Primjeri formula
(a) Rješenje sustava jednadžbi:<br>
\(x = §§V0(1,8,1)§§\)<br>
\(y = §§V1(1,8,1)§§\)<br>
\(z = §§V2(1,8,1)§§\)<br><br>
(b) Izvod funkcije:<br>
\(f'(x) = 2\cos(2x) + e^x\)<br><br>
(c) Rješenje diferencijalne jednadžbe:<br>
\(y(x) = x^3 + x^2 - x + 2\)<br><br>
(d) Određeni integral:<br>
\(\int_0^1 x^2 \cos(x) \,dx = \frac{1}{3}(\sin(1) + \frac{2}{3})\)<br><br>
(e) Razlaganje na parcijalne razlomke:<br>
\(f(x) = \frac{1}{3} \cdot \frac{1}{x-§§V3(1,8,1)§§} - \frac{1}{2} \cdot \frac{1}{x+§§V4(1,8,1)§§}\)<br><br>
(f) Laplaceova transformacija:<br>
\(F(s) = \frac{12}{(s-§§V5(1,8,1)§§)^2 + 16}\)<br><br>
(g) Determinanta matrice:<br>
\(-10\)<br><br>
(h) Invertibilnost:<br>
\(f(x) = x^3\) je invertibilna na \(\mathbb{R}\)<br><br>
(i) Integralna jednadžba:<br>
\(y(x) = e^{x^2}\)<br><br>
(j) Kompleksni broj:<br>
\(\frac{(1+2i)(3-4i)}{2-3i} = §§V6(1,8,1)§§ + §§V7(1,8,1)§§i\)<br><br>
<hr>
<strong>Drugi dio</strong><br><br>
(a) Diferencijalna jednadžba:<br>
\(y(x) = e^{-§§V8(1,8,1)§§x} \cdot (x^2 + 2x + 3)\)<br><br>
(b) Kompleksni korijeni:<br>
\(z_1 = \pm i, z_2 = \pm i\sqrt{3}\)<br><br>
(c) Sustav diferencijalnih jednadžbi:<br>
\(x(t) = e^{-§§V9(1,8,1)§§t} \cdot \cos(2t)\)<br>
\(y(t) = e^{-§§V10(1,8,1)§§t} \cdot \sin(2t)\)<br><br>
(d) Fourierova transformacija:<br>
\(F(\omega) = \sqrt{\pi}/4 \cdot (\delta(\omega - 1) + \delta(\omega + 1) - \frac{1}{2}\delta(\omega))\)<br><br>
(e) Gradijent:<br>
\(\nabla f(1, -1, 2) = (§§V11(1,8,1)§§, §§V12(1,8,1)§§, §§V13(1,8,1)§§)\)<br><br>
(f) Linearna regresija:<br>
\(y = 2x + 1\)<br><br>
(g) Inverzna Laplaceova transformacija:<br>
\(f(t) = e^{-§§V14(1,8,1)§§t} \cdot (3t + 2)\)<br><br>
(h) Vektori:<br>
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{ §§V1(6,49,1)§§ }{ §§V6(5,9,1)§§ } = \quad \large\square + \frac{ \large\square}{ \large\square } \normalsize \qquad
(b) \quad \frac{ §§V2(-60,-6,1)§§ }{ §§V7(5,9,1)§§ } = \quad \large\square + \frac{ \large\square}{ \large\square } \normalsize \qquad
(c) \quad \frac{ §§V3(1,10,1)§§ }{ §§V8(1,5,1)§§ } = \quad \large\square + \frac{ \large\square}{ \large\square } \normalsize \ && \\
&(d) \quad \frac{ §§V4(-10,-1,1)§§ }{ §§V9(1,5,1)§§ } = \quad \large\square + \frac{ \large\square}{ \large\square } \normalsize \qquad
(e) \quad \frac{ §§V5(30,50,1)§§ }{ §§V1(1,5,1)§§ } = \quad \large\square \large\square + \frac{ \large\square}{ \large\square } \normalsize \qquad
(f) \quad \frac{ §§V6(50,99,1)§§ }{ §§V2(1,9,1)§§ } = \quad \large\square \large\square + \frac{ \large\square}{ \large\square } \normalsize && \\
&(g) \quad \frac{ §§V1(10,50,1)§§ }{ §§V6(1,9,1)§§ } = \quad \large\square + \frac{ \large\square}{ \large\square } \normalsize \qquad
(h) \quad \frac{ §§V2(2,50,2)§§ }{ §§V7(3,9,3)§§ } = \quad \large\square + \frac{ \large\square}{ \large\square } \normalsize \qquad
(i) \quad \frac{ §§V3(3,66,3)§§ }{ §§V8(1,7,2)§§ } = \quad \large\square \large\square + \frac{ \large\square}{ \large\square } \normalsize && \\
&(j) \quad \frac{ §§V4(-50,50,1)§§ }{ §§V9(2,8,2)§§ } = \quad \large\square \large\square + \frac{ \large\square}{ \large\square } \normalsize \qquad
(k) \quad \frac{ §§V5(-50,50,1)§§ }{ §§V1(1,9,1)§§ } = \quad \large\square \large\square + \frac{ \large\square}{ \large\square } \normalsize \quad
