Question

(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as $t$ increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. $x=\sqrt{t}, \quad y=1-t$

Name: Problem 9, Chapter 10: Calculus
Uploaded: 2021-08-24T22:21:14-08:00
Duration: 1 min 22 s
Channel: Carson Merrill
Description: (a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. x=√(t), y=1-t

   (a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as $t$ increases.
(b) Eliminate the parameter to find a Cartesian equation of the curve.
$x=\sqrt{t}, \quad y=1-t$

Calculus

James Stewart 7th Edition

Chapter 10, Problem 9

Instant Answer

Solved by Expert Carson Merrill

08/24/2021

Step 1

Since $ x = \sqrt{t} $, $ t $ must be non-negative. Therefore, $ t \geq 0 $. Show more…

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(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as $t$ increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. $x=\sqrt{t}, \quad y=1-t$

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Key Concepts

Parametric Equations

Parametric equations express the coordinates of the points on a curve as functions of a parameter, usually denoted by t. This way of representing curves is useful for describing motion or other phenomena where a single independent quantity influences multiple dependent variables. They allow for straightforward plotting by computing coordinates for various parameter values and help in analyzing the behavior of curves, such as directionality and the passage of time.

Elimination of the Parameter

Eliminating the parameter involves rewriting a set of parametric equations as a relationship solely between the variables, typically x and y. This process simplifies the analysis by converting the curve into its Cartesian form, making it easier to understand its properties and to perform operations like finding intersections or determining symmetry without explicitly referencing the parameter.

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