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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.3.48
Capitolo 11, Problema 11.3.48

Polar to Cartesian Equations


Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.


r = 3 cos θ

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Recall the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\), and also \(r^2 = x^2 + y^2\).
Start with the given polar equation: \(r = 3 \cos \theta\).
Multiply both sides of the equation by \(r\) to eliminate the \(\cos \theta\) term: \(r \cdot r = 3r \cos \theta\), which gives \(r^2 = 3r \cos \theta\).
Substitute the Cartesian equivalents: replace \(r^2\) with \(x^2 + y^2\) and \(r \cos \theta\) with \(x\), resulting in the equation \(x^2 + y^2 = 3x\).
Rewrite the equation in standard Cartesian form by moving all terms to one side: \(x^2 - 3x + y^2 = 0\). Then, complete the square for the \(x\) terms to identify the graph type.

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Polar and Cartesian Coordinate Systems

Polar coordinates represent points using a radius and an angle (r, θ), while Cartesian coordinates use (x, y) positions on a plane. Understanding how these systems relate is essential for converting equations between them.
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Intro to Polar Coordinates

Conversion Formulas Between Polar and Cartesian Coordinates

The key formulas are x = r cos θ and y = r sin θ, with r = √(x² + y²) and θ = arctan(y/x). These allow rewriting polar equations in terms of x and y, enabling analysis in the Cartesian plane.
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Intro to Polar Coordinates

Graphing and Identifying Curves from Equations

After converting to Cartesian form, recognizing the type of curve (circle, line, etc.) involves analyzing the equation's structure. For example, r = 3 cos θ corresponds to a circle in Cartesian coordinates.
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Determining Concavity from the Graph of f
Pratica correlata
Domanda del libro di testo

Centroids


Find the coordinates of the centroid of the curve x = cos t, y = t + sin t, 0 ≤ t ≤ π.

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Finding Cartesian from Parametric Equations


In Exercises 19–24, match the parametric equations with the parametric curves labeled A through F.


x = cos t, y = sin 3t


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Tangent Lines to Parametrized Curves


In Exercises 1−14, 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 d²y/dx² at this point.


x = sec² t − 1, y = tan t, t = −π/4

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Parabolas


Exercises 9-16 give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch.


x² = 6y

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Shifting Conic Sections


You may wish to review Section 1.2 before solving Exercises 39-56.


Exercises 53-56 give equations for hyperbolas and tell how many units up or down and to the right or left each hyperbola is to be shifted. Find an equation for the new hyperbola, and find the new center, foci, vertices, and asymptotes.


x²/4 − y²/5 = 1, right 2, up 2

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Hyperbolas


Exercises 27-34 give equations for hyperbolas. Put each equation in standard form and find the hyperbola's asymptotes. Then sketch the hyperbola. Include the asymptotes and foci in your sketch.


8x² − 2y² = 16

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