Convexity, Curvature, and Radius of Curvature

For what values of $x$ is the curve $y=x^{3}+ax+b$ convex upward?

Determine the direction of convexity of the curve $x(x^{2}-y^{2})+y^{2}=0$ at the points where $x=\dfrac {3}{2}$.

Investigate the nature of the concavity of the curve $x^{4}=y(x^{2}-y^{2})$ at points where $y<0$.

Investigate the nature of the convexity of the curve $r\cos^{3}3\varphi = 1$ for $-\dfrac {\pi }{6}<\varphi <\dfrac {\pi }{6}$.

Show that the curve $x^{4}+y^{4}=x^{2}+y^{2}$ is turned with its convex side toward the origin at the points where it meets the axes, and with its concave side at the points where it meets the bisectors of the angles.

Show that the curve $x=\varphi (t)$, $y=\psi (t)$ is turned with its concave side toward the origin if

Starting from the definition of curvature, find the length of an arc of $1^{\circ }$ of the Earth's meridian at the pole and at the equator. The meridian may be taken as an ellipse with semi-axes $a=6378$ km and $b=a-21$ km (the polar semi-axis).

Find the radius of curvature of the curve $3ay^{2}=2x^{3}$.

Find the smallest radius of curvature of the parabola $y^{2}=2px$.

Find the radius of curvature of the curve $y=x^{3}$ at the point $(1,1)$.

Find the radius of curvature of the curve $x=t^{2}$, $y=t^{3}$ at the point $(1,1)$.

Find the radius of curvature of the curve $y=a\ch \dfrac {x}{\alpha }$ (the catenary) at the point $(0,a)$.

Find the radius of curvature of the curve $y=a\ln \cos \dfrac {x}{a}$.

Find the radius of curvature of the curve $x^{n}+y^{n}=2a^{n}$ at the point $(a,a)$.

Find the radius of curvature of the curve $x=a\cos^{4}t$, $y=a\sin^{4}t$ at the point $(a,0)$.

Find the radius of curvature of the curve $x=a(\cos t+t\sin t)$, $y=a(\sin t-t\cos t)$.

Find the radius of curvature of the curve $x=a(t-\sin t)$, $y=a(1-\cos t)$.

Find the radius of curvature of the curve $r=a(1+\cos \varphi )$.

Find the radius of curvature of the curve $r^{2}=a^{2}\cos 2\varphi$, at $\varphi =0$.

Find the radius of curvature of the curve

Find the radius of curvature of the curve

Find the radius of curvature of the curve

Find the greatest curvature of the curve $y=\ln x$.

Find the greatest curvature of the curve $x=a\cos t$, $y=b\sin t$ (an ellipse).

Find the greatest curvature of the curve $y=a\ln \left(1-\dfrac {x^{2}}{a^{2}}\right)$.

Find the greatest curvature of the curve $y=a\ch \dfrac {x}{a}$.

If, along the normal from a point of a curve, one lays off in the direction of concavity of the curve a length equal to the radius of curvature, one obtains the center of curvature. Prove that its coordinates for the curve $x=\varphi (t)$, $y=\psi (t)$ are given by the formulas:

Find the coordinates of the center of curvature of the hyperbola $xy=a^{2}$ at the point $(a,a)$.

Find the coordinates of the center of curvature of the strophoid $y^{2}=x^{2}\dfrac {a+x}{a-x}$.

Find the coordinates of the center of curvature of the curve $y=x\ln x$ at the point where $y'=0$.

Find the coordinates of the center of curvature of the curve $r=a(1+\cos \varphi )$.

Find the coordinates of the center of curvature of the curve $r=a\cos^{3}\varphi$.

Prove that for points of the Archimedean spiral $r=a\varphi$, as $\varphi \rightarrow \infty$, the difference between the radius vector and the radius of curvature tends to zero.

Prove that under the same conditions (see the preceding problem) the center of curvature moves along a curve that tends to coincide with the circle $r=a$.

Prove that for the curves $r^{n}=a^{n}\cos n\varphi$, the polar normal is $n+1$ times the radius of curvature.

Prove that for the curves $r^{n}=a^{n}\sin n\varphi$, the part of the radius vector lying inside the circle of curvature equals $\dfrac {2r}{n+1}$.

Prove the equality $R=\dfrac {r\, dr}{dp}$, where $R$ is the radius of curvature, $r$ is the radius vector, and $p$ is the perpendicular from the pole to the tangent.

Prove that on an ellipse there exist, in general, three points such that the circles of curvature of the ellipse at these points pass through a given point of the ellipse.

Prove that the centers of curvature at points of the Archimedean spiral $r=a\varphi$ lying on one ray lie on an ellipse whose dimensions do not depend on the choice of the ray.

Find the smallest distance from the center of curvature of an ellipse to the center of the ellipse.

The coordinates of the points of a curve can be expressed in terms of the arc length of the curve: $x=x(s)$, $y=y(s)$. Denoting by $\Delta s$ a small increment of $s$, and by $\Delta l$ the corresponding chord, prove the equality

where $R$ is the radius of curvature of the curve.

The point of tangency is taken as the origin, the tangent as the axis $Ox$, and the axis $Oy$ is directed from the point of tangency toward the center of curvature. Prove that in a neighborhood of the point of tangency, i.e.for small values of $x$, the following formulas hold:

where $R$ is the radius of curvature at the point $(0,0)$, $y_{1}$ is the ordinate of points of the curve, and $y_{2}$ is the ordinate of points of the circle of curvature.

Find a parabola joining the origin $O$ to the point $B(a+b,0)$ so that the arc $OB$ of the parabola, together with the lower half of the circle $(x-a)^{2}+y^{2}=b^{2}$, forms a smooth curve with continuous curvature.

Find the parabola $y=ax^{2}+bx+c$ that has, with the sinusoid $y=\sin x$, at the point $\left(\dfrac {\pi }{2}, 1\right)$, a common tangent and a common curvature.

Find the fifth-degree parabola

which at the points $(a,h)$, $(-a,-h)$ is tangent to the lines $y=h$ and $y=-h$, having contact of the second order.

Prove that the circle $(x-3a)^{2}+(y-3a)^{2}=8a^{2}$ and the parabola $\sqrt{x}+\sqrt{y}=2\sqrt{a}$ have contact of the third order at the point $(a,a)$.

Prove that the osculating circle (circle of curvature) of a second-order curve at the vertex of the curve has contact of the third order.

Prove that for every point at which a curve is sufficiently smooth, there exists a second-order curve having, with the given curve at the given point, contact of the fourth order.

Prove that for a cycloid, the second-order curve with contact of the fourth order is always an ellipse.

Prove that for a logarithmic spiral, the second-order curves having contact of the fourth order with it are ellipses whose major axes make a constant angle with the radius vector of the point of tangency.

Find the locus of the foci of parabolas having, at a given point, contact of the second order with a given curve.

Find the locus of the midpoints of the chords common to the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ and its osculating circle.

Find the locus of the feet of the perpendiculars dropped from the center of the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ onto the chords common to the ellipse and its osculating circle.