Question

What is the instantaneous rate of growth of \(y\) in each of the following? \((a) ; y=e^{0.07 t}\) \((b) y=15 e^{0.03 t}\) (c) \(y=A e^{0.4 t}\) \((d) ; y=0.03 e^{t}\)

Short answer

(a) 0.07e^{0.07t}; (b) 0.45e^{0.03t}; (c) 0.4Ae^{0.4t}; (d) 0.03e^{t}

Step-by-step solution

Understanding Instantaneous Rate of Growth

The instantaneous rate of growth of a function at any point is given by its derivative with respect to time. For an exponential function of the form \(y = Ae^{kt}\), the derivative \(\frac{dy}{dt}\) represents the rate of change of \(y\) with respect to time \(t\).

Derivative of y = e^{0.07t}

We take the derivative of \(y = e^{0.07t}\) with respect to \(t\). Applying the chain rule, the derivative is \(\frac{dy}{dt} = 0.07e^{0.07t}\).

Derivative of y = 15 e^{0.03t}

For \(y = 15 e^{0.03t}\), apply the chain rule to find the derivative: \(\frac{dy}{dt} = 15 \times 0.03 e^{0.03t} = 0.45e^{0.03t}\).

Derivative of y = A e^{0.4t}

In the case of \(y = A e^{0.4t}\), the derivative will be \(\frac{dy}{dt} = A \times 0.4 e^{0.4t} = 0.4Ae^{0.4t}\).

Derivative of y = 0.03 e^{t}

For the function \(y = 0.03 e^{t}\), the derivative is straightforward: \(\frac{dy}{dt} = 0.03 e^{t}\). This is obtained by multiplying the constant 0.03 with the derivative of \(e^{t}\) which is itself.

Key concepts

Instantaneous Rate of Change

The concept of instantaneous rate of change is a fundamental idea in calculus. It refers to how fast a function's value is changing at any specific point. Imagine you're driving a car and you want to know your speed at a particular instant, not the average speed over 10 minutes. That's the type of measurement this concept addresses. In mathematics, this is done by calculating the derivative of a function. For the exponential growth functions like the ones in our exercise, the instantaneous rate of change gives us an insight into how quickly the quantity represented by the function is increasing at any given point in time. In other words, the derivative of a function provides the instantaneous rate of growth at a specific time, enabling us to predict future values and understand the behavior of natural processes and phenomena.

Derivative

The derivative is a core concept in calculus used to find the instantaneous rate of change. It is essentially the slope of the tangent line to the curve of a function at any given point. This slope tells us how steep the curve is at that point, which translates to how quickly the function's value is changing. For a simple linear function, the derivative is constant, because the slope is constant. But for more complex functions, like exponential ones, the derivative changes depending on the value of the input variable. Calculating derivatives involves specific rules, such as the power rule, product rule, and chain rule, the latter being crucial for dealing with complex exponential functions. Understanding how to compute derivatives allows us to solve a variety of practical problems, from predicting population growth to optimizing resource allocation.

Chain Rule

When dealing with complex functions, particularly in calculus, the chain rule is an essential tool. It allows us to differentiate compositions of functions. Take the function in the exercise, for instance: each function is a product of a constant and an exponential function. To differentiate it, we need to apply the chain rule. The rule essentially states that the derivative of a composite function is the derivative of the outer function multiplied by the derivative of the inner function. For the exponential function part, we know that the derivative of \(e^{kt}\) is \(ke^{kt}\). By the chain rule, to differentiate \(Ae^{kt}\), we multiply the derivative of the inner function \(e^{kt}\) by \(k\) and then by the constant \(A\), providing us with the solution for the instantaneous rate of change for each problem in the exercise.

Exponential Function

Exponential functions play a significant role in both mathematics and real-world applications, especially in processes that involve growth or decay. In an exponential function like \(y = Ae^{kt}\), \(A\) is the initial amount (or the value when \(t = 0\)), and \(kt\) represents the growth rate multiplied by time. These functions are characterized by constant percentage growth, which is why they're used to model phenomena like population growth, radioactive decay, and even interest compounding in finance. The formula shows that the function's rate of change is proportional to its current value, resulting in rapid increases or decreases as time progresses. This unique characteristic is what differentiates exponential functions from other types of growth functions, making them uniquely suited for specific scientific, economic, and technical challenges.