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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 12.1.89c
Chapter 12, Problem 12.1.89c

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


c. The parametric equations x=t, y=t², for t≥0, describe the complete parabola y=x².

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1
Identify the given parametric equations: \(x = t\) and \(y = t^{2}\), with the parameter constraint \(t \geq 0\).
Recall that the parabola \(y = x^{2}\) includes all points where \(y\) is the square of \(x\), for all real values of \(x\).
Analyze the range of \(x\) values generated by the parametric equations: since \(x = t\) and \(t \geq 0\), \(x\) only takes non-negative values (i.e., \(x \geq 0\)).
Check if the parametric equations cover the entire parabola: the parabola \(y = x^{2}\) extends for all real \(x\), including negative values, but the parametric form only covers \(x \geq 0\).
Conclude that the parametric equations describe only the right half (where \(x \geq 0\)) of the parabola \(y = x^{2}\), not the complete parabola.

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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, often denoted t. Instead of y as a function of x, both x and y depend on t, allowing the description of more general curves and motions.
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Parameterizing Equations

Domain and Range in Parametric Curves

The domain of the parameter t restricts which points on the curve are traced. For example, if t ≥ 0, only points corresponding to nonnegative t values are included, which may represent only part of the full curve.
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Differentiation of Parametric Curves

Graph of a Parabola y = x²

The parabola y = x² includes all points where y equals the square of x, for all real x. To describe the complete parabola parametrically, the parameter must cover all real x values, not just nonnegative ones.
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Properties of Parabolas
Related Practice
Textbook Question

11–14. Working with parametric equations Consider the following parametric equations.

c. Eliminate the parameter to obtain an equation in x and y.

d. Describe the curve.


x=−t+6, y=3t−3; −5≤t≤5 

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Textbook Question

Spiral arc length Consider the spiral r=4θ, for θ≥0.


c. Show that L′(θ)>0. Is L″(θ) positive or negative? Interpret your answer.

Textbook Question

67–72. Derivatives Consider the following parametric curves.

b. Make a sketch of the curve showing the tangent line at the point corresponding to the given value of t.


x = 2 + 4t, y = 4 − 8t; t = 2

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Textbook Question

11–14. Working with parametric equations Consider the following parametric equations.

c. Eliminate the parameter to obtain an equation in x and y.

d. Describe the curve.


x=2 t,y=3t−4;−10≤t≤10 

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Textbook Question

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.


c. x = 1 + 3s, y = 4 + 2s and x = 4 - 3t, y = 6 + 4t

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Textbook Question

Regions bounded by a spiral: Let Rₙ be the region bounded by the nth turn and the (n+1)st turn of the spiral r = e⁻ᶿ in the first and second quadrants, for θ ≥ 0 (see figure).

c. Evaluate lim(n→∞) Aₙ₊₁/Aₙ.


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