- Started with
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
22.8
- Using strategy
rm 22.8
- Applied exp-neg to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
22.8
- Applied un-div-inv to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
22.8
- Applied exp-neg to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{red}{e^{-\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
22.8
- Applied flip-+ to get
\[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2}{1 - \frac{1}{\varepsilon}}} \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
23.0
- Applied frac-times to get
\[\frac{\color{red}{\frac{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2}{1 - \frac{1}{\varepsilon}} \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\frac{\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot 1}{\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
23.3
- Applied frac-sub to get
\[\frac{\color{red}{\frac{\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot 1}{\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\color{blue}{\frac{\left(\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - \left(\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)}{\left(\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
23.3
- Applied simplify to get
\[\frac{\frac{\left(\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - \left(\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)}{\color{red}{\left(\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\frac{\left(\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - \left(\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)}{\color{blue}{{\left(e^{x}\right)}^{\left(\left(1 - 0\right) + 1\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)}}}{2}\]
23.3
- Applied taylor to get
\[\frac{\frac{\left(\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - \left(\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)}{{\left(e^{x}\right)}^{\left(\left(1 - 0\right) + 1\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)}}{2} \leadsto \frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2}\]
0.1
- Taylor expanded around 0 to get
\[\frac{\color{red}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2} \leadsto \frac{\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2}\]
0.1
- Applied simplify to get
\[\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2} \leadsto \frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}\]
0.2
- Applied final simplification
- Applied simplify to get
\[\color{red}{\frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}} \leadsto \color{blue}{\frac{{x}^3 \cdot \frac{2}{3} + \left(2 - x \cdot x\right)}{2}}\]
0.2
