Implicitly Defined Parametrizations
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.
x sin t + 2x = t, t sin t − 2t = y, t = π
Symmetries and Polar Graphs
Identify the symmetries of the curves in Exercises 1–12. Then sketch the curves in the xy-plane.
r = 1 + 2 sin θ
Finding Polar Areas
Find the areas of the regions in Exercises 9–18.
Inside the circle r = 4 sin θ and below the horizontal line r = 3 csc θ
Shifting Conic Sections
Find the center, foci, vertices, asymptotes, and radius, as appropriate, of the conic sections in Exercises 57-68.
9x² + 6y² + 36y = 0
Finding Cartesian from Parametric Equations
Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.
x = 2 sinh t, y = 2 cosh t, −∞<t<∞
Circles
Sketch the circles in Exercises 53–56. Give polar coordinates for their centers and identify their radii.
r = −2 cos θ
