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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.1.50
Chapter 12, Problem 12.1.50

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The left half of the parabola y=x ² +1, originating at (0, 1)

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Identify the given curve: the parabola is defined by the equation \(y = x^2 + 1\).
Since we want parametric equations, choose a parameter, typically \(t\), to represent \(x\). Let \(x = t\).
Express \(y\) in terms of \(t\) using the original equation: \(y = t^2 + 1\).
Determine the interval for \(t\) to represent the left half of the parabola. The left half corresponds to \(x \leq 0\), so \(t \leq 0\).
Write the parametric equations and specify the interval: \(x = t\), \(y = t^2 + 1\), with \(t \in (-\infty, 0]\).

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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 are defined in terms of t, allowing more flexibility in describing curves, especially those that are not functions in the traditional sense.
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Parameterizing Equations

Parabola and Its Properties

A parabola is a curve defined by a quadratic equation, such as y = x² + 1. Understanding its shape and symmetry is essential; here, the parabola opens upward with vertex at (0,1). The 'left half' refers to the portion where x ≤ 0, which guides the choice of parameter intervals.
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Properties of Parabolas

Parameter Interval and Curve Orientation

Choosing an appropriate interval for the parameter ensures the parametric equations trace the desired portion of the curve. For the left half of the parabola, the parameter should cover values corresponding to x ≤ 0, and the interval defines the start and end points, such as originating at (0,1).
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Finding Area Between Curves that Cross on the Interval
Related Practice
Textbook Question

Second derivative Assume a curve is given by the parametric equations x=f(t) and y=g(t), where f and g are twice differentiable. Use the Chain Rule to show that y″x=(fʹ(t)g″(t)−gʹ(t)f″(t))/(fʹ(t))³.  

Textbook Question

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.


r² = 4 sin θ  

Textbook Question

Air drop—inverse problem A plane traveling horizontally at 100 m/s over flat ground at an elevation of 4000 m must drop an emergency packet on a target on the ground. The trajectory of the packet is given by

x = 100t, y = −4.9t² + 4000, t ≥ 0

where the origin is the point on the ground directly beneath the plane at the moment of the release. How many horizontal meters before the target should the packet be released in order to hit the target?

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Textbook Question

63–74. Arc length of polar curves Find the length of the following polar curves.


{Use of Tech} The complete limaçon r=4−2cosθ

Textbook Question

What is the polar equation of a circle of radius √(a²+b²) centered at (a, b)?

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Textbook Question

9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.


(-1, -π/3)

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