Ch. 11 - Parametric Equations and Polar Coordinates

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 11 - Parametric Equations and Polar Coordinates

Problem 11.1.32

Chapter 11, Problem 11.1.32

Finding Parametric Equations

In Exercises 31–36, find a parametrization for the curve.

the line segment with endpoints (-1,3) and (3,-2)

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Identify the two endpoints of the line segment: \(A(-1, 3)\) and \(B(3, -2)\).

Recall that a parametric equation for a line segment between points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) can be written as: \(x(t) = x_1 + t(x_2 - x_1)\) \(y(t) = y_1 + t(y_2 - y_1)\), where the parameter \(t\) varies between 0 and 1.

Substitute the coordinates of points \(A\) and \(B\) into the parametric formulas: \(x(t) = -1 + t(3 - (-1))\) \(y(t) = 3 + t(-2 - 3)\).

Simplify the expressions inside the parentheses: \(x(t) = -1 + t(4)\) \(y(t) = 3 + t(-5)\).

Write the final parametric equations for the line segment as: \(x(t) = -1 + 4t\) \(y(t) = 3 - 5t\), with \(t\) in the interval \([0, 1]\) to trace the segment from \(A\) to \(B\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parametric Equations of a Line Segment

Parametric equations express the coordinates of points on a line segment as functions of a parameter, usually t. For a segment between points A and B, the equations are x = x_A + t(x_B - x_A) and y = y_A + t(y_B - y_A), where t varies from 0 to 1 to trace the segment.

Vector Representation of Points

Points in the plane can be represented as vectors from the origin. The vector from point A to point B is found by subtracting their coordinates, which helps in defining direction and magnitude for parametrization of the line segment.

Parameter Interval and Its Role

The parameter t typically ranges between 0 and 1 to represent points along the line segment from the start to the end point. When t=0, the position corresponds to the first endpoint, and when t=1, it corresponds to the second endpoint.

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Related Practice

Textbook Question

Cartesian to Polar Equations

Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.

x - y = 3

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Textbook Question

Polar to Cartesian Equations

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

r² = 4r sin θ

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Textbook Question

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)

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Textbook Question

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

Textbook Question

Symmetries and Polar Graphs

Identify the symmetries of the curves in Exercises 1–12. Then sketch the curves in the xy-plane.

r = 1 + 2 sin θ

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Textbook Question

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

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