(l) \quad \frac{ §§V6(-50,50,1)§§ }{ §§V2(1,9,1)§§ } = \quad \large\square + \frac{ \large\square}{ \large\square } \normalsize \quad && \\
& \textbf{} && \\
& \textbf{Part 2} && \\
& \quad \text{ - negative zahlen} && \\
&(a) \quad \frac{ §§V1(-6,-49,-1)§§ }{ §§V6(-5,-9,-1)§§ } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \qquad
(b) \quad \frac{ §§V6(-50,99,1)§§ }{ §§V2(1,9,1)§§ } = \quad \large\square + \large\square + \frac{\large\square}{\large\square} \normalsize \qquad
(c) \quad \frac{ §§V2(-60,6,1)§§ }{ §§V7(-5,-9,-1)§§ } = \quad \large\square +
\frac{\large\square}{\large\square} \normalsize && \\
&(d) \quad \frac{ §§V4(-10,-1,1)§§ }{ §§V9(1,5,1)§§ } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \qquad
(e) \quad \frac{ §§V5(-30,-50,-1)§§ }{ §§V1(1,5,1)§§ } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize \qquad
(f) \quad \frac{ §§V6(50,99,1)§§ }{ §§V2(1,9,1)§§ } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize && \\
\\&(g) \quad \frac{ §§V1(-10,-50,-1)§§ }{ §§V6(1,9,1)§§ } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \qquad
(h) \quad \frac{ §§V2(2,50,2)§§ }{ §§V7(3,9,3)§§ } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \qquad
(i) \quad \frac{ §§V3(3,66,3)§§ }{ §§V8(1,7,2)§§ } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize && \\
&(j) \quad \frac{ §§V4(-50,50,1)§§ }{ §§V9(2,8,2)§§ } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize \qquad
(k) \quad \frac{ §§V5(-50,50,1)§§ }{ §§V1(1,9,1)§§ } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize \qquad
(l) \quad \frac{ §§V6(-50,50,1)§§ }{ §§V2(1,9,1)§§ } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \quad && \\
& \textbf{} && \\
& \textbf{Part 3} && \\
& \quad \text{ - Ein Geschenk meines Vaters :-) } && \\
&(a) \quad \frac{ §§V1(-6,-49,-1)§§ }{ §§V6(-5,-9,-1)§§ } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \qquad
(b) \quad \frac{ §§V6(-50,99,1)§§ }{ §§V2(1,9,1)§§ } = \quad \large\square + \large\square + \frac{\large\square}{\large\square} \normalsize \qquad
(c) \quad \frac{ §§V2(-60,6,1)§§ }{ §§V7(-5,-9,-1)§§ } = \quad \large\square +
\frac{\large\square}{\large\square} \normalsize && \\
&(d) \quad \frac{ §§V4(-10,-1,1)§§ }{ §§V9(1,5,1)§§ } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \qquad
(e) \quad \frac{ §§V5(-30,-50,-1)§§ }{ §§V1(1,5,1)§§ } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize \qquad
(f) \quad \frac{ §§V6(50,99,1)§§ }{ §§V2(1,9,1)§§ } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize && \\
\\&(g) \quad \frac{ §§V1(-10,-50,-1)§§ }{ §§V6(1,9,1)§§ } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \qquad
(h) \quad \frac{ §§V2(2,50,2)§§ }{ §§V7(3,9,3)§§ } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \qquad
(i) \quad \frac{ §§V3(3,66,3)§§ }{ §§V8(1,7,2)§§ } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize && \\
&(j) \quad \frac{ §§V4(-50,50,1)§§ }{ §§V9(2,8,2)§§ } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize \qquad
(k) \quad \frac{ §§V5(-50,50,1)§§ }{ §§V1(1,9,1)§§ } = \quad \large\square \large\square + \frac{\large\square}{\large\square} \normalsize \qquad
(l) \quad \frac{ §§V6(-50,50,1)§§ }{ §§V2(1,9,1)§§ } = \quad \large\square + \frac{\large\square}{\large\square} \normalsize \quad && \\
\end{flalign*}