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Finding Parametric Equations and Tangent Lines
Find parametric equations for the given curve.
9x² + 4y² = 36
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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.PE.12a
Hass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.PE.12aCapitolo 11, Problema 11.PE.12a
Finding Parametric Equations and Tangent Lines
Find parametric equations for the given curve.
Line through (1,-2) with slope 3
Guida verificata passo dopo passo1
Recall that a parametric equation for a line can be expressed as $x = x_0 + at$ and $y = y_0 + bt$, where \((x_0, y_0)\) is a point on the line and \((a, b)\) is a direction vector parallel to the line.
Identify the given point on the line as \((1, -2)\), so \(x_0 = 1\) and \(y_0 = -2\).
Use the slope of the line, which is 3, to find the direction vector. Since slope \(m = \frac{b}{a} = 3\), you can choose \(a = 1\) and \(b = 3\) for simplicity.
Write the parametric equations using the point and direction vector: \(x = 1 + 1 \cdot t\) and \(y = -2 + 3 \cdot t\).
These parametric equations describe the line passing through \((1, -2)\) with slope 3, where \(t\) is the parameter.
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Parametric Equations
Parametric equations express a curve by defining both x and y coordinates as functions of a third variable, usually t. This allows representation of curves that are difficult to describe with a single function y = f(x). For example, a line can be represented as x = x0 + at and y = y0 + bt, where (x0, y0) is a point on the line and (a, b) relates to the direction.
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08:02
Parameterizing Equations
Equation of a Line Using Point-Slope Form
The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. This form is useful for quickly writing the equation of a line when a point and slope are known. It can be rearranged or converted into parametric form for further analysis.
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Percorso guidato
05:13Slopes of Tangent Lines
Tangent Lines and Their Slopes
A tangent line to a curve at a point touches the curve without crossing it and has the same slope as the curve at that point. For parametric curves, the slope of the tangent line is found by dy/dx = (dy/dt) / (dx/dt). Understanding tangent lines helps in analyzing the behavior and direction of curves.
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Percorso guidato
05:13Slopes of Tangent Lines
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Identifying Conic Sections
Complete the squares to identify the conic sections in Exercises 69-76. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.
x² + y² + 4x + 2y = 1
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Identifying Parametric Equations in the Plane
Exercises 1–6 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 and indicate the direction of motion and the portion traced by the particle.
x = √t, y = 1 − √t, t ≥ 0
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Domanda del libro di testo
Identifying Parametric Equations in the Plane
Exercises 1–6 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 and indicate the direction of motion and the portion traced by the particle.
x = 4 cos t, y = 9 sin t, 0 ≤ t ≤ 2π
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Domanda del libro di testo
Graphing Conic Sections
Find the eccentricities of the ellipses and hyperbolas in Exercises 59–62. Sketch each conic section. Include the foci, vertices, and asymptotes (as appropriate) in your sketch.
5y² − 4x² = 20
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Domanda del libro di testo
Polar to Cartesian Equations
Sketch the lines in Exercises 23-28. Also, find a Cartesian equation for each line.
r cos (θ − 3π/4) = (√2)/2
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