Name: using first derivatives to find graph changes
Uploaded: 2026-01-12T12:24:54.000Z
Duration: 1 min 22 s
Channel: JoVE

The shape of a graph&#8212;whether it rises, falls, or levels off&#8212;depends on its first derivative. The first derivative gives the slope of the tangent line at any point on a graph.A positive derivative means the function is increasing and the graph slopes upward. A negative derivative means it is decreasing and the graph slopes downward.When the derivative is zero or undefined, the point is called a critical point. These points mark locations where the function may change from increasing to decreasing, or vice versa.They divide the domain into intervals where the first derivative test can be used to find the function&#8217;s behavior.If the derivative is positive before a critical point and negative after, the function reaches a local maximum. If the derivative is negative before the critical point and positive after, the function reaches a local minimum. If the sign does not change, the point is neither a maximum nor a minimum.First derivatives also help study how diseases spread. A positive derivative shows cases increasing at that moment, while a negative derivative shows cases decreasing and the spread slowing.

eerste afgeleiden en de vorm van een grafiek

First Derivatives and the Shape of a Graph

In de differentiaalrekening speelt het begrip eerste afgeleide een cruciale rol bij het begrijpen van het gedrag van een functie op haar domein. De eerste afgeleide, aangeduid als f&#x27;(x), verschaft inzicht in de manier waarop een functie op een bepaald punt verandert, vergelijkbaar met een fietser die zijn snelheid aanpast langs een bochtig traject. Door de eerste afgeleide te analyseren, kunnen wiskundigen vaststellen waar een functie toeneemt, afneemt of kritieke punten bereikt.
De eerste afgeleide biedt een nauwkeurige methode voor het classificeren van intervallen waarin een functie toeneemt of afneemt. Een functie f(x) wordt stijgend genoemd op een interval wanneer haar afgeleide positief is, wat betekent dat de helling van de raaklijn in ieder punt van dat interval positief is. Omgekeerd is een functie dalend op een interval waarin de afgeleide negatief is, wat wijst op een negatieve helling. Deze informatie helpt bij het schetsen van de algemene vorm van de grafiek van de functie en bij het voorspellen van haar gedrag in verschillende delen van het domein.
Kritieke punten doen zich voor wanneer de eerste afgeleide gelijk is aan nul of niet gedefinieerd is. Deze punten zijn van bijzonder belang, omdat zij kunnen overeenkomen met lokale maxima of minima. De eerste afgeleidentest biedt een systematische methode om dergelijke lokale extrema te identificeren. Indien f&#x27;(x) bij een kritisch punt verandert van positief naar negatief, neemt de functie daar een lokaal maximum aan. Indien f&#x27;(x) daarentegen verandert van negatief naar positief, neemt de functie een lokaal minimum aan. Wanneer er geen tekenverandering in f&#x27;(x) optreedt, komt het kritieke punt niet overeen met een lokaal extremum.
De eerste afgeleide kent talrijke toepassingen in uiteenlopende disciplines, waaronder natuurkunde, economie en de techniek. In de natuurkunde wordt snelheid beschreven als de veranderingssnelheid van de positie ten opzichte van de tijd. In de economie helpt de eerste afgeleide bij de analyse van winstfuncties door punten te identificeren waarop de marginale kosten gelijk zijn aan de marginale opbrengsten. In de techniek ondersteunt zij het optimaliseren van ontwerpparameters door effici&#xEB;ntiepunten te bepalen. Inzicht in de rol van de eerste afgeleide is derhalve fundamenteel voor het interpreteren van veranderingen in de werkelijkheid en voor het nemen van weloverwogen beslissingen op basis van wiskundige modellen.

Eerste afgeleiden en de vorm van een grafiek

In de differentiaalrekening speelt het begrip eerste afgeleide een cruciale rol bij het begrijpen van het gedrag van een functie op haar domein. De eerste ...

first-derivatives-and-the-shape-of-a-graph

Education

JoVE Core

Calculus

Unit

Eenheid

Applications of Differentiation

Toepassingen van differentiatie

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN CONCEPTEN

using first derivatives to find graph changes

The first derivative shows how a function changes across its domain. In calculus, the derivative is written as f'(x). It tells us whether the graph is rising, falling, or turning at a given point.

