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

Eğik asimptote, x çok büyük veya çok küçük olduğunda bir fonksiyonun grafiğinin yaklaştığı eğik bir çizgidir. Rasyonel bir fonksiyonda, eğik asimptot vardır; eğer paydanın derecesi tam olarak bir oran daha yüksek olursa.Eğik asimptotu belirlemek için, polinom uzun bölme uygulanır ve payda ile payda bölünür.Bölüm doğrusal bir ifadedir ve kalan bir kesir oluşturur. x pozitif veya negatif sonsuzluğa yaklaştıkça, o kesim sıfıra yaklaşır.Fonksiyon, bölüm tarafından verilen doğruya, yani eğik asimptote'a yaklaşır.Dikey asimptotlar, paydayı sıfıra eşit olarak ayarlayarak bulunur ve kesişmeler grafiğin eksenleri nerede kesiştiğini gösterir.Ancak eğik asimptot, grafiğin uzun vadeli davranışını tanımlar.Gösterici derecesi yüksek olduğundan, yatay asimptot yoktur. Grafik, dikey asimptota yakın keskin bir şekilde kıvrılır ve eğimli asimptotla her iki uçta hizalanır.Gerçek hayatta, eğik asimptot ortalama bir maliyet fonksiyonunu modelleyebilir. Makine aşınması gibi faktörler toplam maliyetin dörtlü olarak artmasına neden olursa, eşya başına ortalama maliyet eğik asimptota yaklaşır.

