Parametric Equations and Polar Coordinates

By utilizing unique x and y functions, Parametric equations can illustrate a set's mathematical characteristics. On the other hand, Polar coordinates provide a two-dimensional framework, utilizing the distance from a specific point and an angle from a predetermined direction to pinpoint a location. These methods offer different approaches for depicting mathematical scenarios.

Eliminating the Parameter from the Function

Find an equation for the line tangent to the curve at the point defined by the given value of

t

. Also, find the value of

\frac{d^{2} y}{d x^{2}}

at this point.

x = 8 \cos t, y = 4 \sin t, t = \frac{π}{4}

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Converting to Polar Coordinates

Find the exact length of the polar curve

r = \cos^{4} (θ / 4)

. Length

=

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Identifying and Graphing Circles

A circle is described by the parametric equations:

x = 5 c o s (t)

and

y = 5 s i n (t)

. What is the radius of the circle, and graph the circle.

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Identifying and Graphing Limacons

Find the slope of the tangent to the curve

r = 4 - 4 \cos θ

at the value

θ = π / 2

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Identifying and Graphing Roses

Identify and graph the rose given by the polar equation

r = 5 \cos (3 θ)

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Identifying and Graphing Cardioids

Consider the polar equation

r = 1 + \cos (θ)

. Identify this equation as a cardioid and sketch its graph.

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Parametric Equations and Polar Coordinates

Eliminating the Parameter from the Function

Converting to Polar Coordinates

Identifying and Graphing Circles

Identifying and Graphing Limacons

Identifying and Graphing Roses

Identifying and Graphing Cardioids

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