91Ó°ÊÓ

Prove that the part of the polar curve $r=\frac{1}{\mathrm{θ}}$that lies inside the circle defined by the polar equation r = 1 has infinite length.

Prove that, for every even integer n, the graph of $r=\sin n\mathrm{θ}$ is symmetrical with respect to the $x-$axis.

use the result of Exercise 54 to find parametric equations for the line segments connecting the given pairs of points in the direction indicated.

From $(1,4)$ to$(−3,5)$

Convert the equations given in polar coordinates to equations in rectangular coordinates.

$r=k$for each positive integer k less than 10.

Prove Theorem 9.20 (a). That is, show that the graph of the equation $\frac{x^2}{A^2}-\frac{y^2}{B^2}=1$satisfies Definition 9.19, where the points with coordinates $(Â\pm \sqrt{A^2+B^2},0)$are the foci of the hyperbola

Complete the proof of Theorem 9.14 by verifying that

${\left(f\text{'}\left(\mathrm{θ}\right)\cos \mathrm{θ}-f\left(\mathrm{θ}\right)\sin \mathrm{θ}\right)}^2+{\left(f\text{'}\left(\mathrm{θ}\right)\sin \mathrm{θ}+f\left(\mathrm{θ}\right)\cos \mathrm{θ}\right)}^2$is equal to${\left(f\text{'}\left(\mathrm{θ}\right)\right)}^2+{\left(f\left(\mathrm{θ}\right)\right)}^2$

Prove that, for every integer n, the graph of $r=\sin n\mathrm{θ}$ is symmetrical with respect to the $y-$axis.

Use the result of Exercise $54$to find parametric equations for the line segments connecting the given pairs of points in the direction indicated

From$(−3,5)to(1,4)$

Prove Theorem 9.20 (b). That is, show that the graph of the equation satisfies$\frac{y^2}{B^2}-\frac{x^2}{A^2}=1$ Definition 9.19, where the points with coordinates $\left(0,Â\pm \sqrt{A^2+B^2}\right)$ are the foci of the hyperbola

convert the equations given in polar coordinates to equations in rectangular coordinates

$\mathrm{θ}=0,\mathrm{θ}=\frac{\mathrm{π}}{4},\mathrm{θ}=-\frac{\mathrm{π}}{2}$