Final answer:
da/dt = 8π when r = 2
Step-by-step explanation:
Differentiation is a fundamental concept in calculus that deals with the rate at which a quantity changes. It involves finding the derivative of a function, which represents the instantaneous rate of change of the function with respect to its independent variable. The derivative provides information about the slope or gradient of the function at any given point.
To find da/dt, the rate of change of the area a of a circle with respect to time t, we need to differentiate the area formula with respect to t and then substitute the given value of r and dr/dt.
The area formula of a circle is A = πr^2. Differentiating with respect to t, we get da/dt = 2πr(dr/dt).
Substituting r = 2 and dr/dt = 2, we have da/dt = 2π(2)(2) = 8π when r = 2.