Here’s a simple high-school problem that my daughter and I worked on today — I write these exercises down to help us now and in future:
If the height h of a cone increases at a constant rate of 0.2cm/s and the initial height is 2cm, express the volume V of cone in terms of t and find the rate of change of V at time t.
So, the following are given:
V = pi * h**3 / 12 h = 0.2t + 2
Differentiating V w.r.t. h using rules of differentiation, we get:
(dV / dh) = pi * h**2 / 4
Similarly, differentiating h w.r.t. to t, we get:
(dh / dt) = 0.2
We know rate of change of V in terms of h, and h in turn in terms of t — from above, so, we could do this following to get rate of change V in terms of t:
(dV / dt) = (dV / dh) * (dh / dt) = (pi * h**2 / 4) * 0.2
Substituting the value of h in the above, we get the following:
(dV / dt) = (pi / 4) * (0.2t + 2)**2 * 0.2 = 0.1571 * (0.2t + 2)**2
This following is how we verified the result in Calca:
V = (pi / 12) * h**3 h(t) = 0.2t + 2 der(V, t) => 0.1571*(0.2 t + 2)**2
We’ve been using symbolic math to solve the aforementioned problem. For the programmatically inclined, there are numerical methods like finite difference method, automatic differentiation method, or symbolic computation that could be put to use (hat-tip: MRocklin).