Question

Find a parametrization. $$x=x(t), \quad y=y(t) \quad t \in[0,1]$$ for the given curve. The line segment from (3,7) to (8,5).

Name: Problem 27, Chapter 10: Calculus: One Variable
Uploaded: 2021-12-17T18:33:42-08:00
Duration: 1 min 4 s
Channel: Lucas Finney
Description: Find a parametrization. x=x(t), y=y(t) t ∈[0,1] for the given curve. The line segment from (3,7) to (8,5).

   Find a parametrization.
$$x=x(t), \quad y=y(t) \quad t \in[0,1]$$
for the given curve.
The line segment from (3,7) to (8,5).

Calculus: One Variable

Garret J. Etgen,… 10th Edition

Chapter 10, Problem 27

Instant Answer

Solved by Expert Lucas Finney

12/17/2021

Step 1

The standard parametrization for a line segment from point $(x_1, y_1)$ to point $(x_2, y_2)$ is given by: $$x(t) = x_1 + t(x_2 - x_1)$$ $$y(t) = y_1 + t(y_2 - y_1)$$ where $t$ is a parameter that varies from 0 to 1. Show more…

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Find a parametrization. $$x=x(t), \quad y=y(t) \quad t \in[0,1]$$ for the given curve. The line segment from (3,7) to (8,5).

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Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter. This approach is widely used because it allows precise control of the path traced by the curve and easily handles curves that are not functions in the traditional y = f(x) sense.

Linear Interpolation

Linear interpolation is a method for finding intermediate points along a straight line between two given endpoints. By scaling the difference between the endpoints with a parameter that ranges from 0 to 1, you can generate all the points on the line segment.

Vector Representation of a Line

This concept involves representing a line segment using a starting point and a direction vector. The line is described by adding the scaled direction vector to the starting point, which naturally leads to a parametrization of the line segment using a parameter that determines the position along the line.

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