Exponential Growth and Decay Models

Introduction to Exponential Models

Exponential growth and decay models are used to describe processes where the rate of change of a quantity is proportional to the current amount. These models are fundamental in calculus and have applications in biology, chemistry, physics, and finance.

• Exponential Growth: Occurs when the rate of increase of a quantity is proportional to its current value.

• Exponential Decay: Occurs when the rate of decrease of a quantity is proportional to its current value.

General Formula:

$\text{Amount} = \text{initial amount} \times (\text{rate})^{t/\text{time unit}}$

Example: Bacterial Growth

Suppose a single bacterium of E. coli divides every 20 minutes. Under perfect conditions, the number of E. coli cells doubles every 20 minutes. If we start with one cell, the amount after time $t$ (in minutes) is:

• Initial amount: $1$

• Rate: $2$ (since the population doubles)

• Formula: $m(t) = 1 \times 2^{t/20}$

Application: How long until the mass of E. coli equals the mass of the Earth ($6 \times 10^{24}$ kg), given each cell has a mass of $10^{-12}$ kg?

• Number of cells needed: $\frac{6 \times 10^{24}}{10^{-12}} = 6 \times 10^{36}$

• Set $2^{t/20} = 6 \times 10^{36}$

• Take logarithms: $\log_2(6 \times 10^{36}) = \frac{t}{20}$

• Solve for $t$: $t = 20 \log_2(6 \times 10^{36}) \approx 2642.8$ minutes

Differentiation of Exponential Functions

To find the rate of change of an exponential function, differentiate with respect to time:

• If $m(t) = a \cdot b^{t/k}$, then $m'(t) = a \cdot b^{t/k} \cdot \ln(b) \cdot \frac{1}{k}$

For the bacterial example:

• $m(t) = 2^{t/20}$

• $m'(t) = 2^{t/20} \cdot \ln(2) \cdot \frac{1}{20}$

Half-Life Models

Definition and Formula

The half-life of a substance is the time required for half of the initial amount to decay. This concept is widely used in radioactive decay and pharmacology.

• General Formula: $\text{Amount} = \text{initial amount} \times \left(\frac{1}{2}\right)^{t/\text{half-life}}$

Example: Radioactive Decay

An isotope has a half-life of 10 hours. If there are initially 12 grams, the amount after $t$ hours is:

• $m(t) = 12 \left(\frac{1}{2}\right)^{t/10}$

• Rate of decay at $t = 6$ hours:

• $m'(t) = 12 \left(\frac{1}{2}\right)^{t/10} \ln\left(\frac{1}{2}\right) \frac{1}{10}$

• At $t = 6$: $m'(6) = 12 \left(\frac{1}{2}\right)^{6/10} \ln\left(\frac{1}{2}\right) \frac{1}{10}$

Differential Equations for Exponential Growth and Decay

Introduction to Differential Equations

A differential equation relates a function to its derivatives. In exponential growth and decay, the rate of change of a quantity is proportional to the quantity itself.

• General form: $\frac{dy}{dt} = k y(t)$, where $k$ is a constant

• Population growth: $k > 0$

• Population decay: $k < 0$

Solving Differential Equations

The solution to $\frac{dy}{dt} = k y(t)$ is:

• $y(t) = C e^{kt}$, where $C$ is a constant determined by initial conditions

Example: Given $y'(t) = 0.12y$ and $y(0) = 12$, the solution is:

• $y(t) = 12 e^{0.12t}$

Initial Value Problems (IVP)

An initial value problem specifies the value of the function at a particular time, allowing for a unique solution.

• If $y'(t) = k y(t)$ and $y(0) = y_0$, then $y(t) = y_0 e^{kt}$

Application: Population Growth

Suppose a population $P$ grows at a rate equal to $0.3$ times the population. If $P(0) = 100,000$:

• General solution: $P(t) = 100,000 e^{0.3t}$

• Population after 4 years: $P(4) = 100,000 e^{0.3 \times 4}$

Summary Table: Exponential Growth and Decay Models

Key Concepts and Definitions

• Exponential Function: A function of the form $f(x) = a b^{x}$, where $a$ and $b$ are constants.

• Half-Life: The time required for a quantity to reduce to half its initial value.

• Differential Equation: An equation involving derivatives of a function.

• Initial Value Problem: A differential equation together with a specified value at a given point.

Summary

Exponential growth and decay models, along with their associated differential equations, are essential tools in calculus for modeling real-world phenomena. Understanding how to set up, solve, and interpret these models is crucial for applications in science and engineering.