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

Considere uma função; suas primeiras e segundas derivadas são usadas para esboçar sua curva.A primeira derivada é zero quando x é igual a quatro, e indefinida quando x é igual a zero ou x é igual a seis. Esses pontos críticos definem os intervalos para análise.O sinal da primeira derivada determina se a curva está aumentando ou diminuindo.Uma mudança de sinal de negativo para positivo em x igual a zero fornece um mínimo local.Uma mudança de positivo para negativo em x igual a quatro gera um máximo local. Se o sinal não mudar em x igual a seis, não existe extremo ali.A segunda derivada é negativa nos intervalos de menos infinito até 0 e de 0 a 6, o que mostra que o gráfico é côncavo para baixo nesses intervalos.Mas a segunda derivada é positiva no intervalo x maior que seis, o que dá o gráfico côncavo para cima nesse intervalo.Por exemplo, considere uma função de lucro. A primeira derivativa acompanha a subida ou queda do lucro e identifica seus valores extremos.A segunda derivada mostra como a taxa de lucro muda, o que ajuda a esboçar a função.

esboço de curvas e derivadas

Curve Sketching and Derivatives

Compreender o comportamento de uma fun&#xE7;&#xE3;o por meio de suas primeiras e segundas derivadas &#xE9; essencial para analisar seu gr&#xE1;fico. As derivadas fornecem informa&#xE7;&#xF5;es sobre onde uma fun&#xE7;&#xE3;o cresce ou decresce, onde atinge m&#xE1;ximos ou m&#xED;nimos locais e como sua curvatura se comporta em diferentes intervalos.A primeira derivada de uma fun&#xE7;&#xE3;o revela a inclina&#xE7;&#xE3;o da linha tangente em qualquer ponto. Pontos onde a derivada &#xE9; zero ou indefinida s&#xE3;o considerados cr&#xED;ticos, pois frequentemente indicam poss&#xED;veis extremos ou mudan&#xE7;as no comportamento. Um m&#xED;nimo local ocorre quando a primeira derivada muda de negativa para positiva, enquanto um m&#xE1;ximo local &#xE9; encontrado quando ela muda de positiva para negativa. Se a derivada n&#xE3;o mudar de sinal em um ponto cr&#xED;tico, n&#xE3;o h&#xE1; extremo presente.A segunda derivada oferece informa&#xE7;&#xF5;es sobre a concavidade da fun&#xE7;&#xE3;o. Quando a segunda derivada &#xE9; negativa, o gr&#xE1;fico &#xE9; c&#xF4;ncavo para baixo, indicando que a inclina&#xE7;&#xE3;o da primeira derivada est&#xE1; diminuindo. Isso resulta em uma curva que se arqueia para baixo. Por outro lado, uma segunda derivada positiva implica que a fun&#xE7;&#xE3;o &#xE9; c&#xF4;ncava para cima, com a inclina&#xE7;&#xE3;o da primeira derivada aumentando e a curva se arqueando para cima. Mudan&#xE7;as na concavidade sugerem a presen&#xE7;a de pontos de inflex&#xE3;o, nos quais a curva passa de um tipo de concavidade para outro.Ao examinar os sinais e comportamentos da primeira e da segunda derivadas em diferentes intervalos, &#xE9; poss&#xED;vel construir um esbo&#xE7;o preciso da fun&#xE7;&#xE3;o. Essa abordagem combinada ajuda a identificar regi&#xF5;es de crescimento ou decrescimento, localizar m&#xE1;ximos e m&#xED;nimos locais e determinar a forma geral do gr&#xE1;fico da fun&#xE7;&#xE3;o.Aplica&#xE7;&#xF5;esPor exemplo, considere um modelo de lucro representado por uma fun&#xE7;&#xE3;o quadr&#xE1;tica: \begin{equation} P(x) = -2x^2 + 12x - 10 \end{equation}O gr&#xE1;fico representa uma curva de lucro, na qual x representa o gasto com publicidade em milhares de d&#xF3;lares americanos e P(x) representa o lucro, tamb&#xE9;m em milhares de d&#xF3;lares americanos. A curva apresenta a forma de um U invertido, que lembra uma colina. Em termos matem&#xE1;ticos, esse formato &#xE9; descrito como c&#xF4;ncavo para baixo.Em n&#xED;veis baixos de gasto com publicidade, o aumento desse gasto impulsiona as vendas e o lucro de forma eficiente. No entanto, &#xE0; medida que os gastos continuam a aumentar, o lucro adicional gerado por cada unidade adicional de publicidade come&#xE7;a a diminuir. Isso reflete rendimentos decrescentes. Se os gastos se tornarem muito altos, o lucro eventualmente diminui porque o custo da publicidade adicional excede a receita gerada.A inclina&#xE7;&#xE3;o da curva &#xE9; dada pela primeira derivada, e o lucro m&#xE1;ximo ocorre no ponto em que essa inclina&#xE7;&#xE3;o se torna zero. Nesse n&#xED;vel de gasto, a curva atinge seu valor m&#xE1;ximo.A segunda derivada da fun&#xE7;&#xE3;o &#xE9; um n&#xFA;mero negativo constante. Uma segunda derivada negativa indica que toda a curva &#xE9; c&#xF4;ncava para baixo, confirmando que a fun&#xE7;&#xE3;o de lucro se curva para baixo em todos os pontos.

Esboço de curvas e derivadas

Compreender o comportamento de uma fun&#xE7;&#xE3;o por meio de suas primeiras e segundas derivadas &#xE9; essencial para analisar seu gr&#xE1;fico. As ...

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JoVE Core

Calculus

Cálculo

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Applications of Differentiation

Aplicações da Diferenciação

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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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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...

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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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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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optimization problems

Pipe Length Around a Corner

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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

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Antiderivative Graph Shape and Slope

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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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