Textbook Question
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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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 11, Problem 11.PE.12a
Finding Parametric Equations and Tangent Lines
Find parametric equations for the given curve.
Line through (1,-2) with slope 3
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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Guided course
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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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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05:13Slopes of Tangent Lines
Related Practice
Textbook Question
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
Textbook Question
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
Textbook Question
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
Textbook Question
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
Textbook Question
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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