Let T be the time constant of the curve y=C e^-λt as defined in Fig. 5 . Show that T=1 / λ. [Hin

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Let T be the time constant of the curve y=C e^-λt as defined...

Let $T$ be the time constant of the curve $y=C e^{-\lambda t}$ as defined in Fig. $5 .$ Show that $T=1 / \lambda$. [Hint: Express the slope of the tangent line in Fig. 5 in terms of $C$ and $T$. Then, set this slope equal to the slope of the curve $y=C e^{-\lambda t}$ at $\left.t=0 .\right]$

Key Concepts

Exponential Decay Function

This function models processes where the rate of change is proportional to the current value. In the form y = Ce^(–?t), C represents the initial value, ? is the decay rate, and t is time. This function is foundational in many fields for describing how quantities decrease over time, such as radioactive decay, cooling processes, and charging/discharging in circuits.

Time Constant

The time constant, usually denoted as T, is a measure of the characteristic time over which an exponential process evolves. It is defined as the time required for the function to decay to 1/e (approximately 36.8%) of its initial value. In exponential decay, it provides a single value that quantifies the speed of the decay process.

Derivative of the Exponential Function

The derivative of an exponential function like y = Ce^(–?t) with respect to time gives the rate at which y changes at any moment. At t = 0, the derivative is –?C, connecting the decay rate ? with the initial slope. This concept is critical in understanding how the instantaneous rate of change relates to the overall decay behavior of the function.

Tangent Line Slope

The tangent line to a curve at a given point represents the instantaneous rate of change of the function at that point. Equating the slope of the tangent line expressed in terms of the time constant with the derivative of the exponential function at t = 0 allows one to derive the relationship between the time constant and the decay rate, showing that T = 1/?.

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