Exponential Growth, Decay Models, and Differential Equations

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:

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

Example: Bacterial Growth

Suppose a single bacterium of E. coli divides every 20 minutes. Under perfect conditions, the number of E. coli doubles every 20 minutes. If we start with one cell, how long will it take for the mass of E. coli to equal the mass of the Earth?

• Given: Doubling time = 20 minutes, initial amount = 1 cell

• Mass of Earth: $6 \times 10^{24}$ kg

• Mass of one E. coli cell: $1 \times 10^{-12}$ kg

• Number of cells needed: $\frac{6 \times 10^{24}}{1 \times 10^{-12}} = 6 \times 10^{36}$ cells

• Exponential equation: $2^{\frac{t}{20}} = 6 \times 10^{36}$

• Solving for t:

$\log_2(6 \times 10^{36}) = \frac{t}{20}$ $t = 20 \log_2(6 \times 10^{36}) \approx 2642.8$ minutes

Exponential Functions and Their Derivatives

Exponential functions have the form $m(t) = a \cdot b^{\frac{t}{T}}$, where $a$ is the initial amount, $b$ is the growth/decay factor, and $T$ is the time interval.

• Derivative of Exponential Function:

$m'(t) = a \cdot b^{\frac{t}{T}} \cdot \ln(b) \cdot \frac{1}{T}$

Half-Life Models

Understanding Half-Life

Half-life is the time required for a quantity to reduce to half its initial value. It is commonly used in radioactive decay and pharmacology.

• General Formula:

$\text{Amount after } n \text{ half-lives} = \text{initial amount} \times \left(\frac{1}{2}\right)^n$

Example: Radioactive Decay

An isotope has a half-life of 10 hours. If there are initially 12 grams, what is the rate of decay at $t = 6$ hours?

• Half-life: 10 hours

• Initial amount: 12 grams

• Exponential decay model: $m(t) = 12 \left(\frac{1}{2}\right)^{\frac{t}{10}}$

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

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

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

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

Example: Population Growth

Given $y'(t) = 0.12y$ and $y(0) = 12$, find the solution.

• General solution: $y(t) = C e^{0.12t}$

• Using initial condition: $y(0) = 12 \Rightarrow C = 12$

• Final solution: $y(t) = 12 e^{0.12t}$

Initial Value Problems (IVP)

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

• General IVP: $y'(t) = k y(t)$, $y(t_0) = y_0$

• Solution: $y(t) = y_0 e^{k(t-t_0)}$

Example: Population Model

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

• a) Formula for $P(t)$: $P(t) = 100,000 e^{0.3t}$

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

Summary Table: Exponential Growth and Decay Models

Model	General Formula	Derivative	Example Application
Exponential Growth	$y(t) = y_0 e^{kt}$	$y'(t) = k y_0 e^{kt}$	Population growth, compound interest
Exponential Decay	$y(t) = y_0 e^{-kt}$	$y'(t) = -k y_0 e^{-kt}$	Radioactive decay, cooling
Half-Life	$y(t) = y_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}$	$y'(t) = y_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \ln\left(\frac{1}{2}\right) \frac{1}{T_{1/2}}$	Radioactive isotopes, drug elimination

Key Concepts

• Exponential models describe processes where change is proportional to current value.

• Differential equations are used to model growth and decay mathematically.

• Initial conditions allow for unique solutions to differential equations.

• Half-life is a specific case of exponential decay.

Additional info: The notes include both general theory and worked examples, making them suitable for exam preparation in a college calculus course, specifically covering material from chapters on differential equations and applications of integration.