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 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.
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
Ellipses
Exercises 25 and 26 give information about the foci and vertices of ellipses centered at the origin of the xy−plane. In each case, find the ellipse's standard−form equation from the given information.
Foci: ( ±√2, 0) Vertices: (±2,0)
