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

Betrachten wir eine Funktion; seine erste und zweite Ableitung werden verwendet, um seine Kurve zu skizzieren.Die erste Ableitung ist null, wenn x gleich vier ist, und undefiniert, wenn x gleich null oder x gleich sechs ist. Diese kritischen Punkte definieren die Intervalle für die Analyse.Das Vorzeichen der ersten Ableitung bestimmt, ob die Kurve steigt oder abnimmt.Eine Vorzeichenänderung von negativ zu positiv bei x gleich null ergibt ein lokales Minimum.Eine Änderung von positiv zu negativ bei x ergibt vier ein lokales Maximum. Wenn das Vorzeichen bei x nicht gleich sechs ist, existiert dort kein Extremum.Die zweite Ableitung ist negativ auf den Intervallen von minus unendlich bis 0 und von 0 bis 6, was zeigt, dass das Diagramm auf diesen Intervallen konkav nach unten ist.Die zweite Ableitung ist jedoch positiv auf dem Intervall x größer als sechs, was das konkave Aufwärtsdiagramm auf diesem Intervall ergibt.Betrachten wir zum Beispiel eine Gewinnfunktion. Das erste Derivat verfolgt den Anstieg oder Fall des Gewinns und identifiziert dessen Extremwerte.Die zweite Ableitung zeigt, wie sich die Gewinnrate verändert, was hilft, die Funktion zu skizzieren.

kurvendiskussion und ableitungen

Curve Sketching and Derivatives

Das Verst&#xE4;ndnis des Verhaltens einer Funktion anhand ihrer ersten und zweiten Ableitungen ist f&#xFC;r die Analyse ihres Graphen unerl&#xE4;sslich. Ableitungen geben Aufschluss dar&#xFC;ber, in welchen Bereichen eine Funktion steigt oder f&#xE4;llt, wo sie lokale Maxima oder Minima erreicht und wie sich ihre Kr&#xFC;mmung &#xFC;ber unterschiedliche Intervalle verh&#xE4;lt.Die erste Ableitung einer Funktion zeigt die Steigung der Tangentengerade an einem beliebigen Punkt. Punkte, an denen die Ableitung null oder nicht definiert ist, gelten als kritische Stellen, da sie h&#xE4;ufig auf potenzielle Extrema oder &#xC4;nderungen im Funktionsverhalten hinweisen. Ein lokales Minimum liegt vor, wenn die erste Ableitung von negativ nach positiv wechselt, w&#xE4;hrend ein lokales Maximum dort auftritt, wo sie von positiv nach negativ wechselt. &#xC4;ndert die Ableitung an einem kritischen Punkt ihr Vorzeichen nicht, liegt kein Extremum vor.Die zweite Ableitung gibt Aufschluss &#xFC;ber die Konkavit&#xE4;t der Funktion. Ist die zweite Ableitung negativ, ist der Graph konkav nach unten gerichtet, was bedeutet, dass die Steigung der ersten Ableitung abnimmt. Dies f&#xFC;hrt zu einer nach unten gekr&#xFC;mmten Kurve. Eine positive zweite Ableitung hingegen zeigt eine konkav nach oben verlaufende Funktion an, bei der die Steigung der ersten Ableitung zunimmt und die Kurve nach oben gekr&#xFC;mmt ist. &#xC4;nderungen der Konkavit&#xE4;t deuten auf Wendepunkte hin, an denen sich die Art der Kr&#xFC;mmung der Kurve &#xE4;ndert.Durch die Analyse der Vorzeichen und der Verl&#xE4;ufe der ersten und zweiten Ableitungen &#xFC;ber verschiedene Intervalle l&#xE4;sst sich eine pr&#xE4;zise Skizze der Funktion erstellen. Dieser kombinierte Ansatz unterst&#xFC;tzt dabei, Bereiche mit zunehmender oder abnehmender Funktion zu identifizieren, lokale Maxima und Minima zu bestimmen und die Gesamtform des Funktionsgraphen zu erfassen.AnwendungenBetrachten Sie beispielsweise ein Gewinnmodell, das durch eine quadratische Funktion beschrieben wird: \begin{equation} P(x) = -2x^2 + 12x - 10 \end{equation}Der Graph stellt eine Gewinnkurve dar, wobei x die Werbeausgaben in Tausend US-Dollar und P(x) den Gewinn, ebenfalls in Tausend US-Dollar, bezeichnet. Die Kurve hat die Form eines umgekehrten U und &#xE4;hnelt einem H&#xFC;gel. Mathematisch wird diese Form als konkav nach unten beschrieben.Bei geringen Werbeausgaben steigert eine Erh&#xF6;hung der Ausgaben Umsatz und Gewinn effizient. Mit zunehmendem Werbeaufwand nimmt der zus&#xE4;tzliche Gewinn pro weiterer Werbeeinheit jedoch ab, was abnehmende Grenzertr&#xE4;ge widerspiegelt. Bei zu hohen Ausgaben sinkt der Gewinn schlie&#xDF;lich, da die Kosten der zus&#xE4;tzlichen Werbung die erzielten Einnahmen &#xFC;bersteigen.Die Steigung der Kurve wird durch die erste Ableitung erfasst, und der maximale Gewinn tritt an der Stelle auf, an der diese Steigung null ist. An diesem Punkt erreicht die Kurve ihren H&#xF6;chstwert.Die zweite Ableitung der Funktion ist eine konstante negative Zahl. Eine negative zweite Ableitung zeigt, dass die gesamte Kurve konkav nach unten verl&#xE4;uft, wodurch best&#xE4;tigt wird, dass die Gewinnfunktion an jedem Punkt nach unten gekr&#xFC;mmt ist.

Kurvendiskussion und Ableitungen

Das Verst&#xE4;ndnis des Verhaltens einer Funktion anhand ihrer ersten und zweiten Ableitungen ist f&#xFC;r die Analyse ihres Graphen unerl&#xE4;sslich. ...

Education

Lehre

JoVE Core

Calculus

Analysis

Unit

Einheit

Applications of Differentiation

Anwendungen der Differenzialrechnung

KEY TERMS AND CONCEPTS

SCHLÜSSELBEGRIFFE UND KONZEPTE

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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Using First Derivatives to Find Graph Changes

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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

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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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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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

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Maximizing Bakery Revenue with Calculus

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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) /...

the antiderivative of a function

Finding Original Functions with Antiderivatives

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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...

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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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