Name: bacterial doubling in exponential growth
Uploaded: 2026-01-12T12:24:51.000Z
Duration: 1 min 29 s
Channel: JoVE

Le popolazioni cellulari nelle colture batteriche sono spesso modellate come funzioni esponenziali. Qui, P di t mostra la funzione della popolazione al tempo t, C mostra la popolazione iniziale e k è la costante di crescita.La popolazione iniziale al tempo zero è indicata; sostituendo questo nella funzione di popolazione si ottiene il valore di C.Ora questo C viene sostituito nuovamente nella funzione di popolazione, lasciando un parametro sconosciuto.Dividendo entrambi i lati per 50 e applicando il logaritmo naturale si ottiene il valore di k.La popolazione a 1,5 ore è nota. Sostituendo valori noti nell'equazione si ottiene il valore di k.Questo fornisce la funzione popolazione con le costanti note.L'obiettivo è trovare il numero di batteri e la velocità di crescita a 3 ore.Per trovare il tasso di crescita, si prende la derivata della funzione popolazione, che è uguale alla costante di crescita moltiplicata per la funzione popolazione stessa.Per prima cosa, per trovare il numero di batteri a 3 ore, sostituisci tre nella funzione di popolazione. Questo fornisce una stima del conteggio cellulare.Poi, sostituendo questo valore nella derivata della funzione popolazione si ottiene il tasso di crescita.

crescita e decrescita esponenziali

Exponential Growth

Le popolazioni batteriche presentano una crescita esponenziale quando condizioni quali la disponibilit&#xE0; di nutrienti e la temperatura sono favorevoli. In questa fase, le cellule si riproducono mediante fissione binaria, in cui ciascuna cellula si divide in due cellule figlie geneticamente identiche. Questo processo fa s&#xEC; che la popolazione raddoppi a intervalli regolari, determinando un tasso di crescita direttamente proporzionale al numero attuale di cellule. Con l&#x2019;aumentare della popolazione, cresce anche il numero di nuove cellule prodotte a ogni generazione, con conseguente aumento rapido del numero totale di cellule nel tempo.
La crescita esponenziale &#xE8; altamente prevedibile in condizioni costanti, poich&#xE9; il tempo di raddoppio rimane invariato indipendentemente dalla dimensione iniziale della popolazione. Questa propriet&#xE0; consente agli scienziati di stimare con quale rapidit&#xE0; le colture batteriche aumenteranno in laboratorio o nei processi industriali. Ad esempio, una piccola popolazione iniziale pu&#xF2; raggiungere rapidamente milioni di cellule in poche ore se i nutrienti restano abbondanti.
Comprendere la crescita esponenziale &#xE8; essenziale in molti ambiti, tra cui la medicina, dove aiuta a spiegare come le infezioni possano peggiorare rapidamente se non trattate. &#xC8; inoltre cruciale nelle applicazioni biotecnologiche che si basano su colture batteriche per la produzione. Riconoscendo i fattori che sostengono o limitano la crescita esponenziale, i ricercatori possono gestire meglio le popolazioni batteriche e progettare strategie efficaci per il controllo o la coltivazione.

Crescita e decrescita esponenziali

Le popolazioni batteriche presentano una crescita esponenziale quando condizioni quali la disponibilit&#xE0; di nutrienti e la temperatura sono ...

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bacterial doubling in exponential growth

Bacterial populations can grow by doubling when conditions are favorable. When nutrients and temperature support growth, cells divide by binary fission. In binary fission, one cell splits into two identical daughter cells. This makes the population rise quickly.

During exponential growth, the growth rate depends on how many cells are already present. As the population gets larger, more new cells are formed in each generation. The result is a steep increase in total cell count over time. The doubling time stays the same as long as conditions remain constant.

This pattern is predictable, which helps scientists estimate how fast bacterial cultures will expand in lab or industrial settings. A small starting population can reach millions of cells within a few hours if nutrients stay abundant. Exponential growth also matters in medicine because infections can escalate quickly if they are not treated. It is important in biotechnology as well, where bacteria are used for production. By understanding what supports or limits this growth, researchers can better control or culture bacterial populations.

Bacterial Doubling in Exponential Growth

derivatives of simple functions

Slope Rules for Simple Functions

Derivatives show how fast a function changes, and they also describe slope. A derivative can tell the slope of a straight line or the slope of a tangent line to a curve at one point. In this lesson, the idea is tied to a roller coaster track, where the derivative of the horizontal position tells how steep the ride is at each spot. A flat roller coaster section can be modeled with a constant function, f(x) = c, where c is a fixed height. Its derivative is f'(x) = 0, so the slope is zero and...

the product rule

Derivative of a Changing Rectangle

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the quotient rule

Differentiating Ratios with the Quotient Rule

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derivatives of the trigonometric functions

derivatives-of-the-trigonometric-functions

Trig Derivatives in Ferris Wheel Motion

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limits on trigonometric functions

Trigonometric Limits for Derivatives

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the chain rule

Chain Rule in a Gear System

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the chain rule: problem solving

Chain Rule in Thermal Expansion

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implicit differentiation

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second derivatives of implicit functions

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implicit differentiation: problem solving

Tangent Lines with Implicit Differentiation

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derivatives of logarithmic functions

Natural Log Derivatives and Real-World Uses

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logarithmic differentiation

Differentiating Tire Stress with Logs

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the number e as a limit

e from Natural Log Derivatives

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derivatives of inverse trigonometric functions

derivatives-of-inverse-trigonometric-functions

Inverse Trig Derivatives in Tracking

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application of rates of change

Velocity and Acceleration in Motion Graphs

Velocity and acceleration describe how a car moves along a highway. The car’s position changes over time, so it can be written as a function of time. Rates of change help measure motion in a clear mathematical way. Velocity tells how position changes with time. Average velocity is found by dividing the change in position by the change in time. It gives the overall speed of the car during a time interval, but it does not show changes inside that interval. To find the velocity at one exact...

exponential growth

Bacterial populations can grow by doubling when conditions are favorable. When nutrients and temperature support growth, cells divide by binary fission. In binary fission, one cell splits into two identical daughter cells. This makes the population rise quickly. During exponential growth, the growth rate depends on how many cells are already present. As the population gets larger, more new cells are formed in each generation. The result is a steep increase in total cell count over time. The...

related rates

Balloon Inflation and Changing Rates

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linearization and approximation

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application of linearization and approximation

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Estimating Altitude Changes with Linearization

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hyperbolic functions

Modeling Hanging Cables with Hyperbolic Curves

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inverse hyperbolic functions and their derivatives

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Deriving the Slope of a Catenary Curve

A catenary curve describes the shape of a suspension bridge cable hanging under its own weight. The curve is modeled with the hyperbolic cosine function, which helps show how gravity and tension act along the cable. When the vertical position on the cable is known, the matching horizontal position can be found with the inverse hyperbolic cosine function. Inverse hyperbolic functions are a family of functions that includes inverse hyperbolic sine, cosine, and tangent. Their corresponding...

hyperbolic and inverse hyperbolic functions: problem solving

hyperbolic-and-inverse-hyperbolic-functions-problem-solving

Tangent Lines on a Hyperbolic Arch

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key terms and concepts

Learn how bacteria double by binary fission during exponential growth, why the population rises so fast, and how this matters in labs, medicine, and biotechnology.

Bacterial Doubling in Exponential Growth | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Bacterial Doubling in Exponential Growth

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