Mathematical modelling with functions

A quantity measured every hour gives $5, 15, 45, 135, 405$. Which kind of model fits these values exactly?

A table of a quadratic model shows $2, 5, 10, 17, 26$ at $n = 0, 1, 2, 3, 4$. What value does the model give at $n = 5$?

A $3$ km taxi ride costs $340$ shillings and an $8$ km ride costs $740$ shillings. Assuming a fixed fee plus a constant charge per km, what is the fixed fee, in shillings?

A town's population grows continuously at $3.5\%$ a year, $P(t) = P_0 e^{0.035t}$. About how long does it take to double?

In the model $d(v) = v\,t_r + \dfrac{v^2}{2a}$, the braking part is $\dfrac{v^2}{2a}$. If a driver doubles the speed (same road, same brakes), what happens to the braking distance?

A student fits a polynomial of degree $9$ through all $10$ data points of a sales record, with zero error. What can you conclude about its forecast for next year?

In the proof that the best intercept is $\bar r$ we expand $\sum (r_i - c)^2 = \sum (r_i - \bar r)^2 + 2(\bar r - c)\sum (r_i - \bar r) + n(\bar r - c)^2$. Why does the middle term vanish?

Which Lean statement says “if each value is $b$ times the previous one, the model is exponential”? (Here u : ℕ → ℝ lists the measurements.)

Write fit_line(xs, ys) that returns the least-squares line as a pair (m, c), using $m = S_{xy}/S_{xx}$ and $c = \bar y - m\bar x$, where $S_{xx} = \sum (x_i - \bar x)^2$ and $S_{xy} = \sum (x_i - \bar x)(y_i - \bar y)$.