Plotting Curves

Find the vertices, i.e.the points where the tangent is parallel to one of the coordinate axes and where there is, moreover, no point of inflection, of the curve $x^{2}+xy+y^{2}=3$.

Find the vertices, i.e.the points where the tangent is parallel to one of the coordinate axes and where there is, moreover, no point of inflection, of the curve $x^{4}+y^{4}=x^{2}+y^{2}$.

Find the vertices, i.e.the points where the tangent is parallel to one of the coordinate axes and where there is, moreover, no point of inflection, of the curve $(x^{2}+y^{2})^{2}=xy$.

Find the vertices, i.e.the points where the tangent is parallel to one of the coordinate axes and where there is, moreover, no point of inflection, of the curve $x^{4}+4x^{2}y^{2}-6a^{2}x^{2}+a^{4}=0$.

Find the vertices, i.e.the points where the tangent is parallel to one of the coordinate axes and where there is, moreover, no point of inflection, of the curve $y=\sin \dfrac {1}{x}$.

Find the vertices, i.e.the points where the tangent is parallel to one of the coordinate axes and where there is, moreover, no point of inflection, of the curve $y=\sin x^{2}$.

Find the horizontal distance between successive vertices of the curve $y=e^{-ax}\cos bx$.

Find the points of inflection of the curve $y=x^{4}-6x^{2}$.

Find the points of inflection of the curve $x^{3}+y^{3}=a^{3}$.

Find the points of inflection of the curve $x^{3}+y^{3}=x^{2}$.

Find the points of inflection of the curve $(a^{2}+x^{2})y=ax^{2}$.

Find the points of inflection of the curve $y=1+\sqrt[3]{x^{5}}$.

Find the points of inflection of the curve $y=x^{4}e^{-x}$.

Find the points of inflection of the curve $y=e^{1/x}$.

Find the points of inflection of the curve $y=x^{2}\ln x$.

Find the points of inflection of the curve $y=\tg x$.

Find the points of inflection of the curve $y=\sin^{4}x+\cos^{4}x$.

Find the points of inflection of the curve $r\cos^{3}\varphi =1$.

Find the points of inflection of the curve $r=a(\tg \varphi -1)$.

In investigating the shape of a curve near a given point $(a,b)$ of it, an expansion in a Taylor series is useful. If $f(x,y)=0$ is the equation of the curve, then, expanding the left-hand side in powers of the differences $x-a=\xi$, $y-b=\eta$, we obtain, by virtue of the equality $f(a,b)=0$:

where $p$, $q$, $r$, $s$, and $t$ denote, respectively, the values of the derivatives $\dfrac {\partial f}{\partial x}$, $\dfrac {\partial f}{\partial y}$, $\dfrac {\partial^{2}f}{\partial x^{2}}$, $\dfrac {\partial^{2}f}{\partial x\, \partial y}$, and $\dfrac {\partial^{2}f}{\partial y^{2}}$ at $x=a$, $y=b$.

If $p^{2}+q^{2}>0$, then for small values of $\left|\xi \right|$ and $\left|\eta \right|$ the curve is close to the line $p\xi +q\eta =0$, i.e.to the line $p(x-a)+q(y-b)=0$. This line is the tangent to the curve at the point $(a,b)$. If, however, $p=0$ and $q=0$, i.e.if at the given point both derivatives $\dfrac {\partial f}{\partial x}$ and $\dfrac {\partial f}{\partial y}$ vanish, then the equation above becomes:

The point $(a,b)$ is a singular point of the curve.

If, moreover, $s^{2}-rt>0$, then the point $(a,b)$ is a double point. Two branches of the curve intersect there. The slopes of the tangents to these branches equal the roots of the quadratic equation $t\sigma^{2}+2s\sigma +r=0$.

If $s^{2}-rt<0$, then the point $(a,b)$ is an isolated point: sufficiently near it there are no points of the curve. If $s^{2}-rt=0$, then to fully investigate the nature of the singular point one must take into account further terms of the Taylor expansion. Usually in this case $(a,b)$ is a cusp of the curve.

Investigate the singular points of the algebraic curve $y^{2}=x^{3}$.

