Question: 7
z = cos(x)\cdot \ln(y) + 2y \text{ with } \{ x = |s| + e^t, y = s^2 + 3t \}
1. **State the problem:**
We are given the function $$z = \cos(x) \cdot \ln(y) + 2y$$ where $$x = |s| + e^t$$ and $$y = s^2 + 3t$$.
2. **Understand the variables and formula:**
- The function $$z$$ depends on $$x$$ and $$y$$.
- $$x$$ and $$y$$ themselves depend on $$s$$ and $$t$$.
- The goal is to understand or evaluate $$z$$ given $$s$$ and $$t$$.
3. **Important rules:**
- $$\cos(x)$$ is the cosine function, which takes an angle in radians.
- $$\ln(y)$$ is the natural logarithm, defined only for $$y > 0$$.
- $$|s|$$ is the absolute value of $$s$$.
- $$e^t$$ is the exponential function.
4. **Substitute $$x$$ and $$y$$ into $$z$$:**
$$
z = \cos\big(|s| + e^t\big) \cdot \ln\big(s^2 + 3t\big) + 2\big(s^2 + 3t\big)
$$
5. **Interpretation:**
- To evaluate $$z$$ for specific $$s$$ and $$t$$, first compute $$x = |s| + e^t$$.
- Then compute $$y = s^2 + 3t$$.
- Ensure $$y > 0$$ to compute $$\ln(y)$$.
- Finally, compute $$z$$ using the formula above.
6. **Example evaluation (optional):**
If $$s=1$$ and $$t=0$$:
- $$x = |1| + e^0 = 1 + 1 = 2$$
- $$y = 1^2 + 3\cdot0 = 1$$
- $$z = \cos(2) \cdot \ln(1) + 2 \cdot 1 = \cos(2) \cdot 0 + 2 = 2$$
**Final formula:**
$$
z = \cos\big(|s| + e^t\big) \cdot \ln\big(s^2 + 3t\big) + 2\big(s^2 + 3t\big)
$$