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 ray (half line) with initial point (-1,2) that passes through the point (0,0)
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
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
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.