Investigate the singular points of the algebraic curve $y^{2}=bx^{3}+ax^{2}$.

Investigate the singular points of the algebraic curve $y^{2}=ax^{2}+bx^{5}$.

Investigate the singular points of the algebraic curve $x^{3}+y^{3}-xy=0$.

Investigate the singular points of the algebraic curve $x^{4}-2ax^{2}y+2ay^{3}+a^{2}y^{2}=0$; $a\neq 0$.

Investigate the singular points of the algebraic curve $x^{6}-2a^{2}x^{3}y-b^{3}y^{3}=0$.

Investigate the singular points of the algebraic curve $2y^{3}x-y^{4}=x(y-x)^{2}$.

Investigate the singular points of the algebraic curve $x^{4}+y^{4}=x^{2}+y^{2}$.

For what relation between $a$ and $b$ does the curve

have a double point?

Prove that for $4a^{3}+27b^{2}<0$ the curve $y^{2}=x^{3}+ax+b$ consists of two separate parts, while for $4a^{3}+27b^{2}>0$ it is a single connected curve.

In studying transcendental curves, Taylor's formula is not always applicable, so transcendental curves have singular points of kinds not found on algebraic curves (such are, for example, points of cessation and corner points). Investigate the singular points of the transcendental curve $y=x^{x}$.

Investigate the singular points of the transcendental curve $y=x\ln x$.

Investigate the singular points of the transcendental curve $y\ln x=1$.

Investigate the singular points of the transcendental curve $y=e^{1/x}$.

Investigate the singular points of the transcendental curve $y=\dfrac {1}{1+e^{1/x}}$.

Investigate the singular points of the transcendental curve $y=\dfrac {x}{1+e^{1/x}}$.

Investigate the singular points of the transcendental curve $y=\arctg \dfrac {1}{x}$.

Investigate the singular points of the transcendental curve $y=x\arctg \dfrac {1}{x}$.

Investigate the singular points of the transcendental curve $y=\sin \dfrac {1}{x}$.

Investigate the singular points of the transcendental curve $y=x\sin \dfrac {1}{x}$.

Investigate the singular points of the transcendental curve $y=x^{2}\sin \dfrac {1}{x}$.

Investigate the singular points of the transcendental curve $x^{y}=y^{x}$.

In finding the asymptotes of a curve one should distinguish two cases — asymptotes parallel to the axis $Oy$, and asymptotes not parallel to it. The former have equations $x=x_{0}$, where $x_{0}$ is the value of $x$ at which $y=f(x)$ becomes infinite. A line $y=ax+b$, not parallel to the axis $Oy$, will be an asymptote of the curve if, as $x\rightarrow \infty$, the points of the curve satisfy the equality

It follows that $a=\lim \limits_{x\rightarrow \infty }\dfrac {y}{x}$, $b=\lim \limits_{x\rightarrow \infty }(y-ax)$.

Find the asymptotes of the curve $y=\dfrac {x}{x-1}$.

Find the asymptotes of the curve $y=\dfrac {1}{(x-3)^{2}}$.

Find the asymptotes of the curve $y=\pm \sqrt{\dfrac {x-1}{x+1}}$.

Find the asymptotes of the curve $y=x+\dfrac {1}{x}$.

Find the asymptotes of the curve $y=\dfrac {1}{x}+\dfrac {1}{x-1}+\dfrac {1}{x-2}$.

Find the asymptotes of the curve $y^{2}=\dfrac {x^{3}+ax^{2}}{x-a}$.

Find the asymptotes of the curve $y^{2}=\dfrac {x^{3}-a^{3}}{x+b}$.

Find the asymptotes of the curve $y^{3}-x^{3}=x^{2}+y^{2}$.

Find the asymptotes of the curve $x^{3}+y^{3}=a^{3}$.

Find the asymptotes of the curve $x^{3}+y^{3}=3axy$.

Find the asymptotes of the curve $x^{3}+y^{3}+2y-x=0$.

Find the asymptotes of the curve $r\varphi =a$.

Study the shape of the curve $y^{2}=x^{3}(2-x)$, first solving the equation of the curve for one of the coordinates.

Study the shape of the curve $x^{4}+y^{4}=a^{4}$, first solving the equation of the curve for one of the coordinates.

