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:

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: