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

Consideremos una función; sus primera y segunda derivadas se utilizan para esbozar su curva.La primera derivada es cero cuando x es igual a cuatro, y no definida cuando x es igual a cero o x es igual a seis. Estos puntos críticos definen los intervalos para el análisis.El signo de la primera derivada determina si la curva está aumentando o disminuyendo.Un cambio de signo de negativo a positivo en x igual a cero da un mínimo local.Un cambio de positivo a negativo en x igual a cuatro da un máximo local. Si el signo no cambia en x igual a seis, no existe extremo allí.La segunda derivada es negativa en los intervalos desde menos infinito hasta 0 y de 0 a 6, lo que muestra que el gráfico es cóncavo hacia abajo en estos intervalos.Pero la segunda derivada es positiva en el intervalo x mayor que seis, lo que da el gráfico cóncavo hacia arriba en este intervalo.Por ejemplo, consideremos una función de beneficio. El primer derivado sigue el aumento o la bajada del beneficio e identifica sus valores extremos.La segunda derivada muestra cómo cambia la tasa de beneficio, lo que ayuda a esbozar la función.

dibujo de curvas y derivadas

Curve Sketching and Derivatives

Comprender el comportamiento de una funci&#xF3;n a trav&#xE9;s de sus primeras y segundas derivadas es esencial para analizar su gr&#xE1;fica. Las derivadas permiten comprender d&#xF3;nde aumenta o disminuye una funci&#xF3;n, d&#xF3;nde alcanza m&#xE1;ximos o m&#xED;nimos locales y c&#xF3;mo se comporta su curvatura en diferentes intervalos.La primera derivada de una funci&#xF3;n revela la pendiente de la recta tangente en cualquier punto dado. Los puntos donde la derivada es cero o indefinida se consideran cr&#xED;ticos, ya que suelen indicar posibles extremos o cambios en el comportamiento. Un m&#xED;nimo local se produce cuando la primera derivada cambia de negativa a positiva, mientras que un m&#xE1;ximo local se encuentra cuando cambia de positiva a negativa. Si la derivada no cambia de signo en un punto cr&#xED;tico, no existe ning&#xFA;n extremo.La segunda derivada ofrece informaci&#xF3;n sobre la concavidad de la funci&#xF3;n. Cuando la segunda derivada es negativa, la gr&#xE1;fica es c&#xF3;ncava hacia abajo, lo que indica que la pendiente de la primera derivada es decreciente. Esto da como resultado una curva que se curva hacia abajo. Por el contrario, una segunda derivada positiva implica que la funci&#xF3;n es c&#xF3;ncava hacia arriba, con la pendiente de la primera derivada aumentando y la curva arque&#xE1;ndose hacia arriba. Los cambios en la concavidad sugieren la presencia de puntos de inflexi&#xF3;n, donde la naturaleza de la curva cambia de un tipo de concavidad a otro.Al examinar los signos y el comportamiento de las primeras y segundas derivadas en diferentes intervalos, es posible construir un bosquejo preciso de la funci&#xF3;n. Este enfoque combinado ayuda a identificar regiones de aumento o disminuci&#xF3;n, localizar m&#xE1;ximos y m&#xED;nimos locales y determinar la forma general del gr&#xE1;fico de la funci&#xF3;n.AplicacionesPor ejemplo, considere un modelo de ganancias representado por una funci&#xF3;n cuadr&#xE1;tica: \begin{equation} P(x) = -2x^2 + 12x - 10 \end{equation}El gr&#xE1;fico representa una curva de beneficio, donde x representa la inversi&#xF3;n publicitaria en miles de USD y P(x) representa el beneficio, tambi&#xE9;n en miles de USD. La curva tiene forma de U invertida, que recuerda a una colina. En t&#xE9;rminos matem&#xE1;ticos, esta forma se describe como c&#xF3;ncava hacia abajo.Con niveles bajos de publicidad, aumentar el gasto impulsa las ventas y el beneficio de manera eficiente. Sin embargo, a medida que el gasto contin&#xFA;a aumentando, el beneficio adicional generado por cada nueva unidad de publicidad comienza a disminuir. Esto refleja rendimientos decrecientes. Si el gasto se vuelve demasiado alto, el beneficio acaba disminuyendo porque el coste de la publicidad adicional excede los ingresos que genera.La pendiente de la curva viene dada por la primera derivada, y el beneficio m&#xE1;ximo se produce cuando est&#xE1; pendiente se vuelve cero. Con este nivel de gasto, la curva alcanza su valor m&#xE1;s alto.La segunda derivada de la funci&#xF3;n es un n&#xFA;mero negativo constante. Una segunda derivada negativa indica que toda la curva es c&#xF3;ncava hacia abajo, lo que confirma que la funci&#xF3;n de beneficio se curva hacia abajo en todos sus puntos.

Dibujo de curvas y derivadas

Comprender el comportamiento de una funci&#xF3;n a trav&#xE9;s de sus primeras y segundas derivadas es esencial para analizar su gr&#xE1;fica. Las ...

Education

Educación

JoVE Core

Calculus

Cálculo

Unit

Unidad

Applications of Differentiation

Aplicaciones de la derivación

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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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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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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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The Mean Value Theorem links average change to instantaneous slope. It shows that when a function changes smoothly over an interval, the average change across the whole interval must match the change at one specific point inside it. The theorem is easy to see in a motion example. Imagine a vehicle traveling between two points on a road. Its velocity can change during the trip, beginning slowly, speeding up during acceleration, and slowing down near the end. Even when the speed is not...

first derivatives and the shape of a graph

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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...

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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

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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

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

Indeterminate forms appear when a limit gives an expression that cannot be read directly. Common examples are 0 divided by 0 and infinity divided by infinity. These results do not show the true behavior of the function near the point of interest, so more analysis is needed. L’Hôpital’s Rule is a useful method for handling these ambiguous limits. It applies when two differentiable functions approach 0 or infinity at the same time near the point being studied. Instead of finding the limit of the...

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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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application of differentiation to business

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Maximizing Bakery Revenue with Calculus

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Newton’s Method for Finding Roots

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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...

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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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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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