Study the shape of the curve $x^{2n}+y^{2n}=a^{2n}$, where $n$ is a large positive integer, first solving the equation of the curve for one of the coordinates.

Study the shape of the curve $x^{4}-4x^{2}y^{2}-6x^{2}-4y^{2}=0$, first solving the equation of the curve for one of the coordinates.

Study the shape of the curve $x^{4}-6x^{2}y+25y^{2}-16x^{2}=0$, first solving the equation of the curve for one of the coordinates.

Study the shape of the curve $y^{2}-x^{4}+x^{6}=0$, first solving the equation of the curve for one of the coordinates.

Study the shape of the curve $x^{4}+x^{2}y^{2}-18x^{2}y+9y^{2}=0$, first solving the equation of the curve for one of the coordinates.

Study the shape of the curve $x^{4}+y^{4}-3x^{3}-4x^{2}=0$, first solving the equation of the curve for one of the coordinates.

Study the shape of the curve $y^{4}-2xy^{2}-3x^{2}+x^{4}=0$, first solving the equation of the curve for one of the coordinates.

Study the shape of the curve $x^{4}+y^{4}-6y^{3}+8x^{2}y=0$, first solving the equation of the curve for one of the coordinates.

Study the shape of the curve $x^{4}+y^{4}-8x^{2}-10y^{2}+16=0$, first solving the equation of the curve for one of the coordinates.

Study the shape of the closed curve $x^{4}+y^{4}=2xy$, by passing to polar coordinates.

Study the shape of the closed curve $(x^{2}+y^{2}-6x)^{2}=x^{2}+y^{2}$ (the limaçon of Pascal), by passing to polar coordinates.

Study the shape of the closed curve $(x^{2}+y^{2}-4x)^{2}=16(x^{2}+y^{2})$ (a cardioid), by passing to polar coordinates.

Study the shape of the closed curve $x^{4}+y^{4}=8xy^{2}$, by passing to polar coordinates.

Study the shape of the closed curve $x^{4}+y^{4}=x^{2}+y^{2}$, by passing to polar coordinates.

Study the shape of the closed curve $(x^{2}+y^{2})^{3}=27x^{2}y^{2}$, by passing to polar coordinates.

Study the shape of the closed curve $(x^{2}+y^{2})^{2}=xy$, by passing to polar coordinates.

Plot the curve (take $a>0$) $y^{2}=x^{3}-2x^{2}+x$.

Plot the curve (take $a>0$) $4y^{2}=4x^{2}y+x^{5}$.

Plot the curve (take $a>0$) $x^{5}+5ax^{4}-16a^{3}y^{2}=0$.

Plot the curve (take $a>0$) $y^{3}-x^{2}y+x^{5}=0$.

Plot the curve (take $a>0$) $y^{5}+x^{4}=xy^{2}$.

Plot the curve (take $a>0$) $x^{6}+2x^{3}y-y^{3}=0$.

Plot the curve (take $a>0$) $x^{4}+2y^{3}=4x^{2}y$.

Plot the curve (take $a>0$) $y^{3}-x^{3}+y-2x=0$.

Plot the curve (take $a>0$) $(x^{2}-y^{2})^{2}=2x$.

Plot the curve (take $a>0$) $x^{2}(x-y)^{2}+y=0$.

Plot the curve (take $a>0$) $x^{3}y^{3}=x-y$.

Plot the curve (take $a>0$) $x^{2}y^{2}+x-2y=0$.

Plot the curve (take $a>0$) $(2x+y)^{2}(x+y)=x$.

Plot the curve (take $a>0$) $(x^{2}-y^{2})(x-y)=1$.

Plot the curve (take $a>0$) $3x^{4}-y^{4}+2x^{2}y^{2}+2x=0$.

Plot the curve (take $a>0$) $xy(x-y)+x+y=0$.

Plot the curve (take $a>0$) $x^{2}y^{2}+y=1$.

Plot the curve (take $a>0$) $xy(x^{2}-y^{2})+1=0$.

Plot the curve (take $a>0$) $xy(x^{2}+y^{2})-2x^{2}y^{2}=12$.

