Name: exponential growth and decay models
Uploaded: 2025-11-26T05:43:46.000Z
Duration: 1 min 29 s
Channel: JoVE

Funkcja wykładnicza jest zdefiniowana przez dodatnią podstawę — nie równą jedności — podniesioną do wykładnika liczby rzeczywistej.Jego ogólna forma to wartość początkowa pomnożona przez podstawę podniesioną do wykładnika.Gdy podstawa jest większa niż jeden, funkcja modeluje wzrost. Gdy mieści się w przedziale od zera do jednego, niewłącznie, modeluje zanik.Weźmy na przykład pod uwagę lek, który po wstrzyknięciu osiąga początkowe stężenie 30 mikrogramów na mililitr w krwiobiegu i zachowuje 75 procent swojego stężenia co godzinę.W tym przypadku funkcja wykładnicza to 30 pomnożone przez 0,75 podniesione do wykładnika t, reprezentującego upływ godzin.Po godzinie stężenie spada do 22,5 mikrograma na mililitr. Po dwóch godzinach staje się 16,9 mikrograma na mililitr i wzór jest kontynuowany.Na wykresie krzywa najpierw gwałtownie opada, a następnie stopniowo się spłaszcza, zbliżając się do poziomej asymptoty na zero.Ponieważ czas nie może być ujemny, dziedziną są nieujemne liczby rzeczywiste. Ponieważ stężenie nigdy nie osiąga zera, zakres obejmuje tylko wartości dodatnie.

wprowadzenie do funkcji wykładniczych

Funkcje wyk&#322;adnicze maj&#261; fundamentalne znaczenie w modelowaniu proces&#243;w dynamicznych, w kt&#243;rych tempo zmian jest proporcjonalne do warto&#347;ci bie&#380;&#261;cej. Definiowane wzorem f(x) = b^x, gdzie b jest dodatni&#261; sta&#322;&#261; r&#243;&#380;n&#261; od jedno&#347;ci, stanowi&#261; podstaw&#281; opisu proces&#243;w wzrostu i zaniku zale&#380;nie od tego, czy podstawa b jest wi&#281;ksza, czy mniejsza od jedno&#347;ci.Modele wyk&#322;adnicze opisuj&#261; sytuacje, w kt&#243;rych zmiany zachodz&#261; z szybko&#347;ci&#261; proporcjonaln&#261; do warto&#347;ci bie&#380;&#261;cej. Nale&#380;&#261; do nich takie zjawiska, jak namna&#380;anie bakterii, rozpad substancji radioaktywnych oraz deprecjacja aktyw&#243;w. Og&#243;ln&#261; posta&#263; modelu wyk&#322;adniczego mo&#380;na zapisa&#263; jako:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/84;' src='https://cloudfront.jove.com/files/ftp_upload/40061/40061_Equation_1_editor_40061.svg' alt='40061_Equation_1.svg' /></figure>W tym uj&#281;ciu A_0 to warto&#347;&#263; pocz&#261;tkowa, b to podstawa okre&#347;laj&#261;ca czynnik wzrostu (b &gt; 1) lub spadku (0 &lt; b &lt; 1), a t reprezentuje czas lub inn&#261; zmienn&#261; ci&#261;g&#322;&#261;.Przekszta&#322;cenia, takie jak przesuni&#281;cia, odbicia i rozci&#261;gni&#281;cia, zmieniaj&#261; graficzn&#261; reprezentacj&#281; funkcji wyk&#322;adniczych, lecz zachowuj&#261; ich zasadniczy kszta&#322;t i w&#322;asno&#347;ci asymptotyczne. Ponadto w&#322;asno&#347;&#263; r&#243;&#380;nowarto&#347;ciowo&#347;ci funkcji wyk&#322;adniczych zapewnia, &#380;e je&#347;li b^x = b^y, to x = y, co umo&#380;liwia rozwi&#261;zywanie r&#243;wna&#324; z wyk&#322;adnikami poprzez por&#243;wnanie podstaw.

Wprowadzenie do funkcji wykładniczych

Funkcje wyk&#322;adnicze maj&#261; fundamentalne znaczenie w modelowaniu proces&#243;w dynamicznych, w kt&#243;rych tempo zmian jest proporcjonalne do ...

Education

JoVE Core

Math Fundamentals

Podstawy matematyki

Unit

Jednostka

Exponential and Logarithmic Functions

Funkcje wykładnicze i logarytmiczne

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

Introduction to Exponential Functions

exponential growth and decay models

Exponential functions model growth and decay when the rate of change depends on the current value. They are written as f(x) = b^x, where b is a positive constant and not equal to 1. When the base b is greater than 1, the function shows growth. When 0 &lt; b &lt; 1, the function shows decay.

