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

Asymptota skośna to przechylona linia, do której graf funkcji zbliża się, gdy x staje się bardzo duże lub bardzo małe. W funkcji wymiernej asymptota skośna istnieje, gdy stopień licznika jest dokładnie o jeden wyższy niż mianownika.Aby wyznaczyć asymptotę skośną, stosuje się wielomianowe dzielenie długie, dzieląc licznik przez mianownik.Iloraz jest wyrażeniem liniowym, a reszta tworzy ułamek. Gdy x zbliża się do dodatniej lub ujemnej nieskończoności, ta frakcja zbliża się do zera.Funkcja zbliża się do prostej danej przez iloraz, która jest asymptotą skośną.Pionowe asymptoty wyznacza się, ustawiając mianownik równy zero, a przecięcia pokazują, gdzie graf przecina osi.Asymptota skośna jednak definiuje długoterminowe zachowanie grafu.Ze względu na wyższy stopień w liczniku nie istnieje pozioma asymptota. Graf ostro zakrzywia się w pobliżu asymptoty pionowej i wyrównuje się z asymptotą skośną na obu końcach.W rzeczywistości asymptota skośna mogłaby modelować funkcję średniego kosztu. Jeśli czynniki takie jak zużycie maszyn powodują wzrost całkowitych kosztów kwadratowo, średni koszt na przedmiot zbliża się do asymptoty skośnej.

asymptoty ukośne

Slant Asymptotes

Zachowanie funkcji cz&#x119;sto jest determinowane przez ograniczenia asymptotyczne, w kt&#xF3;rych jeden sk&#x142;adnik dominuje nad drugim, definiuj&#x105;c trend graniczny. W podanym scenariuszu zale&#x17C;no&#x15B;&#x107; matematyczna jest zgodna z funkcj&#x105; wymiern&#x105;: cz&#x142;on sze&#x15B;cienny w liczniku jest dzielony przez wyraz kwadratowy w mianowniku. W rezultacie powstaje funkcja o odr&#x119;bnych cechach, w tym asymptota uko&#x15B;na, punkty krytyczne oraz obszary nieokre&#x15B;lono&#x15B;ci.
Okre&#x15B;lono&#x15B;&#x107; funkcji zale&#x17C;y od mianownika, kt&#xF3;ry musi by&#x107; r&#xF3;&#x17C;ny od zera. To ograniczenie dzieli dziedzin&#x119; na oddzielne obszary, pozwalaj&#x105;c unikn&#x105;&#x107; osobliwo&#x15B;ci, w kt&#xF3;rych mog&#x105; wyst&#x119;powa&#x107; warto&#x15B;ci skrajne. Punkty krytyczne pojawiaj&#x105; si&#x119; tam, gdzie pierwsza pochodna jest r&#xF3;wna zeru, co oznacza przej&#x15B;cia mi&#x119;dzy kierunkami zmian funkcji. Poniewa&#x17C; funkcja przecina o&#x15B; poziom&#x105;, gdy licznik jest r&#xF3;wny zeru, pierwiastki rzeczywiste wyrazu sze&#x15B;ciennego wyznaczaj&#x105; te miejsca przeci&#x119;cia. Brakuje jednak punktu przeci&#x119;cia z osi&#x105; OY, co wskazuje, &#x17C;e funkcja nie przechodzi przez pocz&#x105;tek uk&#x142;adu wsp&#xF3;&#x142;rz&#x119;dnych.
Kluczow&#x105; cech&#x105; tej funkcji jest obecno&#x15B;&#x107; asymptoty uko&#x15B;nej, kt&#xF3;ra powstaje w wyniku dzielenia wyraz&#xF3;w wielomianowych. Poniewa&#x17C; licznik ma wy&#x17C;szy stopie&#x144; ni&#x17C; mianownik, dzielenie wielomian&#xF3;w ujawnia iloraz liniowy reprezentuj&#x105;cy trend przewodni. Funkcja zbli&#x17C;a si&#x119; do tej asymptoty dla skrajnych warto&#x15B;ci argumentu, cho&#x107; nigdy jej nie przecina ani jej nie przekracza.
Tempo zmiany, opisane przez pierwsz&#x105; pochodn&#x105;, potwierdza monotoniczno&#x15B;&#x107; malej&#x105;c&#x105; w jednym obszarze a&#x17C; do osi&#x105;gni&#x119;cia minimum lokalnego, po kt&#xF3;rym nast&#x119;puje monotoniczno&#x15B;&#x107; rosn&#x105;ca. P&#x142;ynne przej&#x15B;cie mi&#x119;dzy tymi zachowaniami odzwierciedla ci&#x105;g&#x142;o&#x15B;&#x107; funkcji. Dodatkowo druga pochodna pozostaje dodatnia, co wskazuje, &#x17C;e funkcja zachowuje wypuk&#x142;o&#x15B;&#x107; skierowan&#x105; w g&#xF3;r&#x119;. Ta w&#x142;a&#x15B;ciwo&#x15B;&#x107; zapewnia, &#x17C;e funkcja nie wykazuje nieprzewidywalnych oscylacji, lecz zachowuje stabiln&#x105; i sp&#xF3;jn&#x105; krzywizn&#x119;. Takie funkcje cz&#x119;sto pojawiaj&#x105; si&#x119; w fizyce i in&#x17C;ynierii, modeluj&#x105;c zjawiska, w kt&#xF3;rych le&#x17C;&#x105;ca u ich podstaw granica strukturalna ogranicza wzrost. Asymptota uko&#x15B;na przedstawia teoretyczn&#x105; granic&#x119; analogiczn&#x105; do sytuacji ze &#x15B;wiata rzeczywistego, w kt&#xF3;rych obiekt zbli&#x17C;a si&#x119; do granicy, lecz nigdy jej nie osi&#x105;ga.

Asymptoty ukośne

Zachowanie funkcji cz&#x119;sto jest determinowane przez ograniczenia asymptotyczne, w kt&#xF3;rych jeden sk&#x142;adnik dominuje nad drugim, ...

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Calculus

Rachunek różniczkowy i całkowy

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Applications of Differentiation

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KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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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