Name: using derivatives to read a graph
Uploaded: 2026-01-12T12:24:56.000Z
Duration: 1 min 22 s
Channel: JoVE

Rozważmy funkcję; jej pierwsza i druga pochodna służą do szkicowania krzywej.Pierwsza pochodna jest równa się zero, gdy x równa się czterem, a nieokreślona, gdy x równa się zero lub x równa się sześć. Te krytyczne punkty definiują przedziały do analizy.Znak pierwszej pochodnej określa, czy krzywa rośnie czy maleje.Zmiana znaku z ujemnego na dodatniego w x równa się zero daje lokalne minimum.Zmiana z dodatniego na ujemny w x równa się cztery, daje lokalne maksimum. Jeśli znak nie zmienia się w x równa się sześć, nie istnieje tam ekstremum.Druga pochodna jest ujemna na przedziałach od minus nieskończoności do 0 oraz od 0 do 6, co pokazuje, że wykres jest wklęsły w dół na tych przedziałach.Ale druga pochodna jest dodatnia na przedziale x większym niż sześć, co daje wklęsły wykres w górę na tym przedziale.Na przykład rozważmy funkcję zysku. Pierwsza pochodna śledzi wzrost lub spadek zysku oraz wskazuje jego ekstremalne wartości.Druga pochodna pokazuje, jak zmienia się stopa zysku, co pomaga narysować funkcję.

szkicowanie krzywych i pochodne

Curve Sketching and Derivatives

Zrozumienie zachowania funkcji na podstawie jej pierwszej i drugiej pochodnej jest niezb&#x119;dne do analizy jej wykresu. Pochodne dostarczaj&#x105; informacji o tym, gdzie funkcja ro&#x15B;nie lub maleje, gdzie osi&#x105;ga lokalne maksima lub minima oraz jak zachowuje si&#x119; jej krzywizna w r&#xF3;&#x17C;nych przedzia&#x142;ach.Pierwsza pochodna funkcji ujawnia nachylenie stycznej w dowolnym punkcie. Punkty, w kt&#xF3;rych pochodna jest r&#xF3;wna zero lub nieokre&#x15B;lona, uznaje si&#x119; za krytyczne, poniewa&#x17C; cz&#x119;sto wskazuj&#x105; na potencjalne ekstrema lub zmiany w zachowaniu funkcji. Minimum lokalne wyst&#x119;puje, gdy pierwsza pochodna zmienia si&#x119; z ujemnej na dodatni&#x105;, natomiast maksimum lokalne wyst&#x119;puje, gdy zmienia si&#x119; z dodatniej na ujemn&#x105;. Je&#x15B;li pochodna nie zmienia znaku w punkcie krytycznym, ekstremum nie wyst&#x119;puje.Druga pochodna dostarcza informacji o wkl&#x119;s&#x142;o&#x15B;ci i wypuk&#x142;o&#x15B;ci funkcji. Gdy druga pochodna jest ujemna, wykres jest wkl&#x119;s&#x142;y w d&#xF3;&#x142;, co wskazuje, &#x17C;e nachylenie pierwszej pochodnej maleje. W rezultacie krzywa wygina si&#x119; w d&#xF3;&#x142;. Odwrotnie, dodatnia druga pochodna oznacza, &#x17C;e funkcja jest wkl&#x119;s&#x142;a ku g&#xF3;rze, przy czym nachylenie pierwszej pochodnej ro&#x15B;nie, a krzywa wygina si&#x119; ku g&#xF3;rze. Zmiany wkl&#x119;s&#x142;o&#x15B;ci sugeruj&#x105; obecno&#x15B;&#x107; punkt&#xF3;w przegi&#x119;cia, w kt&#xF3;rych charakter krzywej zmienia si&#x119; z jednego typu krzywizny w inny.Analizuj&#x105;c znaki i zachowanie pierwszej i drugiej pochodnej w r&#xF3;&#x17C;nych przedzia&#x142;ach, mo&#x17C;na sporz&#x105;dzi&#x107; dok&#x142;adny szkic funkcji. To po&#x142;&#x105;czone podej&#x15B;cie pomaga zidentyfikowa&#x107; obszary wzrostu lub spadku, zlokalizowa&#x107; lokalne maksima i minima oraz okre&#x15B;li&#x107; og&#xF3;lny kszta&#x142;t wykresu funkcji.ZastosowaniaNa przyk&#x142;ad mo&#x17C;na rozwa&#x17C;y&#x107; model zysku reprezentowany przez funkcj&#x119; kwadratow&#x105;: \begin{equation} P(x) = -2x^2 + 12x - 10 \end{equation}Wykres przedstawia krzyw&#x105; zysku, gdzie x oznacza wydatki na reklam&#x119; w tysi&#x105;cach USD, a P(x) r&#xF3;wnie&#x17C; reprezentuje zysk w tysi&#x105;cach USD. Krzywa tworzy kszta&#x142;t odwr&#xF3;conej litery U, przypominaj&#x105;cy wzg&#xF3;rze. W terminologii matematycznej kszta&#x142;t ten opisuje si&#x119; jako wkl&#x119;s&#x142;y w d&#xF3;&#x142;.Przy niskim poziomie wydatk&#xF3;w reklamowych zwi&#x119;kszanie nak&#x142;ad&#xF3;w skutecznie podnosi sprzeda&#x17C; i zysk. Jednak wraz z dalszym wzrostem wydatk&#xF3;w dodatkowy zysk generowany przez ka&#x17C;d&#x105; kolejn&#x105; jednostk&#x119; reklamy zaczyna male&#x107;. Odzwierciedla to zjawisko malej&#x105;cych korzy&#x15B;ci kra&#x144;cowych. Je&#x15B;li wydatki stan&#x105; si&#x119; zbyt wysokie, zysk ostatecznie spadnie, poniewa&#x17C; koszt dodatkowej reklamy przekroczy generowany przez ni&#x105; przych&#xF3;d.Nachylenie krzywej jest opisywane przez pierwsz&#x105; pochodn&#x105;, a maksymalny zysk wyst&#x119;puje w punkcie, w kt&#xF3;rym nachylenie to staje si&#x119; r&#xF3;wne zeru. Przy tym poziomie wydatk&#xF3;w krzywa osi&#x105;ga najwy&#x17C;sz&#x105; warto&#x15B;&#x107;.Druga pochodna funkcji jest sta&#x142;&#x105; liczb&#x105; ujemn&#x105;. Ujemna druga pochodna wskazuje, &#x17C;e ca&#x142;a krzywa jest wkl&#x119;s&#x142;a w d&#xF3;&#x142;, co potwierdza, &#x17C;e funkcja zysku jest wygi&#x119;ta w d&#xF3;&#x142; w ka&#x17C;dym punkcie.

