Question

the function ( q ) defined above models the electric charge, measured in coulombs, inside a lightbulb ( t ) seconds after it is turned on. which of the following presents the method for finding the instantaneous rate of change of the lightbulbs electric charge, in coulombs per second, at time ( t = 4 )?

(a) ( q(4)=-0.213 )

(b) ( q(4)=-0.171 )

(c) ( \frac{q(4)-q(0)}{4 - 0}=0.053 )

(d) ( q(4)=0.194 )

Explanation:

Step1: Recall the definition of the instantaneous rate of change

The instantaneous rate of change of a function \(y = f(x)\) at \(x=a\) is given by \(f^{\prime}(a)\). In the context of the function \(Q(t)\) (where \(Q(t)\) represents electric charge as a function of time \(t\)), the instantaneous rate of change of the electric charge with respect to time is given by the first - derivative of \(Q(t)\) evaluated at \(t = 4\).

Step2: Analyze each option

• Option A: \(Q''(4)\) is the second - derivative of \(Q(t)\) at \(t = 4\). The second - derivative represents the rate of change of the rate of change (acceleration - like concept for the function \(Q(t)\)), not the instantaneous rate of change of \(Q(t)\) itself.
• Option B: \(Q^{\prime}(4)\) is the first - derivative of \(Q(t)\) evaluated at \(t = 4\). By the definition of the derivative as the instantaneous rate of change (for a function \(y = Q(t)\), \(\frac{dQ}{dt}\) gives the rate of change of \(Q\) with respect to \(t\)), this is the correct formula for the instantaneous rate of change of \(Q(t)\) at \(t = 4\).
• Option C: \(\frac{Q(4)-Q(0)}{4 - 0}\) is the average rate of change of \(Q(t)\) over the interval \([0,4]\). The average rate of change is calculated by \(\frac{\Delta Q}{\Delta t}=\frac{Q(t_2)-Q(t_1)}{t_2 - t_1}\) (here \(t_1 = 0\) and \(t_2=4\)), not the instantaneous rate of change.
• Option D: \(Q(4)\) is the value of the function \(Q(t)\) at \(t = 4\), which represents the electric charge at \(t = 4\) seconds, not the rate of change of the electric charge.

So, the method for finding the instantaneous rate of change of the lightbulb's electric charge at \(t = 4\) is \(Q^{\prime}(4)\).

Answer:

B. \(Q'(4)= - 0.171\)