Ch. 11 - Parametric Equations and Polar Coordinates

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 11 - Parametric Equations and Polar Coordinates

Problema 11.6.2

Capitolo 11, Problema 11.6.2

Identifying Graphs

Match the parabolas in Exercises 1−4 with the following equations: x² = 2y, x² = −6y, y² = 8x, y² = −4x

Then find each parabola's focus and directrix.

Guida verificata passo dopo passo

Step 1: Identify the orientation of the parabola from the graph. This parabola opens to the right, which suggests it is of the form \(y^2 = 4px\) or \(y^2 = -4px\).

Step 2: Compare the given equations to the form \(y^2 = 4px\). The equations with \(y^2\) are \(y^2 = 8x\) and \(y^2 = -4x\). Since the parabola opens to the right, the coefficient of \(x\) should be positive, so the equation is \(y^2 = 8x\).

Step 3: From the equation \(y^2 = 8x\), identify \(4p = 8\), so \(p = 2\). This means the focus is at \((p, 0)\), or \((2, 0)\).

Step 4: The directrix is the vertical line \(x = -p\), so the directrix is \(x = -2\).

Step 5: Summarize: The parabola corresponds to the equation \(y^2 = 8x\), with focus at \((2, 0)\) and directrix \(x = -2\).

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Standard Form of Parabolas

Parabolas can be expressed in standard forms such as x² = 4py or y² = 4px, where p represents the distance from the vertex to the focus. The sign and variable squared determine the parabola's orientation (up, down, left, or right). Recognizing these forms helps match equations to their graphs.

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Circles in Standard Form Example 1

Focus and Directrix of a Parabola

The focus is a fixed point inside the parabola, and the directrix is a line outside it. Every point on the parabola is equidistant from the focus and the directrix. The value of p in the standard form determines the location of the focus and directrix relative to the vertex.

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Horizontal Parabolas

Graph Interpretation and Matching

Analyzing the graph's orientation and shape helps identify which equation corresponds to which parabola. For example, a parabola opening right or left corresponds to y² = 4px, while one opening up or down corresponds to x² = 4py. The sign of p indicates direction (positive for right/up, negative for left/down).

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Graphing The Derivative

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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Finding Lengths of Polar Curves

Find the lengths of the curves in Exercises 21–28.

The curve r = cos³(θ/3), 0 ≤ θ ≤ π/4

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Ellipses and Eccentricity

Exercises 9–12 give the foci or vertices and the eccentricities of ellipses centered at the origin of the xy-plane. In each case, find the ellipse’s standard-form equation in Cartesian coordinates.

Vertices: (±10,0)

Eccentricity: 0.24

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Surface Area

Find the areas of the surfaces generated by revolving the curves in Exercises 31-34 about the indicated axes.

x = t + √2, y = (t²/2) + √2t, −√2 ≤ t ≤ √2; y−axis

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Finding Polar Areas

Find the areas of the regions in Exercises 9–18.

Shared by the circles r = 1 and r = 2 sin θ