Name: average and instantaneous change in functions
Uploaded: 2026-06-30T10:40:47.088Z
Duration: 1 min 21 s
Channel: JoVE

Le taux de variation montre comment une quantité varie en réponse à une autre.Par exemple, la température à l’intérieur d’une pièce varie tout au long de la journée — plus basse tôt le matin, montant à midi, puis redescendant la nuit.Cette variation de température peut être représentée par une fonction mathématique, où la température est exprimée en fonction du temps.Le taux moyen de variation sur un intervalle de temps est calculé en divisant le changement de température par le changement de temps.Mathématiquement, cela correspond à la pente d’une ligne sécante reliant deux points sur le graphe.Pour déterminer à quelle vitesse la température change à un instant donné, on utilise le taux de variation instantané.La limite du quotient de différence lorsque l’intervalle de temps diminue à zéro donne le taux instantané exact, qui est la pente de la droite tangente à ce point.Dans la vie réelle, le concept de vitesse de variation est utilisé pour calculer la vitesse de variation de concentration dans les réactions chimiques. Le taux moyen de variation donne la variation de la concentration sur une période, et le taux instantané de variation est le taux de variation à un moment donné.

taux de variation

Le taux de variation est un concept central en math&#xE9;matiques qui quantifie la mani&#xE8;re dont une variable varie par rapport &#xE0; une autre. Il constitue un outil fondamental de mod&#xE9;lisation des syst&#xE8;mes dynamiques dans des disciplines telles que la physique, la biologie, l&#x2019;&#xE9;conomie et l&#x2019;ing&#xE9;nierie. La compr&#xE9;hension des taux de variation moyens et instantan&#xE9;s permet d&#x2019;analyser le comportement des fonctions d&#xE9;crivant des ph&#xE9;nom&#xE8;nes du monde r&#xE9;el.Taux de variation moyenPour une fonction f(x) d&#xE9;finie sur un intervalle [x_1, x_2], le taux de variation moyen se calcule par :\begin{equation*} \jfrac{f(x_2) - f(x_1)}{x_2 - x_1} \end{equation*}Cette formule repr&#xE9;sente la pente de la droite s&#xE9;cante reliant les points (x_1, f(x_1)) et (x_2, f(x_2)) sur le graphique de la fonction. Elle fournit une mesure globale de la variation de la valeur de la fonction par unit&#xE9; d&#x2019;augmentation de la variable d&#x2019;entr&#xE9;e sur l&#x2019;intervalle consid&#xE9;r&#xE9;. Bien que ce taux offre un aper&#xE7;u des tendances g&#xE9;n&#xE9;rales, il ne rend pas compte des variations &#xE0; l&#x2019;int&#xE9;rieur de l&#x2019;intervalle.Taux de variation instantan&#xE9;Pour d&#xE9;terminer la mani&#xE8;re dont une fonction varie en un point pr&#xE9;cis, on utilise le taux de variation instantan&#xE9;. Celui-ci est d&#xE9;fini comme la limite du taux de variation moyen lorsque l&#x2019;intervalle tend vers z&#xE9;ro :\begin{equation*} \lim_{h \to 0} \jfrac{f(x + h) - f(x)}{h} \end{equation*}Cette limite, lorsqu&#x2019;elle existe, d&#xE9;finit la d&#xE9;riv&#xE9;e de la fonction en x, not&#xE9;e f&#x2032;(x). La d&#xE9;riv&#xE9;e correspond &#xE0; la pente de la droite tangente &#xE0; la courbe en ce point et fournit une mesure pr&#xE9;cise de la variation de la valeur de la fonction pour cette valeur exacte de la variable d&#x2019;entr&#xE9;e.Les d&#xE9;riv&#xE9;es jouent un r&#xF4;le essentiel dans la compr&#xE9;hension des processus continus. Elles permettent de mod&#xE9;liser les taux de variation du mouvement, de la croissance, de la d&#xE9;croissance et dans les probl&#xE8;mes d&#x2019;optimisation, et constituent le fondement du calcul diff&#xE9;rentiel ainsi que du raisonnement analytique en sciences.

