Math Problems Vault

Problem 1591

Choose the parameters $k$ and $b$ of the line

y=kx+b

so that it has tangency of order higher than one with the curve

y=x^{3}-3x^{2}+2

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Problem 1592

How should the coefficients $a, b$ and $c$ be chosen so that the parabola

y=ax^{2}+bx+c

has second-order tangency with the curve $y=e^{x}$ at the point $x=x_{0}$ ?

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Problem 1593

(a)(b)(c)

What is the order of tangency of each of the following curves with the $Ox$ axis at the point $x=0$ :

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(a)

$y=1-\cos x$ ；

(b)

$y=\tan x-\sin x$ ；

(c)

$y=e^{x}-\left(1+x+\dfrac {x^{2}}{2}\right)$ .

Problem 1594

Prove that the curve: $y=e^{-\frac{1}{x^{2}}}$ for $x \neq 0$ , and $y=0$ for $x=0$ , has tangency of infinite order with the $Ox$ axis at the point $x=0$ .

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Problem 1595

(a)(b)

Find the radius of curvature and center of curvature of the hyperbola

xy=1

at each of the following points:

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(a)

$M(1,1)$ ；

(b)

$N(100,0.01)$ .

Problem 1596

Find the radius of curvature of the following curve: the parabola $y^{2}=2px$ .

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Problem 1597

Find the radius of curvature of the following curve: the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ .

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Problem 1598

Find the radius of curvature of the following curve: the hyperbola $\dfrac {x^{2}}{a^{2}}-\dfrac {y^{2}}{b^{2}}=1$ .

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Problem 1599

Find the radius of curvature of the following curve: the astroid $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ .

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Problem 1600

Find the radius of curvature of the following curve: the ellipse $x=a\cos t, y=b\sin t$ .

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Problem 1601

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

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Problem 1602

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

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Problem 1603

Prove that for the quadratic curve

y^{2}=2px-qx^{2}

the radius of curvature is proportional to the cube of the length of the normal segment.

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Problem 1604

Write down the formula for the radius of curvature of a curve given in polar coordinates.

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Problem 1605

Find the radius of curvature of the curve given by the following polar equation: the Archimedean spiral $r=a\varphi$ .

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Problem 1606

Find the radius of curvature of the curve given by the following polar equation: the logarithmic spiral $r=ae^{m\varphi }$ .

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Problem 1607

Find the radius of curvature of the curve given by the following polar equation: the cardioid $r=a(1+\cos \varphi )$ .

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Problem 1608

Find the radius of curvature of the curve given by the following polar equation: the lemniscate $r^{2}=a^{2}\cos 2\varphi$ .

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Problem 1609

Find the point of maximum curvature on the curve $y=\ln x$ .

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Problem 1610

The cubic parabola $y=\dfrac {kx^{3}}{6}$ ( $0 \leq x<+\infty , k>0$ ) has maximum curvature equal to $\dfrac {1}{1000}$ ; find the point $x$ at which this maximum curvature is attained.

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Problem 1611

Find the equation of the evolute of each of the following curves: the evolute of the parabola $y^{2}=2px$ .

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Problem 1612

Find the equation of the evolute of each of the following curves: the evolute of the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ .

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Problem 1613

Find the equation of the evolute of each of the following curves: the evolute of the astroid $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ .

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Problem 1614

Find the equation of the evolute of each of the following curves: the evolute of the tractrix $x=a\ln \dfrac {a+\sqrt{a^{2}-y^{2}}}{y}-\sqrt{a^{2}-y^{2}}$ .

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Problem 1615

Find the equation of the evolute of each of the following curves: the evolute of the logarithmic spiral $r=ae^{m\varphi }$ .

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Problem 1616

Prove that the evolute of the cycloid

x=a(t-\sin t), \quad y=a(1-\cos t)

is again a cycloid, differing from the given cycloid only in its position.

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