91Ó°ÊÓ

Find the areas of the regions. Inside the circle \(r=4 \sin \theta\) and below the horizontal line \(r=3 \csc \theta\)

Find the slopes of the curves in Exercises at the given points. Sketch the curves along with their tangent lines at these points. $$\text { Cardioid } \quad r=-1+\sin \theta ; \quad \theta=0, \pi$$

Assuming that the equation define \(x\) and \(y\) implicitly as differentiable functions \(x=f(t), y=g(t),\) find the slope of the curve \(x=f(t), y=g(t)\) at the given value of \(t\) $$x \sin t+2 x=t, \quad t \sin t-2 t=y, \quad t=\pi$$

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26\). $$\theta=\pi / 2, \quad r \geq 0$$

Find the slopes of the curves in Exercises at the given points. Sketch the curves along with their tangent lines at these points. $$\text { Four-leaved rose } \quad r=\sin 2 \theta ; \quad \theta=\pm \pi / 4, \pm 3 \pi / 4$$

Assuming that the equation define \(x\) and \(y\) implicitly as differentiable functions \(x=f(t), y=g(t),\) find the slope of the curve \(x=f(t), y=g(t)\) at the given value of \(t\) $$x=t^{3}+t, \quad y+2 t^{3}=2 x+t^{2}, \quad t=1$$

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26\). $$\theta=\pi / 2, \quad r \leq 0$$

Find the slopes of the curves in Exercises at the given points. Sketch the curves along with their tangent lines at these points. $$\text { Four-leaved rose } r=\cos 2 \theta ; \quad \theta=0, \pm \pi / 2, \pi$$

Find the lengths of the curves. $$\text { The spiral } r=\theta^{2}, \quad 0 \leq \theta \leq \sqrt{5}$$

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26\). $$0 \leq \theta \leq \pi, \quad r=1$$