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

Consideriamo una porta ad arco che segue una precisa funzione coseno iperbolica.Poiché la curva è simmetrica, valutare la funzione all'origine fornisce l'altezza massima.Il compito consiste nell'identificare le posizioni in cui l'altezza raggiunge 5 metri e calcolare le equazioni della retta tangente in queste posizioni per stimare la pendenza richiesta per la progettazione strutturale.Trovare dove la struttura raggiunge orizzontalmente una data altezza richiede di regolare il valore della funzione.Risolvendo per le posizioni orizzontali si identificano due posizioni che soddisfano questa condizione.Ora, per calcolare le equazioni della retta tangente, il primo passo è trovare la pendenza della curva in queste posizioni specifiche.Per farlo, si deriva la funzione passo dopo passo e si calcola la derivata della funzione in questi punti.Sostituendo la posizione orizzontale nell'equazione derivata si ottengono pendenze di circa +0,98 e –0,98, riflettendo la simmetria della struttura.Poi, consideriamo la forma punto-pendenza di una retta. Si sostituiscono queste pendenze e le corrispondenti coordinate per ottenere le equazioni delle rette tangenti.Questo mostra l'applicazione delle funzioni iperboliche nel calcolo e nella modellazione del mondo reale.

funzioni iperboliche e funzioni iperboliche inverse: risoluzione di problemi

Una porta ad arco pu&#xF2; essere modellata efficacemente mediante una funzione del coseno iperbolico, poich&#xE9; questo tipo di funzione &#xE8; liscia e simmetrica rispetto all&#x2019;asse verticale. Quando l&#x2019;arco &#xE8; centrato nell&#x2019;origine, l&#x2019;altezza massima si raggiunge nel punto centrale. Tale simmetria garantisce che qualunque altezza al di sotto della sommit&#xE0; dell&#x2019;arco venga raggiunta in due posizioni orizzontali equidistanti dall&#x2019;asse di simmetria, ma situate su lati opposti.
Per determinare in quali punti la porta raggiunge un&#x2019;altezza di cinque metri, si impone che la funzione assuma il valore di cinque metri e si individuano le corrispondenti posizioni orizzontali. La risoluzione di tale relazione mostra che esistono esattamente due posizioni in cui la struttura raggiunge tale altezza, riflettendo la simmetria intrinseca della forma del coseno iperbolico.
Per costruire le rette tangenti in tali posizioni, occorre valutare l&#x2019;inclinazione dell&#x2019;arco. Ci&#xF2; si ottiene determinando la rapidit&#xE0; con cui l&#x2019;altezza della curva varia lungo la direzione orizzontale. La valutazione di tale tasso di variazione nei due punti in cui l&#x2019;altezza &#xE8; pari a cinque metri consente di ottenere pendenze uguali in valore assoluto ma di segno opposto. Una pendenza &#xE8; positiva, a indicare che il lato sinistro dell&#x2019;arco cresce in direzione del centro, mentre l&#x2019;altra &#xE8; negativa, a indicare che il lato destro decresce allontanandosi dal centro.
Utilizzando ciascun punto dell&#x2019;arco insieme alla pendenza corrispondente, si possono scrivere le rette tangenti che approssimano localmente la forma della struttura in prossimit&#xE0; del livello dei cinque metri. Queste rette forniscono una rappresentazione lineare della curva nelle immediate vicinanze di tali punti e risultano utili per stimare l&#x2019;inclinazione richiesta nella progettazione strutturale e ingegneristica.

Funzioni iperboliche e funzioni iperboliche inverse: risoluzione di problemi

Una porta ad arco pu&#xF2; essere modellata efficacemente mediante una funzione del coseno iperbolico, poich&#xE9; questo tipo di funzione &#xE8; liscia e ...

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

Education

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Calculus

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Differentiation Rules

Regole di derivazione

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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

Slope Rules for Simple Functions

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 quotient rule

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limits on trigonometric functions

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the chain rule

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implicit differentiation

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derivatives of inverse trigonometric functions

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...

exponential growth

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related rates

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linearization and approximation

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application of linearization and approximation

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hyperbolic functions

Modeling Hanging Cables with Hyperbolic Curves

Hyperbolic functions help describe the shape of a hanging cable. When a flexible cable is suspended between two points at the same height, it naturally forms a catenary. That curve comes from the balance between the cable’s weight and the tension along its length. It shows a state of mechanical equilibrium. The true shape of this hanging cable is not captured well by a simple approximation. Instead, it is described with hyperbolic functions. These functions are related to exponential functions.

inverse hyperbolic functions and their derivatives

inverse-hyperbolic-functions-and-their-derivatives

Deriving the Slope of a Catenary Curve

A catenary curve describes the shape of a suspension bridge cable hanging under its own weight. The curve is modeled with the hyperbolic cosine function, which helps show how gravity and tension act along the cable. When the vertical position on the cable is known, the matching horizontal position can be found with the inverse hyperbolic cosine function. Inverse hyperbolic functions are a family of functions that includes inverse hyperbolic sine, cosine, and tangent. Their corresponding...

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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