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Let $\delta >0$ be the delay in your departure time from work. Also, we can model your remaining commute time $T$ as a function of the current time and your current position $(t,x_t): T(t,x_t)$. We will assume that $x_t$ is continuous (no teleportation allowed!) and monotonically increasing (no backtracking). Your coworkers' conjecture (let's call it ...

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This is an educated guess based on experience. For non-linear/non-homogeneous/non-autonomous ODEs there is no unifying approach to solve such an equation in general. Your ODE is a so called Euler equation, for which it is known that the substitution $P=S^\lambda$ "works". Normally for such a guess, the German word "Ansatz" is used, which Wikipedia defines ...

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From what I can see it's correct: $$P_3 = P_0 \underbrace{\cdot e^{r_1} \cdot e^{r_2} \cdot e^{r_3}}_\text{one factor per year} = P_0 \cdot e^{r_1 + r_2 + r_3} = P_0 \cdot e^{0.15} \\ P_0 = P_3 \cdot e^{-0.15} \approx 8607.08€$$

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Mechanical CAD systems typically use a combination of different curve and surface types. The most important ones are quadrics (especially cylinders and cones) and free-form NURBS curves and surfaces. Look up Siemens NX or Catia or SolidWorks. They all use more-or-less the same same types of geometry. For engineering calculations (stress, heat flow, etc.) ...

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By definition, probability measure $W$ has density $f: [a,b] \to \mathbb R$ if and only if $$W(a,x)=\int_a^x f(t) dt,\quad x \in (a,b)$$ Your course may be using slightly different (but equivalent) definition of the density of probability measure; your question does not contain this definition — see more about this at the end of this answer. If ...

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