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

변화율은 한 양이 다른 양에 어떻게 반응하는지를 보여줍니다.예를 들어, 방 안의 온도는 하루 동안 변동합니다—이른 아침에는 낮아지고, 정오에는 올라가고, 밤에는 다시 내려갑니다.이 온도 변화는 수학적 함수로 표현할 수 있으며, 온도는 시간에 따른 함수로 표현됩니다.일정 시간 동안 평균 변화율은 온도 변화를 시간 변화로 나누어 계산합니다.수학적으로 이는 그래프 내 두 점을 연결하는 분선선의 기울기와 일치합니다.특정 순간에 온도가 얼마나 빠르게 변하는지 알아내기 위해 순간 변화율을 사용합니다.시간 구간이 0으로 줄어들면서 차분의 극한은 정확한 순간 비율을 제공하며, 이는 그 지점에서 접선의 기울기입니다.현실에서는 변화율 개념을 사용해 화학 반응에서 농도 변화율을 계산합니다. 평균 변화율은 일정 기간 동안의 농도 변화를 나타내며, 순간적인 변화율은 특정 순간의 변화율입니다.

변화율

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Education

교육

JoVE Core

Calculus

미적분학

Unit

단위

Derivatives

미분

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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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