AP Calculus AB · Unit 3 · Differentiation: Composite, Implicit, and Inverse Functions

At a highway checkpoint, the density of vehicles $k$, in cars per mile, and their average speed $s$, in miles per hour, satisfy $$k^2 s + s^3 = 128$$ for $k > 0$ and $s > 0$. Selected values of the density $k(t)$ and of $k'(t)$, $t$ minutes after monitoring begins, are given in the table.

(a) Find $\dfrac{ds}{dk}$ at the point $(k, s) = (4, 4)$.
(b) Is the graph of $s$ as a function of $k$ concave up or concave down at the point $(4, 4)$? Justify your answer.
(c) Find $\dfrac{ds}{dt}$ at time $t = 5$. Indicate units of measure.
(d) Near the point $(4, 4)$, $k$ can also be regarded as a differentiable function of $s$. Find $\dfrac{dk}{ds}$ at $s = 4$.