4t} \cdot \sin(2t)\)

(d) Fourierova transformacija:
\(F(\omega) = \sqrt{\pi}/4 \cdot (\delta(\omega - 1) + \delta(\omega + 1) - \frac{1}{2}\delta(\omega))\)

(e) Gradijent:
\(\nabla f(1, -1, 2) = (5, 3, 8)\)

(f) Linearna regresija:
\(y = 2x + 1\)

(g) Inverzna Laplaceova transformacija:
\(f(t) = e^{-3t} \cdot (3t + 2)\)

(h) Vektori:
Tangenta i normala u \(t = \pi/4\) \begin{flalign*} & \textbf{Berechnen Sie, indem Sie den Bruch in eine ganze Zahl und einen Rest umwandeln} && \\ & \quad \text{Beispiel } \frac{19}{7} = 2 + \frac{5}{7} && \\ &(a) \quad \frac{ 8 }{ 1 } = \quad \large\square + \frac{ \large\square}{ \large\square } \normalsize \qquad (b) \quad \frac{ 2 }{ 3 } = \quad \large\square + \frac{ \large\square}{ \large\square } \normalsize \qquad (c) \quad \frac{ 7 }{ 8 } = \quad \large\square + \frac{ \large\square}{ \large\square } \normalsize \ && \\ &(d) \quad \frac{ 8 }{ 5 } = \quad \large\square + \frac{ \large\square}{ \large\square } \normalsize \qquad (e) \quad \frac{ 8 }{ 8 } = \quad \large\square \large\square + \frac{ \large\square}{ \large\square } \normalsize \qquad (f) \quad \frac{ 1 }{ 2 } = \quad \large\square \large\square + \frac{ \large\square}{ \large\square } \normalsize && \\ &(g) \quad \frac{ 8 }{ 1 } = \quad \large\square + \frac{ \large\square}{ \large\square } \normalsize \qquad (h) \quad \frac{ 2 }{ 3 } = \quad \large\square + \frac{ \large\square}{ \large\square } \normalsize \qquad (i) \quad \frac{ 7 }{ 8 } = \quad \large\square \large\square + \frac{ \large\square}{ \large\square } \normalsize && \\ &(j) \quad \frac{ 8 }{ 5 } = \quad \large\square \large\square + \frac{ \large\square}{ \large\square } \normalsize \qquad (k) \quad \frac{ 8 }{ 8 } = \quad \large\square \large\square + \frac{ \large\square}{ \large\square } \normalsize \quad (l) \quad \frac{ 1 }{ 2 } = \quad \large\square + \frac{ \large\square}{ \large\square } \normalsize \quad && \\ & \textbf{} && \\ & \textbf{Part 2} && \\ & \quad \text{ - negative zahlen} && \\ &(a) \quad \frac{ 8 }{ 1 } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \qquad (b) \quad \frac{ 1 }{ 2 } = \quad \large\square + \large\square + \frac{\large\square}{\large\square} \normalsize \qquad (c) \quad \frac{ 2 }{ 3 } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize && \\ &(d) \quad \frac{ 8 }{ 5 } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \qquad (e) \quad \frac{ 8 }{ 8 } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize \qquad (f) \quad \frac{ 1 }{ 2 } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize && \\ \\&(g) \quad \frac{ 8 }{ 1 } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \qquad (h) \quad \frac{ 2 }{ 3 } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \qquad (i) \quad \frac{ 7 }{ 8 } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize && \\ &(j) \quad \frac{ 8 }{ 5 } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize \qquad (k) \quad \frac{ 8 }{ 8 } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize \qquad (l) \quad \frac{ 1 }{ 2 } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \quad && \\ & \textbf{} && \\ & \textbf{Part 3} && \\ & \quad \text{ - Ein Geschenk meines Vaters :-) } && \\ &(a) \quad \frac{ 8 }{ 1 } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \qquad (b) \quad \frac{ 1 }{ 2 } = \quad \large\square + \large\square + \frac{\large\square}{\large\square} \normalsize \qquad (c) \quad \frac{ 2 }{ 3 } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize && \\ &(d) \quad \frac{ 8 }{ 5 } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \qquad (e) \quad \frac{ 8 }{ 8 } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize \qquad (f) \quad \frac{ 1 }{ 2 } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize && \\ \\&(g) \quad \frac{ 8 }{ 1 } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \qquad (h) \quad \frac{ 2 }{ 3 } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \qquad (i) \quad \frac{ 7 }{ 8 } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize && \\ &(j) \quad \frac{ 8 }{ 5 } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize \qquad (k) \quad \frac{ 8 }{ 8 } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize \qquad (l) \quad \frac{ 1 }{ 2 } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \quad && \\ \end{flalign*}