Curve 1 Consider the curve described by the parametric equations x(t) = t³/3 and y(t) = 1²12, (this is the middle animation, by the way) Find the arc length along this curve from the origin (t=0) to point T_0 (t= t_0), some arbitrary point. ● Stay ● To {X(+) = 1/3 (74) = 1²/2 To make the rest of the computations easier, assume that the string initially extends a distance of 1/3 beyond the origin. What is the length of the string from the point T_0 to the point P? Y Extm 1/3 X {X(+) = 1/3 (74) = 1²/2 At T_0, the string is tangent to the curve. Find the slope of this tangent line.
Curve 1 Consider the curve described by the parametric equations x(t) = t³/3 and y(t) = 1²12, (this is the middle animation, by the way) Find the arc length along this curve from the origin (t=0) to point T_0 (t= t_0), some arbitrary point. ● Stay ● To {X(+) = 1/3 (74) = 1²/2 To make the rest of the computations easier, assume that the string initially extends a distance of 1/3 beyond the origin. What is the length of the string from the point T_0 to the point P? Y Extm 1/3 X {X(+) = 1/3 (74) = 1²/2 At T_0, the string is tangent to the curve. Find the slope of this tangent line.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Just need some direction.
Transcribed Image Text:tagent line
To
String
r
•Path trand by P
●
●
Draw a right triangle that shows the relationship between P and T_0. Which angle is the same as the angle
that this line makes with the x axis? Label it theta.
String
tagent line
P
useful
tringle
-path traced by P
{X(+) = 1/3
Y(4) = t½
SX (+) =
(1/3
(y(t) = 1²/2
Use the slope of this tangent line, the relationship you described in the Preliminary problems, and the
triangle from above to describe the coordinates of point P in terms of the parameter t_0.
Eliminate the parameter of your result, and identify the kind of curve that is traced by the point P.
Transcribed Image Text:Draw a line with positive slope m. Label the acute angle this line makes with the x-axis as theta. Find a
relationship between m and theta.
Draw a line with negative slope m. Label the acute angle this makes with the x-axis as theta. Show that the
same relationship from above holds.
Curve 1
Consider the curve described by the parametric equations X(t) = t³/3 and y(t) = t1²12, (this is the middle
animation, by the way)
● Find the arc length along this curve from the origin (t=0) to point T_0 (t= t_0), some arbitrary point.
Y
To
Sx (+) = 1/3
=
7(4) = t²₂
To
X
To make the rest of the computations easier, assume that the string initially extends a distance of 1/3
beyond the origin. What is the length of the string from the point T_0 to the point P?
Y
(½/3
SX (+) =
7(4) = t½
{
X
Extra 1/3
P
At T_0, the string is tangent to the curve. Find the slope of this tangent line.
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