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

Les populations cellulaires en cultures bactériennes sont souvent modélisées comme des fonctions exponentielles. Ici, P de t montre la fonction de population au temps t, C montre la population initiale, et k est la constante de croissance.La population initiale au temps zéro est donnée ; en substituant cela à la fonction de population, on obtient la valeur de C.Ce C est alors remis dans la fonction de population, laissant un paramètre inconnu.En divisant les deux côtés par 50 puis en appliquant le logarithme naturel, on obtient la valeur de k.La population est connue à 1,5 heure. En substituant des valeurs connues dans l’équation, on obtient la valeur de k.Cela donne la fonction de population avec les constantes connues.L’objectif est de déterminer le nombre de bactéries et le taux de croissance à 3 heures.Pour déterminer le taux de croissance, on prend la dérivée de la fonction de population, qui est égale à la constante de croissance multiplicée par la fonction de population elle-même.D’abord, pour trouver le nombre de bactéries à 3 heures, substituez trois dans la fonction de population. Cela donne une estimation du nombre de cellules.Ensuite, en substituant cette valeur dans la dérivée de la fonction de population, on obtient le taux de croissance.

croissance et décroissance exponentielles

Exponential Growth

Les populations bact&#xE9;riennes pr&#xE9;sentent une croissance exponentielle lorsque des conditions telles que la disponibilit&#xE9; des nutriments et la temp&#xE9;rature sont favorables. Au cours de cette phase, les cellules se reproduisent par fission binaire, processus dans lequel chaque cellule se divise en deux cellules filles identiques. Ce processus provoque un doublement de la population &#xE0; intervalles r&#xE9;guliers, ce qui se traduit par un taux de croissance directement proportionnel au nombre actuel de cellules. &#xC0; mesure que la population augmente, le nombre de nouvelles cellules form&#xE9;es &#xE0; chaque g&#xE9;n&#xE9;ration cro&#xEE;t &#xE9;galement, ce qui engendre une augmentation rapide du nombre total de cellules au fil du temps.
La croissance exponentielle est hautement pr&#xE9;visible dans des conditions constantes, car le temps de doublement demeure inchang&#xE9;, quelle que soit la taille initiale de la population. Cette propri&#xE9;t&#xE9; permet aux scientifiques d&#x2019;estimer la vitesse &#xE0; laquelle les cultures bact&#xE9;riennes se multiplient en laboratoire ou dans des proc&#xE9;d&#xE9;s industriels. Par exemple, une petite population initiale peut atteindre en quelques heures plusieurs millions de cellules si les nutriments restent abondants.
Comprendre la croissance exponentielle est essentiel dans de nombreux domaines, notamment en m&#xE9;decine, o&#xF9; elle permet d&#x2019;expliquer comment des infections peuvent s&#x2019;aggraver rapidement si elles ne sont pas trait&#xE9;es. Elle est &#xE9;galement cruciale dans les applications biotechnologiques qui reposent sur des cultures bact&#xE9;riennes pour la production. En identifiant les facteurs qui soutiennent ou limitent la croissance exponentielle, les chercheurs peuvent mieux g&#xE9;rer les populations bact&#xE9;riennes et concevoir des strat&#xE9;gies efficaces de contr&#xF4;le ou de mise en culture.

Croissance et décroissance exponentielles

Les populations bact&#xE9;riennes pr&#xE9;sentent une croissance exponentielle lorsque des conditions telles que la disponibilit&#xE9; des nutriments et ...

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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

The Product Rule helps find the derivative of a product of two changing functions. In calculus, it states that the derivative of one function times another equals the first function times the rate of change of the second, plus the second function times the rate of change of the first. This rule shows how to account for both functions when they vary at the same time. A useful way to see the rule is with a rectangle whose width and height both change over time. If the width and height represent...

the quotient rule

Differentiating Ratios with the Quotient Rule

The quotient rule is a key calculus method for differentiating ratios of two differentiable functions. It is used when a function is written as one function divided by another, and the denominator is not zero. The rule gives a direct way to find the derivative of that ratio. The quotient rule is especially useful for rational functions, trigonometric ratios, and exponential functions. For example, it can be applied to a function in ratio form to find its derivative step by step. This makes it...

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

Implicit differentiation is used to find the slope of a curve when x and y cannot be separated easily. The conchoid of Nicomedes is one example of such a curve. Its equation links x and y in a way that makes solving for one variable difficult, so a special method is needed to study the curve. To work with the conchoid, y is treated as a function of x. The chain rule is then applied to any term that contains y. The derivative is taken on both sides of the equation, which introduces dy/dx terms.<br...

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

application-of-linearization-and-approximation

Estimating Altitude Changes with Linearization

Linearization can help a drone estimate small altitude changes from air pressure readings. Atmospheric pressure drops as altitude rises, and this relationship is often modeled with an exponential function. The model is accurate, but the math needed to convert pressure into altitude can be too complex for repeated calculations during flight. To make the calculation easier, the pressure-altitude relationship is approximated locally with a straight line. This line is built near a chosen reference...

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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