Analysis of Planar Curves

 

Parametric Equations

Representing Cartesian coordinate curves using explicit and implicit forms. Representing curves using parametric equations which define x and y in terms of a third variable. Includes examples of parametric equations for a circle, ellipse, and projectile fired at an angle.

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Nine questions which involve finding equations in rectangular coordinates for those given in parametric form, or putting a rectangular equation in parametric form.

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Implicitization

Finding implicit forms for parameterized curves. Uses examples from the previous section of the notes.

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Polar Coordinate Curves

Definition, with examples of circles and a horizontal line defined in polar coordinates.

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Parametric Representation of a Curve

Using parametric equations to define a curve in two or three dimensions and properties of parametric equations.

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Polar Coordinate Graphs

Three multi-part questions which involve converting rectangular coordinates to polar coordinates, converting polar equations to rectangular equations, and graphing curves given in polar coordinates.

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Polar Curves

Sketching a curve in polar coordinates, and labeling the quadrants, endpoints, tangent slopes, and angles for the curve.

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Sketching a curve given in polar coordinates and finding the area swept by a line segment as one of the endpoints moves along this curve.

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Area Inside a Polar Curve

Sketching a curve defined in polar coordinates and finding the area inside it.

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Polar Coordinates

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Sketching a curve given in polar coordinates and finding points of intersection between that and other curves.

Representing a circle using both rectangular and polar coordinates.

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Integrals in Polar Coordinates: Lunar Eclipse

Setting up and evaluating an integral to represent the uncovered area of the two moons involved in a lunar eclipse on another planet.

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Polar Coordinates: Spiral

Sketching a spiral defined in polar coordinates, counting the times it crosses the x-axis, and finding the area of specific regions of the spiral.

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Lines in Polar Coordinates

Finding an equation in polar coordinates and the appropriate range of theta for a line given in rectangular coordinates.

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Polar Plotter

Applet for showing the graph of a function defined in polar coordinates.

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Curves in Two Dimensions

Applet for plotting curves defined in rectangular or parametric form.

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