91Ó°ÊÓ

Find the slopes of the curves in Exercises \(17 - 20\) at the given points. Sketch the curves along with their tangents at these points. Four-leaved rose \(\quad r = \cos 2 \theta ; \quad \theta = 0 , \pm \pi / 2 , \pi\)

The area of the region that lies inside the cardioid curve \(r=\cos \theta+1\) and outside the circle \(r=\cos \theta\) is not $$\frac{1}{2} \int_{0}^{2 \pi}\left[(\cos \theta+1)^{2}-\cos ^{2} \theta\right] d \theta=\pi$$ Why not? What is the area? Give reasons for your answers.

In Exercises \(17-24,\) find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices. $$ y^{2}-x^{2}=4 $$

Give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch. \(2 x^{2}+y^{2}=4\)

Assuming that the equations in Exercises \(15-20\) 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 \(t\) . $$ t=\ln (x-t), \quad y=t e^{t}, \quad t=0 $$

Find the lengths of the curves in Exercises \(21-28\) . The spiral \(r=\theta^{2}, \quad 0 \leq \theta \leq \sqrt{5}\)

In Exercises \(17-24,\) find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices. $$ 8 x^{2}-2 y^{2}=16 $$

Give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch. \(3 x^{2}+2 y^{2}=6\)

Find the area under one arch of the cycloid $$ x=a(t-\sin t), \quad y=a(1-\cos t) $$

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