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

Populações celulares em culturas bacterianas são frequentemente modeladas como funções exponenciais. Aqui, P de t mostra a função populacional no tempo t, C mostra a população inicial e k é a constante de crescimento.A população inicial no tempo zero é dada; substituindo isso na função populacional obtém o valor de C.Agora esse C é substituído de volta pela função populacional, deixando um parâmetro desconhecido.Dividindo ambos os lados por 50 e aplicando o logaritmo natural, obtém-se o valor de k.A população em 1,5 hora é conhecida. Substituindo valores conhecidos na equação, obtém-se o valor de k.Isso dá a função populacional com as constantes conhecidas.O objetivo é encontrar o número de bactérias e a taxa de crescimento em 3 horas.Para encontrar a taxa de crescimento, é feita a derivada da função populacional, que é igual à constante de crescimento multiplicada pela própria função populacional.Primeiro, para encontrar o número de bactérias em 3 horas, substitua três na função populacional. Isso fornece uma estimativa da contagem celular.Então, substituindo esse valor na derivada da função populacional obtém a taxa de crescimento.

crescimento e decaimento exponencial

Exponential Growth

Popula&#xE7;&#xF5;es bacterianas exibem crescimento exponencial quando condi&#xE7;&#xF5;es como a disponibilidade de nutrientes e a temperatura s&#xE3;o favor&#xE1;veis. Nessa fase, as c&#xE9;lulas se reproduzem por fiss&#xE3;o bin&#xE1;ria, na qual cada c&#xE9;lula se divide em duas c&#xE9;lulas-filhas id&#xEA;nticas. Esse processo faz com que a popula&#xE7;&#xE3;o dobre em intervalos regulares, resultando em uma taxa de crescimento que &#xE9; diretamente proporcional ao n&#xFA;mero atual de c&#xE9;lulas. &#xC0; medida que a popula&#xE7;&#xE3;o aumenta, o n&#xFA;mero de novas c&#xE9;lulas formadas a cada gera&#xE7;&#xE3;o tamb&#xE9;m cresce, criando um aumento acentuado no n&#xFA;mero total de c&#xE9;lulas ao longo do tempo.
O crescimento exponencial &#xE9; altamente previs&#xED;vel sob condi&#xE7;&#xF5;es constantes, uma vez que o tempo de duplica&#xE7;&#xE3;o permanece inalterado, independentemente do tamanho inicial da popula&#xE7;&#xE3;o. Essa propriedade permite que os cientistas estimem a rapidez com que as culturas bacterianas se expandir&#xE3;o em processos laboratoriais ou industriais. Por exemplo, uma pequena popula&#xE7;&#xE3;o inicial pode atingir rapidamente milh&#xF5;es de c&#xE9;lulas em poucas horas se os nutrientes permanecerem abundantes.
Compreender o crescimento exponencial &#xE9; essencial em muitas &#xE1;reas, incluindo a medicina, em que ajuda a explicar como as infec&#xE7;&#xF5;es podem se agravar rapidamente se n&#xE3;o forem tratadas. Tamb&#xE9;m &#xE9; fundamental em aplica&#xE7;&#xF5;es biotecnol&#xF3;gicas que dependem de culturas bacterianas para a produ&#xE7;&#xE3;o. Ao reconhecer os fatores que sustentam ou limitam o crescimento exponencial, os pesquisadores podem gerenciar melhor as popula&#xE7;&#xF5;es bacterianas e desenvolver estrat&#xE9;gias eficazes para seu controle ou cultivo.

Crescimento e decaimento exponencial

Popula&#xE7;&#xF5;es bacterianas exibem crescimento exponencial quando condi&#xE7;&#xF5;es como a disponibilidade de nutrientes e a temperatura s&#xE3;o ...

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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

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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

Slope and Curvature of Elliptical Arches

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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

Logarithmic functions and exponential functions are inverses of each other. If y = log b x, it can also be written as b^y = x. That inverse relationship makes implicit differentiation useful for finding derivatives of logarithmic functions. Logarithmic scales are also important for measuring data that cover many orders of magnitude. The Richter scale for earthquake magnitude and decibels for sound intensity are both examples. These scales help compare very small and very large values on the...

logarithmic differentiation

Differentiating Tire Stress with Logs

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the number e as a limit

e from Natural Log Derivatives

The number e is a key constant in calculus, and it helps describe continuous change. It is especially important for exponential growth. Rather than treating e as an arbitrary number, the transcript shows how it can be defined through the natural logarithm and differentiation. The natural logarithm, written as ln x, is the inverse of the exponential function with base e. Its derivative is 1 divided by its input. When that derivative is evaluated at x = 1, the result is exactly 1. This condition...

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

Balloon inflation is a clear example of related rates, where one changing quantity affects another through a single relationship. Related rates study how linked quantities change with respect to time. As air is added, the balloon’s volume rises, and the radius must also increase. For a hot air balloon, the inflated envelope is often modeled as a perfect sphere. This simplification makes the math easier to work with. In that model, the balloon’s volume depends directly on its radius. When air...

linearization and approximation

Tangent Line Estimates Near a Point

Linearization uses a tangent line to estimate a nonlinear function near a chosen reference point. It replaces a difficult function with a simpler linear model when the input changes only a little. This makes it useful for small deviations from a known value. A square root example shows how the method works. The value at 4 is known exactly, so 4 is a good reference point. The function value and the rate of change are both easy to find there. Estimating the value at a nearby input, such as 4.1,...

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

Hyperbolic functions help describe the shape of a hanging cable. When a flexible cable is suspended between two points at the same height, it naturally forms a catenary. That curve comes from the balance between the cable’s weight and the tension along its length. It shows a state of mechanical equilibrium. The true shape of this hanging cable is not captured well by a simple approximation. Instead, it is described with hyperbolic functions. These functions are related to exponential functions.

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

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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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