eğik asimptotlar

Slant Asymptotes

Bir fonksiyonun davran&#x131;&#x15F;&#x131; &#xE7;o&#x11F;u zaman asimptotik s&#x131;n&#x131;rlamalar taraf&#x131;ndan belirlenir; bu durumda bir terim di&#x11F;erine bask&#x131;n gelerek s&#x131;n&#x131;rlay&#x131;c&#x131; bir e&#x11F;ilimi tan&#x131;mlar. Verilen senaryoda matematiksel yap&#x131; bir rasyonel fonksiyon &#x15F;eklindedir: payda ikinci dereceden bir terim i&#xE7;erirken, pay &#xFC;&#xE7;&#xFC;nc&#xFC; dereceden bir terim i&#xE7;erir. Bu durum, e&#x11F;ik asimptot, kritik noktalar ve tan&#x131;ms&#x131;zl&#x131;k b&#xF6;lgeleri gibi ay&#x131;rt edici &#xF6;zelliklere sahip bir fonksiyonun ortaya &#xE7;&#x131;kmas&#x131;na neden olur.
Fonksiyonun tan&#x131;ml&#x131; oldu&#x11F;u de&#x11F;erler, s&#x131;f&#x131;r olmamas&#x131; gereken payda taraf&#x131;ndan belirlenir. Bu k&#x131;s&#x131;tlama, tan&#x131;m k&#xFC;mesini ayr&#x131; b&#xF6;lgelere ay&#x131;rarak u&#xE7; de&#x11F;erlerin g&#xF6;r&#xFC;lebildi&#x11F;i tekillik noktalar&#x131;n&#x131; d&#x131;&#x15F;ar&#x131;da b&#x131;rak&#x131;r. Kritik noktalar, birinci t&#xFC;revin s&#x131;f&#x131;ra e&#x15F;it oldu&#x11F;u durumlarda ortaya &#xE7;&#x131;kar ve fonksiyonun artma/azalma y&#xF6;n&#xFC;ndeki ge&#xE7;i&#x15F;leri ifade eder. Fonksiyon, pay&#x131;n s&#x131;f&#x131;r oldu&#x11F;u noktalarda yatay ekseni kesti&#x11F;inden, k&#xFC;bik terimin ger&#xE7;ek k&#xF6;kleri bu noktalar&#x131; belirler. Buna kar&#x15F;&#x131;l&#x131;k, y-kesi&#x11F;i bulunmamakta; bu da fonksiyonun orijinden ge&#xE7;medi&#x11F;ini g&#xF6;stermektedir.
Bu fonksiyonun temel &#xF6;zelliklerinden biri, polinom terimlerinin b&#xF6;l&#xFC;nmesi sonucunda ortaya &#xE7;&#x131;kan e&#x11F;ik asimptotun varl&#x131;&#x11F;&#x131;d&#x131;r. Pay&#x131;n derecesinin paydan&#x131;n derecesinden b&#xFC;y&#xFC;k olmas&#x131; nedeniyle uygulanan uzun b&#xF6;lme y&#xF6;ntemi, y&#xF6;nlendirici e&#x11F;ilimi temsil eden do&#x11F;rusal bir b&#xF6;l&#xFC;m polinomu elde edilmesini sa&#x11F;lar. Fonksiyon, ba&#x11F;&#x131;ms&#x131;z de&#x11F;i&#x15F;kenin u&#xE7; de&#x11F;erlerinde bu asimptotu yak&#x131;ndan izler; ancak hi&#xE7;bir zaman onu kesmez ya da a&#x15F;maz.
Birinci t&#xFC;rev ile ifade edilen de&#x11F;i&#x15F;im h&#x131;z&#x131;, fonksiyonun bir b&#xF6;lgede yerel bir minimuma ula&#x15F;&#x131;ncaya kadar azalan, sonras&#x131;nda ise artan bir e&#x11F;ilim sergiledi&#x11F;ini do&#x11F;rular. Bu davran&#x131;&#x15F;lar aras&#x131;ndaki yumu&#x15F;ak ge&#xE7;i&#x15F;, fonksiyonun s&#xFC;reklili&#x11F;ini yans&#x131;t&#x131;r. Ayr&#x131;ca ikinci t&#xFC;revin pozitif olmas&#x131;, fonksiyonun konkavl&#x131;&#x11F;&#x131;n&#x131;n yukar&#x131; y&#xF6;nl&#xFC; oldu&#x11F;unu g&#xF6;stermektedir. Bu &#xF6;zellik, fonksiyonun d&#xFC;zensiz sal&#x131;n&#x131;mlar g&#xF6;stermesini engelleyerek istikrarl&#x131; ve tutarl&#x131; bir e&#x11F;rilik sergilemesini sa&#x11F;lar. Bu t&#xFC;r fonksiyonlar, alt&#x131;nda yatan yap&#x131;sal bir s&#x131;n&#x131;r&#x131;n b&#xFC;y&#xFC;meyi k&#x131;s&#x131;tlad&#x131;&#x11F;&#x131; durumlar&#x131; modellemek amac&#x131;yla fizik ve m&#xFC;hendislik alanlar&#x131;nda s&#x131;kl&#x131;kla kullan&#x131;l&#x131;r. E&#x11F;ik asimptot ise, bir niceli&#x11F;in bir s&#x131;n&#x131;ra giderek yakla&#x15F;t&#x131;&#x11F;&#x131; ancak bu s&#x131;n&#x131;ra hi&#xE7;bir zaman tam olarak eri&#x15F;emedi&#x11F;i ger&#xE7;ek d&#xFC;nya senaryolar&#x131;na benzer &#x15F;ekilde, teorik bir s&#x131;n&#x131;r&#x131; temsil eder.

Eğik Asimptotlar

Bir fonksiyonun davran&#x131;&#x15F;&#x131; &#xE7;o&#x11F;u zaman asimptotik s&#x131;n&#x131;rlamalar taraf&#x131;ndan belirlenir; bu durumda bir terim ...

Education

JoVE Core

Calculus

Kalkülüs

Unit

Birim

Applications of Differentiation

Türevin Uygulamaları

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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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Using First Derivatives to Find Graph Changes

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second derivatives and the shape of a graph

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Using Second Derivatives to Read Curvature

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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...

optimization problems

Pipe Length Around a Corner

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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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