Taux de variation

Le taux de variation est un concept central en math&#xE9;matiques qui quantifie la mani&#xE8;re dont une variable varie par rapport &#xE0; une autre. Il ...

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JoVE Core

Calculus

Calcul

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Derivatives

Dérivées

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

Rates of Change

average and instantaneous change in functions

Rates of change describe how one variable changes in response to another. In math, this idea helps model dynamic systems in physics, biology, economics, and engineering. It also helps students study real-world functions that change over time or across different inputs.

The average rate of change measures the overall change in a function over an interval [x1, x2]. It is found by dividing the change in output by the change in input. On a graph, this value is the slope of the secant line, which connects the points (x1, f(x1)) and (x2, f(x2)). This gives a general trend, but it does not show what happens between the two points.

The instantaneous rate of change shows how a function changes at one exact point. It is defined as the limit of the average rate of change as the interval gets smaller and smaller. When this limit exists, it is the derivative of the function at x, written as f'(x). The derivative is also the slope of the tangent line, which touches the curve at that point.

Derivatives are useful for studying continuous processes. They help describe motion, growth, decay, and optimization problems. Because of this, derivatives are a key part of differential calculus and scientific analysis.

Average and Instantaneous Change in Functions

average-and-instantaneous-change-in-functions

tangent line

Secant Slope to Tangent Line

Differential calculus helps describe how a quantity changes at a specific point. A moving car on a winding road gives a clear example. The car’s path is a continuous curve, and the direction it travels at any instant is shown by the tangent to that curve. A secant line, by contrast, crosses the curve at two points. It shows how the car’s position changes over an interval, so it describes average behavior rather than an instant. The slope of the secant line between two points on a function f(x)...

derivatives

Square Root Derivatives and Tangent Slopes

Derivatives describe instantaneous rate of change, which is a key idea in both mathematics and physics. They are used to show how a moving object changes position over time. A derivative at a point is also the slope of the tangent line to a curve at that point. It tells how much the output of a function changes for a tiny change in input. For the square root function, finding this rate of change starts with the limit of the difference quotient. The algebra is harder here because the numerator...

rates of change

Rates of change describe how one variable changes in response to another. In math, this idea helps model dynamic systems in physics, biology, economics, and engineering. It also helps students study real-world functions that change over time or across different inputs. The average rate of change measures the overall change in a function over an interval [x1, x2]. It is found by dividing the change in output by the change in input. On a graph, this value is the slope of the secant line, which...

the derivative as a function

Graphing the Derivative Function

The derivative function describes how a function changes at each point. It gives the instantaneous rate of change, which is the slope of the tangent line to the graph at a specific x-value. When this idea is applied across the whole domain, it creates a new function that records the rate of change everywhere. For a differentiable function f(x), the derivative function is written as f′(x). It assigns each input x the instantaneous rate at which f(x) is changing. Formally, it comes from the...

higher derivatives

Concavity and Motion with Higher Derivatives

Higher derivatives show how a function changes from one step to the next. In calculus, the first derivative gives the slope of a graph, and the second derivative shows how that slope is changing. These ideas help explain both the shape of a curve and the motion of an object. For the example function f(x) = x^3 - x, the first derivative is f'(x) = 3x^2 - 1. The second derivative is f''(x) = 6x. In this case, the second derivative is a linear function, so the slope changes at a steady rate. <br...

rate of change: problem solving

Brook Trout Growth vs Water Temperature

Brook trout growth changes with water temperature. The data show how trout weight shifts over a 24-day period at several temperatures. At cooler water temperatures, such as 15.5 degrees Celsius, the trout gain a lot of weight. As the temperature rises, the amount of weight gained gets smaller. At the warmest temperature tested, 24.4 degrees Celsius, the trout have a net loss in weight. This pattern suggests that brook trout do better in cooler water. Warmer water may create metabolic stress or...

key terms and concepts

Learn how average and instantaneous rates of change describe function behavior, including secant lines, tangent lines, and derivatives in real-world models.

Average and Instantaneous Change in Functions | 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

Average and Instantaneous Change in Functions

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