Name: tangent lines on a hyperbolic arch
Uploaded: 2026-01-12T12:24:18.000Z
Duration: 1 min 30 s
Channel: JoVE

Considere um portão arqueado que segue uma função cosseno hiperbólica precisa.Como a curva é simétrica, avaliar a função na origem fornece a altura máxima.A tarefa é identificar as posições onde a altura atinge 5 metros e calcular as equações da linha tangente nessas posições para estimar a inclinação necessária para o projeto estrutural.Encontrar onde a estrutura atinge horizontalmente uma determinada altura requer ajustar o valor da função.Resolver as posições horizontais identifica dois locais que satisfazem essa condição.Agora, para calcular as equações da reta tangente, o primeiro passo é encontrar a inclinação da curva nessas posições específicas.Para isso, diferencie a função passo a passo e calcule a derivada da função nesses pontos.Substituindo a posição horizontal na equação derivada obtém inclinações de cerca de +0,98 e –0,98, refletindo a simetria da estrutura.Em seguida, considere a forma ponto-inclinação de uma linha. Substitua essas inclinações e as coordenadas correspondentes para obter as equações das retas tangentes.Isso mostra a aplicação de funções hiperbólicas no cálculo e na modelagem do mundo real.

funções hiperbólicas e hiperbólicas inversas: resolução de problemas

Um port&#xE3;o em arco pode ser modelado de forma eficaz usando um perfil de cosseno hiperb&#xF3;lico, pois esse tipo de fun&#xE7;&#xE3;o &#xE9; suave e sim&#xE9;trico no eixo vertical. Quando o arco est&#xE1; centrado na origem, sua altura m&#xE1;xima ocorre no centro. Essa simetria garante que qualquer altura abaixo da coroa do arco seja atingida em duas posi&#xE7;&#xF5;es horizontais equidistantes da linha central, mas situadas em lados opostos.
Para determinar onde o port&#xE3;o atinge uma altura de cinco metros, ajusta-se o valor da fun&#xE7;&#xE3;o para cinco metros e as posi&#xE7;&#xF5;es horizontais correspondentes s&#xE3;o identificadas. Resolver essa rela&#xE7;&#xE3;o mostra que existem exatamente dois locais em que a estrutura atinge essa altura, refletindo a simetria inerente &#xE0; forma do cosseno hiperb&#xF3;lico.
Para construir as retas tangentes nesses locais, a inclina&#xE7;&#xE3;o do arco deve ser avaliada. Isso &#xE9; feito determinando a taxa de varia&#xE7;&#xE3;o da fun&#xE7;&#xE3;o em rela&#xE7;&#xE3;o &#xE0; dire&#xE7;&#xE3;o x. Avaliar essa taxa de varia&#xE7;&#xE3;o nas duas posi&#xE7;&#xF5;es em que a altura &#xE9; de cinco metros resulta em coeficientes angulares de mesma magnitude, mas com sinais opostos. Um dos coeficientes angulares &#xE9; positivo, indicando que o lado esquerdo do arco se eleva em dire&#xE7;&#xE3;o ao centro, enquanto o outro &#xE9; negativo, indicando que o lado direito do arco desce, afastando-se do ponto central.
Utilizando cada ponto do arco juntamente com seu respectivo coeficiente angular, podem ser obtidas retas tangentes que aproximam-se localmente a forma da estrutura pr&#xF3;xima ao n&#xED;vel de cinco metros. Essas retas tangentes fornecem representa&#xE7;&#xF5;es lineares da curva nas imedia&#xE7;&#xF5;es desses pontos e s&#xE3;o &#xFA;teis para estimar o coeficiente angular necess&#xE1;rio para o projeto estrutural e de engenharia.

Funções hiperbólicas e hiperbólicas inversas: resolução de problemas

Um port&#xE3;o em arco pode ser modelado de forma eficaz usando um perfil de cosseno hiperb&#xF3;lico, pois esse tipo de fun&#xE7;&#xE3;o &#xE9; suave e ...

hyperbolic-and-inverse-hyperbolic-functions-problem-solving

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Hyperbolic and Inverse Hyperbolic Functions: Problem Solving

tangent lines on a hyperbolic arch

Hyperbolic cosine functions can model an arched gate with a smooth, symmetric shape. When the arch is centered at the origin, its highest point is at the center. That symmetry means any lower height appears at two horizontal points on opposite sides of the centerline.

The arch is checked at a height of five meters by adjusting the function to match that level. Solving for the horizontal positions shows that there are exactly two points where the gate reaches five meters. These points are equal distances from the center, which matches the symmetry of the hyperbolic cosine curve.

The steepness of the arch at those points is found by evaluating the rate of change with respect to the horizontal direction. This rate of change gives the slope of the curve, which is used to build tangent lines. The two slopes have the same size but opposite signs. One is positive on the left side of the arch, and the other is negative on the right side.

Using each point and its slope, tangent lines can be written for the five-meter level. A tangent line is a straight line that best matches the curve right near one point. These linear forms give a local approximation of the arch and help estimate the incline needed for structural and engineering design.

Tangent Lines on a Hyperbolic Arch

derivatives of simple functions

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Derivatives show how fast a function changes, and they also describe slope. A derivative can tell the slope of a straight line or the slope of a tangent line to a curve at one point. In this lesson, the idea is tied to a roller coaster track, where the derivative of the horizontal position tells how steep the ride is at each spot. A flat roller coaster section can be modeled with a constant function, f(x) = c, where c is a fixed height. Its derivative is f'(x) = 0, so the slope is zero and...

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the chain rule

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derivatives-of-inverse-trigonometric-functions

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application of rates of change

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Velocity and acceleration describe how a car moves along a highway. The car’s position changes over time, so it can be written as a function of time. Rates of change help measure motion in a clear mathematical way. Velocity tells how position changes with time. Average velocity is found by dividing the change in position by the change in time. It gives the overall speed of the car during a time interval, but it does not show changes inside that interval. To find the velocity at one exact...

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linearization and approximation

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hyperbolic functions

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inverse hyperbolic functions and their derivatives

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Deriving the Slope of a Catenary Curve

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hyperbolic and inverse hyperbolic functions: problem solving

Hyperbolic cosine functions can model an arched gate with a smooth, symmetric shape. When the arch is centered at the origin, its highest point is at the center. That symmetry means any lower height appears at two horizontal points on opposite sides of the centerline. The arch is checked at a height of five meters by adjusting the function to match that level. Solving for the horizontal positions shows that there are exactly two points where the gate reaches five meters. These points are equal...

key terms and concepts

Find the two points where a hyperbolic arch reaches five meters, then use slopes to write tangent lines and estimate the local incline.

Tangent Lines on a Hyperbolic Arch | 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

Tangent Lines on a Hyperbolic Arch

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