These models are useful in many real situations. They can describe bacterial proliferation, radioactive substance decay, and the depreciation of assets. In each case, the change is proportional to the current amount. A general exponential model can be written as A₀b^t, where A₀ is the initial amount, b is the growth or decay factor, and t is time or another continuous variable.

Exponential graphs can also be changed with transformations. Translations, reflections, and stretches shift or reshape the graph, but the basic curve and asymptotic behavior stay the same. Exponential functions also have the One-to-One Property. If b^x = b^y, then x = y, which lets you solve equations with exponents by comparing the bases.

Exponential Growth and Decay Models

introduction to exponential functions

Exponential functions model growth and decay when the rate of change depends on the current value. They are written as f(x) = b^x, where b is a positive constant and not equal to 1. When the base b is greater than 1, the function shows growth. When 0 &lt; b &lt; 1, the function shows decay. These models are useful in many real situations. They can describe bacterial proliferation, radioactive substance decay, and the depreciation of assets. In each case, the change is proportional to the...

exponential functions with base e

Exponential Growth and Decay in Real Life

Exponential functions with base e model continuous growth and decay in real life. The constant e is about 2.718. It appears naturally when change depends on the current value. A positive exponent shows continuous growth, while a negative exponent shows continuous decay. These functions are useful when change happens smoothly over time instead of in steps. One example of exponential decay is a hot beverage cooling down. The temperature drops quickly at first. As it gets closer to room...

introduction to logarithmic functions

Logarithms as Exponents Reversed

Logarithmic functions are the inverse of exponential functions. They are used to solve for exponents. In the general form y = log_a(x), the base a must be greater than 0 and not equal to 1. A logarithm tells you the power needed to get a given number. In other words, y = log_a(x) gives the exponent that the base a must be raised to in order to obtain x. The function is defined only for x &gt; 0, but its range includes all real numbers. The graphs of logarithmic and exponential functions are...

laws of logarithms i

Logarithm Rules for Product and Quotient

Logarithm rules help simplify expressions that use powers, multiplication, and division. A logarithm is the inverse of exponentiation, so it shows how many times a base must be raised to reach a number. For base 10, the common logarithm is written as log. If 10^n = x, then log(x) = n. These rules are useful because they turn harder calculations into simpler steps. The product law says that the logarithm of a product equals the sum of the logarithms. For example, log(100) can be rewritten as...

laws of logarithms ii

Power Law and Change-of-Base Formula

Logarithmic laws help simplify exponential expressions and make them easier to evaluate. Two key tools are the power law and the change-of-base formula. These rules are useful in math and in applied settings where powers and repeated multiplication appear often. The power law says that the logarithm of a number raised to an exponent equals the exponent multiplied by the logarithm of the number. This rule is helpful when solving equations that involve growth over time, such as compound interest.

exponential equations with logarithms: problem solving

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Solving Population Growth with Logarithms

Logarithms help solve population growth problems with exponential models. In ecology, these models are used to predict how a population changes over time when conditions stay favorable. The growth rate is proportional to the current population, so the number increases in a continuous and compounding way. The model writes population as a function of time. It combines the initial population with a growth factor raised to an exponent that includes the growth rate and the time. To find when a...

applications of logarithms

Logarithms in Sound and Decibels

Logarithms help describe sound intensity with the decibel scale. They are useful when a quantity changes over a very wide range. In acoustics, the decibel (dB) scale turns sound levels into numbers that are easier to compare and use. Sound can range from a faint whisper to the roar of a jet engine. A linear scale would make those differences hard to manage. The decibel scale is a logarithmic scale, which means it measures sound intensity level relative to a reference value. For air, that...

exponential equations for modeling growth

exponential-equations-for-modeling-growth

Exponential Growth in Population Models

Exponential growth in population models describes rapid, multiplicative change. It is useful when a population grows by the same factor at regular intervals. This makes it a strong fit for early-stage growth in natural systems. A bacterial culture that doubles every three hours can be modeled with an exponential equation. The model is n(t)=n0·2^(t/3), where n(t) is the population at time t and n0 is the starting population. The base 2 matches the doubling pattern. A more general exponential...

key terms and concepts

Explore exponential growth and decay models, graph transformations, real-world examples like bacteria and radioactive decay, and the one-to-one property.

Exponential Growth and Decay Models | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Exponential Growth and Decay Models

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