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Centroids
Find the coordinates of the centroid of the curve x = cos t, y = t + sin t, 0 ≤ t ≤ π.
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Ch. 11 - Parametric Equations and Polar Coordinates
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
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Hass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.1.24
Hass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.1.24Capitolo 11, Problema 11.1.24
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
Guida verificata passo dopo passo1
Step 1: Identify the parametric equations given: \(x = \cos t\) and \(y = \sin 3t\).
Step 2: Understand the behavior of \(x = \cos t\): it oscillates between -1 and 1 with period \(2\pi\).
Step 3: Understand the behavior of \(y = \sin 3t\): it oscillates between -1 and 1 but with a period of \(\frac{2\pi}{3}\), which is three times faster than \(x\).
Step 4: Look for a graph where the \(x\)-values oscillate smoothly between -1 and 1, while the \(y\)-values oscillate more rapidly, creating multiple waves within one period of \(x\). This will create a pattern with three oscillations in \(y\) for every one oscillation in \(x\).
Step 5: Match this behavior to the graphs provided. The graph labeled F shows a wave pattern with multiple oscillations in \(y\) for each oscillation in \(x\), consistent with \(x = \cos t\) and \(y = \sin 3t\).
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Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted as t. Instead of y as a function of x, both x and y depend on t, allowing representation of more complex curves like loops or spirals.
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Parameterizing Equations
Conversion to Cartesian Form
Converting parametric equations to Cartesian form involves eliminating the parameter t to find a direct relationship between x and y. This helps identify the shape of the curve and compare it to standard graphs, such as circles or ellipses.
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Percorso guidato
6:20Circles in General Form
Trigonometric Parametric Curves
When parametric equations involve trigonometric functions like sine and cosine, the resulting curves often represent periodic or oscillatory shapes. Understanding how frequencies and amplitudes affect the curve is essential for matching equations to their graphs.
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Percorso guidato
06:49Differentiation of Parametric Curves
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Domanda del libro di testo
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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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.
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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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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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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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