Growth of bacteria

Definition

Growth of bacteria can be modeled using exponential functions due to their rapid reproduction under ideal conditions. Understanding this concept is essential for solving integrals involving exponential growth in calculus.

5 Must Know Facts For Your Next Test

1. Bacterial growth often follows an exponential pattern, which can be expressed as $N(t) = N_0 e^{kt}$ where $N(t)$ is the population at time $t$, $N_0$ is the initial population, and $k$ is the growth rate constant.
2. The integral of an exponential function describing bacterial growth can be solved using $\int e^{kx} dx = \frac{1}{k} e^{kx} + C$.
3. Logarithmic functions are used to linearize exponential growth data for easier analysis: taking the natural logarithm of both sides of $N(t) = N_0 e^{kt}$ results in $\ln(N(t)) = \ln(N_0) + kt$.
4. Doubling time for bacterial populations can be calculated using the formula $T_d = \frac{\ln(2)}{k}$, where $T_d$ is the doubling time and $k$ is the growth rate constant.
5. The definite integral $\int_{a}^{b} N_0 e^{kt} dt$ calculates the total bacterial population over a time interval from $t=a$ to $t=b$. The result is $\frac{N_0}{k}(e^{kb} - e^{ka})$.

Review Questions

Related terms

Exponential Function:

A function of the form $f(x) = ae^{bx}$ where $e$ is Euler's number, and $a$, $b$ are constants.

Logarithmic Function:

A function that is the inverse of an exponential function, typically written as $\log_b(x)$ or $\ln(x)$ for base $e$.

Integral: A fundamental concept in calculus representing area under a curve or accumulation, commonly denoted by $\int f(x) dx$.