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

Kształt grafu — czy wzrasta, opada, czy się wyrównuje — zależy od jego pierwszej pochodnej. Pierwsza pochodna określa nachylenie prostej stycznej w dowolnym punkcie na grafie.Pochodna dodatnia oznacza, że funkcja rośnie, a wykres nachyla się ku górze. Ujemna pochodna oznacza, że ona maleje, a wykres opada w dół.Gdy pochodna jest zerowa lub niezdefiniowana, punkt nazywa się punktem krytycznym. Punkty te oznaczają miejsca, w których funkcja może zmieniać się od wzrostu do malenka lub odwrotnie.Dzielą dziedzinę na przedziały, w których pierwszy test pochodny może być użyty do określenia zachowania funkcji.Jeśli pochodna jest dodatnia przed punktem krytycznym i ujemna po niej, funkcja osiąga lokalne maksimum. Jeśli pochodna jest ujemna przed punktem krytycznym i dodatnia po nim, funkcja osiąga lokalne minimum. Jeśli znak się nie zmienia, punkt nie jest ani maksimum, ani minimum.Po pierwsze, pochodne pomagają również badać rozprzestrzenianie się chorób. Pochodna dodatnia pokazuje wzrost przypadków w danym momencie, natomiast pochodna ujemna oznacza malejące przypadki i spowalniający spread.

pierwsze pochodne a kształt wykresu

First Derivatives and the Shape of a Graph

W rachunku r&#xF3;&#x17C;niczkowym poj&#x119;cie pierwszej pochodnej odgrywa kluczow&#x105; rol&#x119; w zrozumieniu zachowania funkcji w jej dziedzinie. Pierwsza pochodna, oznaczana jako f&#x2019;(x), dostarcza informacji o tym, jak funkcja zmienia si&#x119; w dowolnym punkcie, podobnie jak rowerzysta dostosowuj&#x105;cy pr&#x119;dko&#x15B;&#x107; na kr&#x119;tej trasie. Analizuj&#x105;c pierwsz&#x105; pochodn&#x105;, matematycy mog&#x105; okre&#x15B;li&#x107;, gdzie funkcja ro&#x15B;nie, maleje lub osi&#x105;ga punkty krytyczne.
Pierwsza pochodna zapewnia precyzyjn&#x105; metod&#x119; klasyfikacji przedzia&#x142;&#xF3;w, w kt&#xF3;rych funkcja ro&#x15B;nie lub maleje. M&#xF3;wi si&#x119;, &#x17C;e funkcja f(x) ro&#x15B;nie w przedziale, je&#x15B;li jej pochodna jest dodatnia, co oznacza, &#x17C;e nachylenie stycznej w dowolnym punkcie tego przedzia&#x142;u jest dodatnie. Z drugiej strony, funkcja maleje w przedziale, w kt&#xF3;rym pochodna jest ujemna, co wskazuje na ujemne nachylenie stycznej. Ta informacja pomaga naszkicowa&#x107; og&#xF3;lny kszta&#x142;t wykresu funkcji i przewidzie&#x107; jej zachowanie w r&#xF3;&#x17C;nych obszarach dziedziny.
Punkty krytyczne wyst&#x119;puj&#x105;, gdy pierwsza pochodna jest r&#xF3;wna zero lub nieokre&#x15B;lona. Punkty te s&#x105; szczeg&#xF3;lnie interesuj&#x105;ce, poniewa&#x17C; mog&#x105; odpowiada&#x107; lokalnym maksimom lub minimom. Test pierwszej pochodnej zapewnia systematyczn&#x105; metod&#x119; identyfikacji tych ekstrem&#xF3;w lokalnych. Je&#x15B;li f&#x2019;(x) zmienia si&#x119; z dodatniej na ujemn&#x105; w punkcie krytycznym, funkcja osi&#x105;ga maksimum lokalne. Je&#x15B;li f&#x2019;(x) zmienia si&#x119; z ujemnej na dodatni&#x105;, funkcja osi&#x105;ga minimum lokalne. Je&#x15B;li jednak nie nast&#x119;puje zmiana znaku f&#x2019;(x), punkt krytyczny nie odpowiada ekstremum lokalnemu.
Pierwsza pochodna ma liczne zastosowania w r&#xF3;&#x17C;nych dziedzinach, w tym w fizyce, ekonomii i in&#x17C;ynierii. W fizyce pr&#x119;dko&#x15B;&#x107; jest opisana jako tempo zmiany po&#x142;o&#x17C;enia wzgl&#x119;dem czasu. W ekonomii pomaga analizowa&#x107; funkcje zysku poprzez identyfikacj&#x119; punkt&#xF3;w, w kt&#xF3;rych koszt kra&#x144;cowy r&#xF3;wna si&#x119; przychodowi kra&#x144;cowemu. W in&#x17C;ynierii pomaga w optymalizacji parametr&#xF3;w projektu poprzez wyznaczanie punkt&#xF3;w efektywno&#x15B;ci. Zrozumienie roli pierwszej pochodnej ma fundamentalne znaczenie dla interpretacji zmian w &#x15B;wiecie rzeczywistym i podejmowania &#x15B;wiadomych decyzji w oparciu o modele matematyczne.

Pierwsze pochodne a kształt wykresu

W rachunku r&#xF3;&#x17C;niczkowym poj&#x119;cie pierwszej pochodnej odgrywa kluczow&#x105; rol&#x119; w zrozumieniu zachowania funkcji w jej dziedzinie. ...

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

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Applications of Differentiation

Zastosowania różniczkowania

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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...

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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...

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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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guidelines for sketching a curve

Using Derivatives to Sketch Function Graphs

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slant asymptotes

Rational Functions with Slant Asymptotes

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...

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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) /...

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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...

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Antiderivative Graph Shape and Slope

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application-of-antiderivatives-linear-motion

Finding Car Stopping Distance with Integration

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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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