Plot the curve (take $a>0$) $x(x^{2}+y^{2})=a(x^{2}-y^{2})$.

Plot the curve (take $a>0$) $(x^{2}-y^{2})^{2}=4xy$.

Plot the curve (take $a>0$) $x^{2}y^{2}+y^{4}=4x^{2}$.

Plot the curve (take $a>0$) $x^{4}-y^{4}+xy=0$.

Plot the curve (take $a>0$) $x^{2}(x^{2}+y^{2})=4(x-y)^{2}$.

Plot the curve (take $a>0$) $x^{4}-y^{4}=4y^{2}-x^{2}$.

Plot the curve (take $a>0$) $x^{2}y^{2}=(y+1)^{2}(4-y^{2})$.

Plot the curve (take $a>0$) $(x^{2}-1)y^{2}=x^{4}-4x^{2}$.

Plot the curve (take $a>0$) $x^{2}(y-2)^{2}+2xy=y^{2}$.

Plot the curve (take $a>0$) $x^{4}-y^{4}=x^{2}-2y^{2}$.

Plot the curve (take $a>0$) $(x+1)(x+2)y^{2}=x^{2}$.

Plot the curve (take $a>0$) $xy(x+y)+x^{2}=2y^{2}$.

Plot the curve (take $a>0$) $(2a-x)y^{2}=x^{3}$.

Plot the curve (take $a>0$) $x^{3}+y^{3}=3x^{2}$.

Plot the curve (take $a>0$) $x^{4}-2x^{2}=y^{3}(x-1)$.

Plot the curve (take $a>0$) $x^{4}-y^{4}=4x^{2}y$.

Plot the curve (take $a>0$) $x^{3}-2x^{2}y-y^{2}=0$.

Plot the curve (take $a>0$) $x^{4}-2x^{2}y^{2}+y^{3}=0$.

Plot the curve (take $a>0$) $x^{2}y^{2}=x^{2}+y^{2}$.

Plot the curve (take $a>0$) $x^{5}+y^{5}=xy^{2}$.

Plot the curve (take $a>0$) $x^{4}+2x^{2}y-xy^{2}+y^{2}=0$.

Study the curve whose equations are given in parametric form: $x=\dfrac {2+t^{2}}{1+t^{2}}$, $y=t-\dfrac {t}{1+t^{2}}$.

Study the curve whose equations are given in parametric form: $x=\dfrac {t^{2}+1}{4(1-t)}$, $y=\dfrac {t}{t+1}$.

Study the curve whose equations are given in parametric form: $x=\dfrac {t^{2}+1}{t^{2}-1}$, $y=\dfrac {t}{t^{4}+1}$.

Study the curve whose equations are given in parametric form: $x=\dfrac {t}{1-t^{2}}$, $y=\dfrac {t(1-2t^{2})}{1-t^{2}}$.

Study the curve whose equations are given in parametric form: $x=\dfrac {t^{2}}{t^{2}-1}$, $y=\dfrac {t^{2}+1}{t+2}$.

Study the curve whose equations are given in parametric form: $x=\dfrac {t^{2}-3}{t^{2}+1}$, $y=\dfrac {t(t^{2}-3)}{t^{2}+1}$.

Study the curve whose equations are given in parametric form: $x=\dfrac {t-t^{2}}{1+t^{2}}$, $y=\dfrac {t^{2}-t^{3}}{1+t^{2}}$.

Study the curve whose equations are given in parametric form: $x=\dfrac {t^{2}}{t-1}$, $y=\dfrac {t}{t^{2}-1}$.

Study the curve whose equations are given in parametric form: $x=\dfrac {(t+2)^{2}}{t+1}$, $y=\dfrac {(t-2)^{2}}{t-1}$.

Study the curve whose equations are given in parametric form: $x=\dfrac {2t-1}{t^{3}(t-1)}$, $y=\dfrac {2t-1}{t^{2}(t-1)}$.

Study the curve $y^{2}=a^{2}\sin \dfrac {y}{x}$.

Study the curve $y^{2}=a^{2}\cos \dfrac {y}{x}$.

Investigate the curve $x^{y}=y^{x}$, noting that it splits into the line $y=x$ and the curve