Consider the following statement. The parametric equations x = X1 + (x2 - X1)t, y = Y1 + (Y2 – Y1)t where 0 st s 1, describe the line segment that joins the points P,(x1, y1) and P2(x2, Y2). The following is a proposed proof for the statement. 1. When t = 0 the curve passes through P, (x,, y,) and when t = 1 the curve passes through P2(x2, Y2). 2. For 0
Consider the following statement. The parametric equations x = X1 + (x2 - X1)t, y = Y1 + (Y2 – Y1)t where 0 st s 1, describe the line segment that joins the points P,(x1, y1) and P2(x2, Y2). The following is a proposed proof for the statement. 1. When t = 0 the curve passes through P, (x,, y,) and when t = 1 the curve passes through P2(x2, Y2). 2. For 0
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![Consider the following statement.
The parametric equations
x = X1 + (X2 - X1)t, y = Y1 + (Y2 - Y1)t
where 0 <t 1, describe the line segment that joins the points P, (x1,Y;) and P2(x2, Y2).
The following is a proposed proof for the statement.
1. When t
= 0 the curve passes through P,(x,, y,) and when t = 1 the
curve passes through P2(x2, Y2).
2. For 0 <t < 1, x is strictly between
y, and y2.
and
X2
and y is strictly between
Y1
3. For every value of t, x and y satisfy the relation
72 71(x – x,), which is the equation of the line through
X2 - X1
y – Y1
=
P;(X1, Y1) and P2(x2, Y2).
У - У1
x – X1
X -
4. Finally, any point (x, y) on that line will satisfy
If
%D
Y2 - Y1
X2 - X1
y - Y1
x - X1
we let t =
then substituting t into the given
Y2 - Y1
X2 - X1
parametric equations yields the point (x, y), and any (x, y) on the line
segment from P;(x1, Y1) to P2(x2, Y2) yields a value of t in [-1, 1].
5. So the given parametric equations exactly specify the line segment
from P, (x1, Y1) to P2(x2, Y2).
Identify the error(s) in the proposed proof. (Select all that apply.)
O The first sentence assumes t
O and t
1, which cannot be assumed based on the given domain.
The second sentence should say 0 <t < 1 instead of 0 < t < 1.
Y2-Y1(x – x,) is a linear equation when this is not the
X2 - X1
The third sentence claims that y - y1
case.
The fourth sentence should say t in [0, 1] instead of t in [-1, 1].
The fifth sentence should say P(x, y) instead of P,(x2, y2).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff0d90012-896b-4f52-8f0b-e38034636c35%2F17d865df-7288-4419-a18f-89ed0b6c5c0f%2F03h4pem_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the following statement.
The parametric equations
x = X1 + (X2 - X1)t, y = Y1 + (Y2 - Y1)t
where 0 <t 1, describe the line segment that joins the points P, (x1,Y;) and P2(x2, Y2).
The following is a proposed proof for the statement.
1. When t
= 0 the curve passes through P,(x,, y,) and when t = 1 the
curve passes through P2(x2, Y2).
2. For 0 <t < 1, x is strictly between
y, and y2.
and
X2
and y is strictly between
Y1
3. For every value of t, x and y satisfy the relation
72 71(x – x,), which is the equation of the line through
X2 - X1
y – Y1
=
P;(X1, Y1) and P2(x2, Y2).
У - У1
x – X1
X -
4. Finally, any point (x, y) on that line will satisfy
If
%D
Y2 - Y1
X2 - X1
y - Y1
x - X1
we let t =
then substituting t into the given
Y2 - Y1
X2 - X1
parametric equations yields the point (x, y), and any (x, y) on the line
segment from P;(x1, Y1) to P2(x2, Y2) yields a value of t in [-1, 1].
5. So the given parametric equations exactly specify the line segment
from P, (x1, Y1) to P2(x2, Y2).
Identify the error(s) in the proposed proof. (Select all that apply.)
O The first sentence assumes t
O and t
1, which cannot be assumed based on the given domain.
The second sentence should say 0 <t < 1 instead of 0 < t < 1.
Y2-Y1(x – x,) is a linear equation when this is not the
X2 - X1
The third sentence claims that y - y1
case.
The fourth sentence should say t in [0, 1] instead of t in [-1, 1].
The fifth sentence should say P(x, y) instead of P,(x2, y2).
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