Maths - Modelling with Differentiation

Differentials as rates of change

What would the differential be called for $A = \pi r^2$?

\[\frac{dA}{dr}\]

\[A = \pi r^2 \\ \frac{dA}{dr}\]

How would you describe the differential?

The rate of change of area with respect to radius.

Can you differentiate $V = \frac{4}{3} \pi r^3$?

\[\frac{dV}{dr} = 4\pi r^2\]

\[V = \frac{4}{3} \pi r^3 \\ \frac{dV}{dr} = 4\pi r^2\]

How could you explain “the rate of change of volume with respect to radius”?

How much additional volume you gain for a small change in the radius.

This cuboid represents a tank with no top and area $54m^2$. What’s the formula for the surface area?

\[54m^2 = 2x^2 + 3xy\]

What’s the formula the volume of this cuboid?

\[x^2y\]

You have the two equations

\[A = 2x^2 + 3xy = 54m^2 \\ V = x^2y\]

How would you find the actual volume of the cuboid?

Rearrange the first formula in terms of $y$ and then substitute back into the volume formula.