Name: rational functions with slant asymptotes
Uploaded: 2026-01-12T12:24:40.000Z
Duration: 1 min 27 s
Channel: JoVE

Una asíntota inclinada es una línea inclinada a la que se acerca el grafo de una función cuando x se vuelve muy grande o muy pequeño. En una función racional, existe una asíntota inclinada cuando el grado del numerador es exactamente uno mayor que el del denominador.Para determinar la asíntota inclinada, se aplica la división larga polinómica, dividiendo el numerador por el denominador.El cociente es una expresión lineal, y el resto forma una fracción. Cuando x se acerca a un infinito positivo o negativo, esa fracción tice a cero.La función se aproxima a la recta dada por el cociente, que es la asíntota inclinada.Las asíntocas verticales se encuentran poniendo el denominador igual a cero, y las intersecciones muestran dónde el gráfico cruza los ejes.Sin embargo, la asíntota inclinada define el comportamiento a largo plazo del grafo.Debido al grado superior en el numerador, no existe asíntota horizontal. El gráfico se curva bruscamente cerca de la asíntota vertical y se alinea con la asíntota inclinada en ambos extremos.En la vida real, una asíntota sesgada podría modelar una función de coste medio. Si factores como el desgaste de la máquina hacen que el coste total aumente cuadráticamente, el coste medio por artículo se acerca a una asíntota inclinada.

asíntotas oblicuas

Slant Asymptotes

El comportamiento de una funci&#xF3;n suele estar determinado por restricciones asint&#xF3;ticas, donde un t&#xE9;rmino predomina sobre otro, definiendo una tendencia limitante. En el escenario dado, el patr&#xF3;n matem&#xE1;tico sigue una funci&#xF3;n racional: un t&#xE9;rmino c&#xFA;bico en el numerador se divide por un t&#xE9;rmino al cuadrado en el denominador. Esto da como resultado una funci&#xF3;n con caracter&#xED;sticas distintivas que incluyen una as&#xED;ntota oblicua, puntos cr&#xED;ticos y regiones no definidas.
La validez de la funci&#xF3;n est&#xE1; determinada por el denominador, que debe ser diferente de cero. Esta restricci&#xF3;n divide el dominio en regiones separadas, evitando singularidades donde la funci&#xF3;n no est&#xE1; definida. Los puntos cr&#xED;ticos surgen donde la primera derivada es igual a cero, lo que indica transiciones en la direcci&#xF3;n de la funci&#xF3;n. Dado que la funci&#xF3;n corta el eje horizontal cuando el numerador es cero, las ra&#xED;ces reales del t&#xE9;rmino c&#xFA;bico definen estos puntos. Sin embargo, no hay intersecci&#xF3;n con el eje vertical, lo que indica que la funci&#xF3;n no pasa por el origen.
Una caracter&#xED;stica clave de esta funci&#xF3;n es la presencia de una as&#xED;ntota oblicua, que surge de la divisi&#xF3;n de t&#xE9;rminos polin&#xF3;micos. Dado que el numerador tiene un grado mayor que el denominador, la divisi&#xF3;n de polinomios revela un cociente lineal que representa la tendencia principal. La funci&#xF3;n sigue de cerca esta as&#xED;ntota para valores de entrada extremos, aunque nunca la corta ni la supera.
La tasa de cambio, determinada por la primera derivada, confirma una tendencia decreciente en una regi&#xF3;n hasta alcanzar un m&#xED;nimo local, seguida de una tendencia creciente. La transici&#xF3;n fluida entre estos comportamientos refleja la continuidad de la funci&#xF3;n. Adem&#xE1;s, la segunda derivada permanece positiva, lo que sugiere que la funci&#xF3;n mantiene una concavidad positiva. Esta propiedad garantiza que la funci&#xF3;n no oscile de forma impredecible, sino que se adhiera a una curvatura estable y constante. Estas funciones son comunes en f&#xED;sica e ingenier&#xED;a y modelan fen&#xF3;menos donde un l&#xED;mite estructural subyacente restringe el crecimiento. La as&#xED;ntota oblicua representa un l&#xED;mite te&#xF3;rico similar a los escenarios del mundo real donde una entidad se aproxima a un l&#xED;mite, pero nunca lo alcanza del todo.

Asíntotas oblicuas

El comportamiento de una funci&#xF3;n suele estar determinado por restricciones asint&#xF3;ticas, donde un t&#xE9;rmino predomina sobre otro, definiendo ...

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Educación

JoVE Core

Calculus

Cálculo

Unit

Unidad

Applications of Differentiation

Aplicaciones de la derivación

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

rational functions with slant asymptotes

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 roots of the cubic give the horizontal intercepts. There is no vertical intercept, so the graph does not pass through the origin.

Long division shows the slant asymptote, which is the linear quotient that the graph follows as x gets very large or very small. The function may approach that line, but it does not intersect or go past it. The graph also has critical points, where the first derivative equals zero. These points mark a change in direction.

The rate of change shows a decreasing region that reaches a local minimum and then rises again. The function is smooth through this transition, so it stays continuous. Its second derivative stays positive, which means the graph has upward concavity. That steady curve helps explain why the function does not wobble unpredictably.

Functions like this are useful in physics and engineering. They can model situations where growth is limited by a structural constraint. In that sense, the slant asymptote acts like a theoretical bound. The graph keeps getting close to the bound, but it never quite reaches it.

Rational Functions with Slant Asymptotes

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slant asymptotes

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key terms and concepts

Explore how a rational function with a cubic numerator and squared denominator creates a slant asymptote, critical points, and vertical restrictions.

Rational Functions with Slant Asymptotes | 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

Rational Functions with Slant Asymptotes

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