Problem 44 - Calculus Early Transcendentals

Question

Suppose that a quantity \(y\) has an exponential growth model \(y=y_{0} e^{k t}\) or an exponential decay model \(y=y_{0} e^{-k t}\), and it is known that \(y=y_{1}\) if \(t=t_{1}\). In each case find a formula for \(k\) in terms of \(y_{0}, y_{1},\) and \(t_{1},\) assuming that \(t_{1} \neq 0\).

Step-by-Step Solution

Answer

The formula for \(k\) is \( k = \frac{1}{t_1} \ln\left(\frac{y_1}{y_0}\right) \).

Step 1: Determine the model (Growth or Decay)

We need to establish whether the model is exponential growth or decay. The formulas are \(y = y_0 e^{kt}\) for growth and \(y = y_0 e^{-kt}\) for decay. However, the process of finding \(k\) will be identical because we are simply solving for \(k\) regardless of the sign of the exponent. For now, we will proceed with the general form: \(y = y_0 e^{kt}\).

Step 2: Substitute known values

We are given that when \(t = t_1\), \(y = y_1\). Plug these values into the general exponential equation: \[ y_1 = y_0 e^{kt_1} \]

Step 3: Isolate the exponential term

Divide both sides of the equation by \(y_0\) to remove \(y_0\) from the right-hand side: \[ \frac{y_1}{y_0} = e^{kt_1} \]

Step 4: Solve for \(k\)

Take the natural logarithm on both sides to solve for \(k\). This gives:\[ \ln\left( \frac{y_1}{y_0} \right) = kt_1 \]Finally, solve for \(k\) by dividing both sides by \(t_1\):\[ k = \frac{1}{t_1} \ln\left( \frac{y_1}{y_0} \right) \]

Step 5: Identify application to decay

The formula derived can be used universally for both growth and decay by considering the sign of \(k\). If \(k\) is positive, it represents growth, and if negative, it represents decay. In an exponential decay case, if you proceed with \(e^{-kt}\), the derivation still holds with \(k\) as calculated, representing a decay rate.

Key Concepts

Exponential Function

An exponential function is a mathematical expression in which a variable appears in the exponent. It is written in the form \( y = y_0 e^{kt} \), where:

• \( y \) is the quantity being measured.
• \( y_0 \) is the initial amount or starting value of the quantity.
• \( e \) is the base of the natural logarithm, approximately equal to 2.718.
• \( k \) is a constant that represents the growth or decay rate.
• \( t \) represents time or any other independent variable.

Exponential functions are powerful because they model many real-world processes like population growth, radioactive decay, or interest calculation. When \( k \) is positive, the function describes exponential growth, while a negative \( k \) indicates exponential decay. This characteristic makes exponential functions particularly useful for predicting trends and changes over time.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base \( e \), where \( e \approx 2.718 \). It is used to decipher the time required for a given quantity to reach a certain size in exponential models. For the equation \( y = y_0 e^{kt} \), taking the natural logarithm of both sides helps isolate the exponent for easier manipulation:\[ \ln(y) = \ln(y_0) + kt \]Natural logarithms serve as an inverse to the exponential function, making them indispensable for solving equations where the variable is an exponent. This property is critical when determining the constant \( k \) in exponential growth or decay because it allows us to bring the exponential factor down to an algebraic one, enabling straightforward calculations and deeper insights into the rate of change.

Calculus

Calculus, a branch of mathematics focused on change, emphasizes derivatives and integrals. It equips students with tools to explore dynamic systems described by exponential functions. When solving for \( k \) in exponential growth or decay, calculus techniques like taking derivatives can be applicable depending on the complexity of the real-world problem.Calculus can help determine rates of change (like \( k \)) and how a quantity changes at any given point in time. By understanding derivatives, one can predict an increasing or decreasing trend, which can then inform long-term planning and decision-making in fields like biology, finance, and physics. Moreover, integral calculus can be used to calculate total accumulation over time, such as total population growth or decay, by evaluating these functions over specific intervals. Mastery of calculus provides a solid foundation for tackling complex calculations involving exponential functions.