Problem 6 - Calculus

Question

Show that, for any constant \(P_{0},\) the function \(P=P_{0} e^{t}\) satisfies the equation $$\frac{d P}{d t}=P.$$

Step-by-Step Solution

Answer

The function \(P = P_0 e^t\) satisfies \(\frac{dP}{dt} = P\).

Step 1: Differentiation of the Function

Start by differentiating the function \(P = P_0 e^t\) with respect to \(t\). Here, \(P_0\) is a constant and \(e^t\) is an exponential function. The derivative of \(e^t\) with respect to \(t\) is \(e^t\). Therefore, the derivative of \(P = P_0 e^t\) is: \[\frac{dP}{dt} = P_0 e^t.\]

Step 2: Relating the Derivative to the Original Function

Now, compare the derivative \(\frac{dP}{dt} = P_0 e^t\) with the original function \(P = P_0 e^t\). Notice that they are identical, therefore the derivative \(\frac{dP}{dt}\) is equal to \(P\).

Step 3: Conclusion

Since the differentiation of the function \(P = P_0 e^t\) results in \(P_0 e^t\), which is the same as the function itself, it is proven that \(\frac{dP}{dt} = P\).

Key Concepts

Differentiation

Differentiation is a key concept in calculus that refers to finding the rate at which a function is changing at any given point. It essentially provides us with the derivative of the function. In the context of the problem, differentiation is used to determine the instantaneous rate of change of the exponential function given by \(P = P_0 e^t\) with respect to time \(t\). The process involves applying rules of differentiation, like the power rule, product rule, or chain rule, to find the derivative. For our function, since \(P_0\) is constant, and \(e^t\) is a well-known exponential term, we use the basic rule that the derivative of \(e^t\) is \(e^t\) itself. This simplicity in differentiation of exponential functions, particularly the natural exponential function \(e^t\), lies at the core of many real-world applications where things grow or decay at rates proportional to their current sizes.

Exponential Functions

Exponential functions are mathematical expressions in which a constant base, such as the Euler's number \(e\), is raised to a variable exponent. Such functions play a crucial role in modeling growth and decay processes. In the function \(P = P_0 e^t\), \(e\) is a mathematical constant approximately equal to 2.71828, and \(P_0\) serves as a constant multiplier that affects the initial size of \(P\). The variable \(t\) is typically time, demonstrating how \(P\) changes over periods. Exponential functions are unique because they have a constant relative rate of change, meaning their derivative is proportional to their current value. This non-linear, continuous growth model is what makes them vital in fields like finance, biology, and physics. For instance, compound interest, population growth, and radioactive decay are often represented using exponential models.

Derivative Verification

Verifying a derivative means confirming that the derivative found actually represents the rate of change of the initial function. This problem guides us through such verification by comparing the derivative of the function to the original function itself. In the step-by-step solution, the derivative of \(P = P_0 e^t\) is taken to be \(P_0 e^t\). During verification, we compare this result with \(P\) itself, and observe that they are identical: both are \(P_0 e^t\). Such equality is not coincidental but is characteristic of exponential growth, where the rate of change or growth is precisely equal to the current function value. Successfully verifying the derivative confirms the function behaves as expected according to the given relation \(\frac{dP}{dt} = P\). This is a critical step ensuring the accuracy and functionality of the model used in applications requiring precise change estimates over time.