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
Finding Parametric Equations
In Exercises 31–36, find a parametrization for the curve.
the line segment with endpoints (-1,3) and (3,-2)
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 = 1 + sin t, y = cos t − 2, 0 ≤ 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
Theory and Examples
Tangents Find equations for the tangents to the circle (x − 2)² + (y − 1)² = 5 at the points where the circle crosses the coordinate axes.
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
