Textbook Question
31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.
A parabola that opens to the right with directrix x = -4
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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 12, Problem 12.1.29
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 = 8 + 2t, y = 1; −∞ < t < ∞
Verified step by step guidance1
Identify the given parametric equations: \(x = 8 + 2t\) and \(y = 1\), where \(t\) is the parameter.
To eliminate the parameter \(t\), solve one of the equations for \(t\). From \(x = 8 + 2t\), isolate \(t\) by subtracting 8 and then dividing by 2: \(t = \frac{x - 8}{2}\).
Substitute the expression for \(t\) into the other equation. Since \(y = 1\) is constant and does not depend on \(t\), the equation in terms of \(x\) and \(y\) is simply \(y = 1\).
Interpret the resulting equation \(y = 1\): this represents a horizontal line in the \(xy\)-plane at the height \(y = 1\).
Determine the positive orientation by considering how \(x\) changes as \(t\) increases. Since \(x = 8 + 2t\), as \(t\) increases, \(x\) increases, so the curve is oriented from left to right along the line \(y = 1\).
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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 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.
Recommended video:
Guided course
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 helps identify the geometric shape of the curve without referencing the parameter.
Recommended video:
Guided course
05:59Eliminating the Parameter
Curve Orientation
Curve orientation refers to the direction in which the curve is traced as the parameter increases. Understanding orientation is important for interpreting motion or direction along the curve, often indicated by arrows or parameter intervals.
Recommended video:
11:41Summary of Curve Sketching
Related Practice
Textbook Question
85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)
A circular corral of unit radius is enclosed by a fence. A goat inside the corral is tied to the fence with a rope of length 0≤a≤2 (see figure). What is the area of the region (inside the corral) that the goat can graze? Check your answer with the special cases a=0 and a=2.
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Textbook Question
81–88. Arc length Find the arc length of the following curves on the given interval.
x = sin t, y = t - cos t; 0 ≤ t ≤ π/2
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Textbook Question
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.
Let L be the latus rectum of the parabola y ² =4px for p>0. Let F be the focus of the parabola, P be any point on the parabola to the left of L, and D be the (shortest) distance between P and L. Show that for all P, D+|FP|+ is a constant. Find the constant.
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Textbook Question
11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.
r = 1 - sin θ; (1/2, π/6)
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Textbook Question
Plot the points with polar coordinates (2, π/6) and (−3, −π/2). Give two alternative sets of coordinate pairs for both points.
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