A function is increasing on an interval when its derivative is positive. That means the slope of the tangent line is positive. A function is decreasing on an interval when its derivative is negative. In that case, the tangent line slopes downward.

These sign changes help describe the overall shape of a graph. They also help predict how a function behaves in different parts of its domain. The first derivative is therefore a useful tool for sketching graphs.

Critical points are the places where the first derivative is zero or undefined. These points are important because they may mark local maxima or local minima. The First Derivative Test checks how the sign of f'(x) changes near each critical point.

If f'(x) changes from positive to negative, the function has a local maximum. If f'(x) changes from negative to positive, the function has a local minimum. If there is no sign change, the critical point is not a local extremum.

First derivatives also appear in real-world settings. In physics, velocity is the rate of change of position with respect to time. In economics, first derivatives help study profit by finding points where marginal cost equals marginal revenue. In engineering, they help optimize design choices and identify efficiency points.

Using First Derivatives to Find Graph Changes

using-first-derivatives-to-find-graph-changes

absolute and local extreme values

Finding Max and Min on Function Graphs

Absolute and local extreme values show the highest and lowest points on a function graph. Absolute maximum and absolute minimum values describe the highest and lowest values across the entire domain. Local maximum and local minimum values describe the highest and lowest values within a smaller part of the graph. Periodic functions such as sine and cosine can have extreme values at many points. Their repeating pattern creates multiple local extrema. In these cases, the derivative becomes zero...

critical numbers and the closed interval method

critical-numbers-and-the-closed-interval-method

Finding Absolute Extrema on a Closed Interval

Absolute extrema on a closed interval are found by checking critical numbers and the endpoints. A function’s maximum and minimum values help describe its overall behavior across its domain. These values are called extrema, and they can be local or absolute. Local extrema are peaks and valleys in a limited region. Absolute extrema are the highest or lowest points on the entire interval. Critical numbers are the domain values where the derivative is zero or does not exist. According to Fermat’s...

rolle’s theorem

Rolle’s Theorem and Zero Slope Points

Rolle’s Theorem explains when a differentiable function must have a flat tangent inside an interval. It applies to a real-valued function on a closed interval, which means the endpoints are included. The function must also be continuous on the whole interval and differentiable on the open interval, where the interior points lie. The theorem uses one more key condition. The function must have the same value at both endpoints. When that happens, Rolle’s Theorem guarantees at least one point...

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first derivatives and the shape of a graph

The first derivative shows how a function changes across its domain. In calculus, the derivative is written as f'(x). It tells us whether the graph is rising, falling, or turning at a given point. A function is increasing on an interval when its derivative is positive. That means the slope of the tangent line is positive. A function is decreasing on an interval when its derivative is negative. In that case, the tangent line slopes downward. These sign changes help describe the overall shape...

first derivative test: problem solving

Finding Turning Points with the First Derivative

The first derivative test is used to find turning points on a smooth curve. It shows where a function changes from increasing to decreasing, or from decreasing to increasing. In this example, the curve can model an asset price that falls to a low point, rebounds, and then gradually declines again. The process starts by finding the first derivative of the function. Because the function is a product of a polynomial term and an exponential term, the product rule is needed. After differentiating,...

second derivatives and the shape of a graph

second-derivatives-and-the-shape-of-a-graph

Using Second Derivatives to Read Curvature

The second derivative shows how a graph bends. It gives key information about curvature and how that curvature changes over an interval. These ideas help with graph analysis in math and in real-world settings such as road elevation, population growth, and economic trends. A function f(x) is concave upward on an interval when its graph lies above all of its tangent lines. In this case, the second derivative is positive. That means the slope of the tangent line is increasing, and the graph bends...

second derivative test: problem solving

Classifying Critical Points with the Second Derivative

The second derivative test helps classify critical points of a function. Critical points are places where the first derivative is zero or undefined. They are possible locations for local maxima and local minima, but they do not always become one or the other. The second derivative shows concavity, or whether a graph bends up or down. If f''(x) &gt; 0, the function is concave up, and the critical point is a local minimum. If f''(x) &lt; 0, the function is concave down, and the critical point is...

