Domanda del libro di testo
11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.
r = 4 cos 2θ; at the tips of the leaves
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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 12, Problema 12.1.15
15–30. Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.
x = 3 + t, y = 1 − t; 0 ≤ t ≤ 1
Guida verificata passo dopo passo1
Identify the given parametric equations: \(x = 3 + t\) and \(y = 1 - t\), with the parameter \(t\) ranging from \(0\) to \(1\).
To eliminate the parameter \(t\), solve one of the equations for \(t\). For example, from \(x = 3 + t\), isolate \(t\) to get \(t = x - 3\).
Substitute the expression for \(t\) into the other equation: replace \(t\) in \(y = 1 - t\) with \(x - 3\), resulting in \(y = 1 - (x - 3)\).
Simplify the equation to express \(y\) solely in terms of \(x\): \(y = 1 - x + 3\), which simplifies further to \(y = 4 - x\).
Interpret the curve: the equation \(y = 4 - x\) represents a straight line. The parameter \(t\) increases from \(0\) to \(1\), so the curve starts at the point when \(t=0\) (which is \((3,1)\)) and ends at \(t=1\) (which is \((4,0)\)). This indicates the positive orientation of the curve is from \((3,1)\) to \((4,0)\).
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Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves and motions.
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08:02
Parameterizing Equations
Eliminating the Parameter
Eliminating the parameter involves manipulating the parametric equations to remove t, resulting in a direct relationship between x and y. This often requires solving one equation for t and substituting into the other, yielding a Cartesian equation of the curve.
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05:59Eliminating the Parameter
Curve Orientation and Interval of Parameter
The orientation of a parametric curve is determined by the direction in which the parameter t increases. The interval for t specifies the portion of the curve traced, and understanding this helps describe the curve's direction and endpoints.
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02:59Finding Area Between Curves that Cross on the Interval
Pratica correlata
Domanda del libro di testo
33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.
The region inside the limaçon r = 2 + cos θ
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Domanda del libro di testo
15–30. Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.
x = 3 cos t, y = 3 sin t; π ≤ t ≤ 2π
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Domanda del libro di testo
13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.
4x = -y²
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Domanda del libro di testo
90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.
The length of the latus rectum of the parabola y ² =4px or x ² =4py is 4|p|.
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Domanda del libro di testo
13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.
x² + y²/9 = 1
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