Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/186

EXAMPLES

1. Find the radius of curvature for each of the following curves, at the point indicated; draw the curve and the corresponding circle of curvature:

(a) $b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2},(a,0).$	Ans.	$R={\frac {b^{2}}{a}}.$
(b) $b^{2}y^{2}+a^{2}y^{2}=a^{2}b^{2},(0,b).$		$R={\frac {a^{2}}{b}}.$
(c) $y=x^{4}-4x^{3}-18x^{2},(0,0).$		$R={\frac {1}{36}}.$
(d) $16y^{2}=4x^{4}-x^{6},(2,0).$		$R=2.$
(e) $y=x^{3},(x_{1},y_{1}).$		$R={\frac {(1+9{x_{1}}^{4})^{\frac {3}{2}}}{6x_{1}}}.$
(f) $y^{2}=x^{3},(4,8).$		$R={\frac {1}{3}}(40)^{\frac {3}{2}}.$
(g) $y^{2}=8x,({\frac {9}{8}},3).$		$R=7{\frac {13}{16}}.$
(h) $\left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{\frac {2}{3}}=1,(0,b).$		$R={\frac {a^{2}}{3b}}.$
(i) $x^{2}=4ay,(0,0).$		$R=2a$
(j) $(y-x^{2})^{2}=x^{5},(0,0).$		$R={\frac {1}{2}}$
(k) $(b^{2}x^{2}-a^{2}y^{2}=a^{2}b^{2},(x_{1},y_{1}).$		$R={\frac {(b^{4}{x_{1}}^{2}+a^{4}{y_{1}}^{2})^{\frac {3}{2}}}{a^{4}b^{4}}}$
(l) $e^{x}=\sin y,(x_{1},y_{1}).$	(p) $9y=x^{3},x=3.$
(m) $y=\sin x,\left({\frac {\pi }{2}},1\right).$	(q) $4y^{2}=x^{3},x=4.$
(n) $y=\cos x,\left({\frac {\pi }{4}},{\sqrt {2}}\right).$	(r) $x^{2}-y^{2}=a^{2},y=0.$
(o) $y=\log x,x=e.$	(s) $x^{2}+2y^{2}=9,(1,-2).$

2. Determine the radius of curvature of the curve $a^{2}y=bx^{2}+cx^{2}y$ at the origin.

Ans. $R={\frac {a^{2}}{2b}}$.

3. Show that the radius of curvature of the witch $y^{2}={\tfrac {a^{2}(a-x)}{x}}$ at the vertex is ${\tfrac {a}{2}}$.

4. Find the radius of curvature of the curve $y=\log \sec x$ at the point $(x_{1},y_{1})$.

Ans. $R=\sec x_{1}$.

5. Find K at any point on the parabola $x^{\tfrac {1}{2}}+y^{\tfrac {1}{2}}=a^{\tfrac {1}{2}}$. Ans. $K={\tfrac {a^{\tfrac {1}{2}}}{2(x+y)^{\tfrac {3}{2}}}}$.

6. Find R at any point on the hypocycloid $x^{\frac {2}{3}}+y^{\frac {2}{3}}=a^{\frac {2}{3}}$. Ans. $R=3(axy)^{\frac {1}{3}}$.

7. Find R at any point on the cycloid $x=r\operatorname {arcvers} {\tfrac {y}{r}}-{\sqrt {2ry-y^{2}}}$. Ans. $R=2{\sqrt {2ry}}$.

Find the radius of curvature of the following curves at any point:

8. The circle $\rho =a\sin \theta$.		Ans.	$R={\frac {a}{2}}$.
9. The spiral of Archimedes $\rho =a\theta$.			$R={\frac {(\rho ^{2}=a^{2})^{\frac {3}{2}}}{\rho ^{2}+2a^{2}}}$.
10. The cardioid $\rho =a(1-\cos \theta )$.			$R={\frac {2}{3}}{\sqrt {2a\rho }}$.
11. The lemniscate $\rho ^{2}=a^{2}\cos 2\theta$.			$R={\frac {a^{2}}{3\rho }}$.
12. The parabola $\rho =a\sec ^{2}{\frac {\theta }{2}}$.		$R=2a\sec ^{3}{\frac {\theta }{2}}$.
13. The curve $\rho =asin^{3}{\frac {\theta }{3}}$.