curve sketching and derivatives

Using Derivatives to Read a Graph

Derivatives help students read the graph of a function by showing where the graph rises, falls, and bends. The first derivative tells the slope of the tangent line at each point. The second derivative shows how the curve changes shape over an interval. Points where the first derivative is zero or undefined are critical points. These points are important because they may mark a local maximum, a local minimum, or no extremum at all. A local minimum appears when the first derivative changes from...

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Limits with a zero-times-infinity product are indeterminate. In this type of limit, one factor approaches zero while the other factor grows toward positive or negative infinity. The result is not obvious at first, because the product may approach zero, infinity, or a finite nonzero value. To evaluate these product limits, the expression is first rewritten as a quotient. This form makes it possible to use L'Hôpital's Rule, which compares the derivatives of the numerator and denominator. A...

guidelines for sketching a curve

Using Derivatives to Sketch Function Graphs

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slant asymptotes

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Rational functions can follow a slant asymptote when the numerator grows faster than the denominator. In this case, a cubic numerator is divided by a squared denominator. That degree difference creates a linear trend that guides the graph at large input values. The function is only defined where the denominator is not zero. Those excluded x-values split the domain into separate regions and avoid singular points. The graph can still cross the x-axis where the numerator equals zero, so the real...

optimization problems

Pipe Length Around a Corner

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application of differentiation to business

application-of-differentiation-to-business

Maximizing Bakery Revenue with Calculus

Calculus helps businesses make pricing decisions that can improve revenue. In this example, a bakery uses a cupcake model to find the best price and daily sales volume. The goal is to see how changes in price affect demand and total revenue. The demand function shows the link between price and how many cupcakes are sold each day. In this model, p(x) gives the price per cupcake when x cupcakes are sold. Lower prices lead to higher demand, so the function captures how sales change with price. <br...

newton’s method

Newton’s Method for Finding Roots

Newton’s Method is an iterative way to find the roots of real-valued, differentiable functions. A root is the value of x that makes a function equal to zero. This method is useful when an exact algebraic answer is hard to find. The process starts with an initial guess. Newton’s Method then uses the function value and its derivative, which is the slope of the function at that point, to build a better estimate. The next approximation is found from the recursive formula x n+1 = x n - f(x n) /...

the antiderivative of a function

Finding Original Functions with Antiderivatives

Antiderivatives help reconstruct an original function from its derivative. In calculus, an antiderivative of a function f(x) is a function F(x) whose derivative gives back f(x). This idea works like reversing differentiation to recover the starting form. Antiderivatives are not unique because differentiation removes constant terms. For that reason, the general form includes an arbitrary constant C. That constant represents unknown initial conditions when a function is rebuilt from its...

graphing antiderivatives

Antiderivative Graph Shape and Slope

Antiderivative graphs show how a function accumulates over time. In calculus, an antiderivative tracks that buildup, much like the changing position of a rolling ball as its velocity changes. The graph of the antiderivative reflects the total effect of the original function's values. The sign of the original function helps determine the antiderivative's direction. When the function is positive, the antiderivative rises because the accumulation grows. When the function is negative, the...

application of antiderivatives: linear motion

application-of-antiderivatives-linear-motion

Finding Car Stopping Distance with Integration

Integration connects acceleration, velocity, and displacement in linear motion. It works as the reverse of a derivative, which tells how one quantity changes with another. In physics, this makes it possible to move from a rate of change back to the motion itself. A car provides a clear example. It is moving at a steady 20 meters per second when an obstacle appears 800 meters ahead. To stop in time, the car slows with a constant acceleration. That acceleration tells how the car’s velocity...

key terms and concepts

Learn how first derivatives show where a function increases, decreases, and reaches critical points, and how the First Derivative Test finds local maxima and minima.

Using First Derivatives to Find Graph Changes | 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

Using First Derivatives to Find Graph Changes

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