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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
asymptote of Lamé's cubic
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(Example)
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We will show that the Lamé's cubic
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(1) |
where $a$ is a positive constant, has the line $$y = \underbrace{-x}_{g(x)}$$ as its asymptote.
Because the equation (1) of the curve is symmetric with respect to $x$ and $y$ , the curve is symmetric about the line $y = x$ . From the solved form
![$\displaystyle y = \underbrace{\sqrt[3]{a^3-x^3}}_{f(x)}$](http://images.planetmath.org:8080/cache/objects/10403/js/img2.png "$\displaystyle y = \underbrace{\sqrt[3]{a^3-x^3}}_{f(x)}$") |
(2) |
of (1) we see that every real value of $x$ gives one point of the curve.
The difference $\Delta = f(x)\!-\!g(x)$ represents the distance of a point $(x,\,y)$ of the curve and the point of the asserted asymptote $y = -x$ with the same abscissa $x$ . We multiply the numerator and denominator with the expression $(\sqrt[3]{a^3-x^3})^2-x\sqrt[3]{a^3-x^3}+x^2$
for being able to utilise the polynomial formula $$(u+v)(u^2-uv+v^2) = u^3+v^3,$$ getting
Thus, $\displaystyle \Delta \to \frac{a^3}{\infty+\infty+\infty} = 0$ when $ |x| \to \infty$ (see the improper limits). According to the definition of asymptote, the line $y = -x$ is asymptote of Lamé's cubic.
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"asymptote of Lamé's cubic" is owned by pahio.
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Cross-references: improper limits, formula, polynomial, expression, denominator, numerator, abscissa, distance, difference, point, real, symmetric about, symmetric, curve, equation, asymptote, line, positive
There is 1 reference to this entry.
This is version 4 of asymptote of Lamé's cubic, born on 2008-03-14, modified 2008-03-16.
Object id is 10403, canonical name is AsymptoteOfLamesCubic.
Accessed 1169 times total.
Classification:
| AMS MSC: | 26C05 (Real functions :: Polynomials, rational functions :: Polynomials: analytic properties, etc.) | | | 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space) |
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Pending Errata and Addenda
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![\begin{pspicture}(-4,-4)(4,4) \psaxes[Dx=9,Dy=9]{->}(0,0)(-3.5,-3.5)(3.5,3.5) \r... ...\rput(0.5,-4){Lam\'e's cubic\, $y = \sqrt[3]{a^3-x^3}$\, (blue)} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/10403/js/img3.png "\begin{pspicture}(-4,-4)(4,4) \psaxes[Dx=9,Dy=9]{->}(0,0)(-3.5,-3.5)(3.5,3.5) \r... ...\rput(0.5,-4){Lam\'e's cubic\, $y = \sqrt[3]{a^3-x^3}$\, (blue)} \end{pspicture}")
![$\displaystyle = \frac{\sqrt[3]{a^3-x^3}+x}{1}$](http://images.planetmath.org:8080/cache/objects/10403/js/img6.png "$\displaystyle = \frac{\sqrt[3]{a^3-x^3}+x}{1}$")
![$\displaystyle = \frac{(\sqrt[3]{a^3-x^3})^3+x^3}{(\sqrt[3]{a^3-x^3})^2-x\sqrt[3]{a^3-x^3}+x^2}$](http://images.planetmath.org:8080/cache/objects/10403/js/img7.png "$\displaystyle = \frac{(\sqrt[3]{a^3-x^3})^3+x^3}{(\sqrt[3]{a^3-x^3})^2-x\sqrt[3]{a^3-x^3}+x^2}$")
![$\displaystyle = \frac{a^3}{(\sqrt[3]{a^3-x^3})^2-x\sqrt[3]{a^3-x^3}+x^2}.$](http://images.planetmath.org:8080/cache/objects/10403/js/img8.png "$\displaystyle = \frac{a^3}{(\sqrt[3]{a^3-x^3})^2-x\sqrt[3]{a^3-x^3}+x^2}.$")