A capacitor is a device in an electrical circuit that stores charge. In one particular circuit, the charge on the capacitor Q varies in time as shown in the figure. <IMAGE>
a. At what time is the rate of change of the charge Q' the greatest?

Verified step by step guidance

1. Step 1: Understand that the rate of change of the charge Q with respect to time is represented by the derivative Q'(t). We are looking for the time at which this derivative is the greatest.
2. Step 2: Analyze the graph of Q(t) provided in the image. The rate of change, Q'(t), is represented by the slope of the tangent line to the curve at any point.
3. Step 3: Identify the point on the graph where the slope of the tangent line is steepest. This point corresponds to the maximum value of Q'(t).
4. Step 4: If the graph is not available, consider the mathematical function Q(t) if provided. Take the derivative of Q(t) to find Q'(t) and determine where this derivative reaches its maximum value.
5. Step 5: Verify the time at which Q'(t) is greatest by checking the concavity of Q(t) around this point, ensuring it is indeed a maximum.

Key Concepts

Rate of Change

The rate of change refers to how a quantity changes over time, often represented mathematically as a derivative. In the context of the capacitor's charge Q, the rate of change Q' indicates how quickly the charge is increasing or decreasing at any given moment. Understanding this concept is crucial for determining when the charge is changing most rapidly.

Derivatives

Derivatives are fundamental tools in calculus that measure how a function changes as its input changes. For the charge Q as a function of time, the derivative Q' provides insight into the instantaneous rate of change of charge. Analyzing the derivative helps identify critical points where the rate of change reaches its maximum.

Graphical Analysis

Graphical analysis involves interpreting the visual representation of a function to understand its behavior. In this case, examining the graph of charge Q over time can reveal where the slope (representing Q') is steepest, indicating the greatest rate of change. This method is essential for visually identifying key points in the function's behavior.