Szkicowanie krzywych i pochodne

Zrozumienie zachowania funkcji na podstawie jej pierwszej i drugiej pochodnej jest niezb&#x119;dne do analizy jej wykresu. Pochodne dostarczaj&#x105; ...

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Applications of Differentiation

Zastosowania różniczkowania

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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 negative to positive. A local maximum appears when it changes from positive to negative.

If the first derivative does not change sign at a critical point, then the function does not have an extremum there. The second derivative adds information about concavity, which describes whether the graph bends upward or downward. A negative second derivative means the graph is concave downward. A positive second derivative means the graph is concave upward.

When the second derivative is negative, the slope of the first derivative is decreasing, so the curve bends downward. When the second derivative is positive, the slope of the first derivative is increasing, so the curve bends upward. A change in concavity can signal an inflection point, where the graph switches from one type of bending to the other.

By checking the signs of the first and second derivatives on different intervals, students can sketch a function more accurately. This process identifies where the function increases or decreases, where it has local maxima and minima, and how its overall shape develops. The same ideas apply to a quadratic profit model where x is advertising spend in thousands of USD and P(x) is profit in thousands of USD.

In that example, the graph looks like a hill, or an inverted U shape. That shape is concave downward, so the profit curve bends downward throughout. At low advertising levels, more spending raises sales and profit efficiently. As spending keeps rising, the extra profit from each additional unit of advertising gets smaller, which shows diminishing returns.

If advertising becomes too high, profit can start to fall because the cost of the extra advertising is greater than the revenue it brings in. The maximum profit occurs where the first derivative is zero, because the slope of the curve becomes flat there. The second derivative is a constant negative number, which confirms that the entire profit function is concave downward.

Using Derivatives to Read a Graph

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Finding Absolute Extrema on a Closed Interval

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Rolle’s Theorem and Zero Slope Points

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Mean Change and Instantaneous Slope

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first derivatives and the shape of a graph

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

Using First Derivatives to Find Graph Changes

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first derivative test: problem solving

Finding Turning Points with the First Derivative

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second derivatives and the shape of a graph

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

Using Second Derivatives to Read Curvature

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second derivative test: problem solving

Classifying Critical Points with the Second Derivative

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curve sketching and derivatives

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Using Derivatives to Resolve Limit Ambiguity

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guidelines for sketching a curve

Using Derivatives to Sketch Function Graphs

Sketching a function graph uses key features of the function to show its overall shape. A function plots as a curve on the coordinate plane, with the input on the horizontal axis and the output on the vertical axis. The first step is to find the domain, which is the set of input values where the function is defined. The domain shows how far the graph can extend left and right. Intercepts are also important reference points. The x-intercept is where the graph crosses the horizontal axis, and...

slant asymptotes

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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

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Newton’s Method for Finding Roots

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the antiderivative of a function

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Antiderivative Graph Shape and Slope

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Finding Car Stopping Distance with Integration

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key terms and concepts

Learn how the first and second derivatives reveal increase, decrease, local maxima, minima, concavity, and inflection points in function graphs.

Using Derivatives to Read a Graph | 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 Derivatives to Read a Graph

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