Solution Found!
Explain why or why not Determine whether the
Chapter 8, Problem 51E(choose chapter or problem)
Determine whether the following statements are true and give an explanation or counterexample.
a. The equations x = -cos t, y = -sin t, for \(0\ \leq\ t\ \leq\ 2\pi\) generate a circle in the clockwise direction.
b. An object following the parametric curve \(x = 2 \cos 2\pi t,\ y =2 \sin 2\pi t \) circles the origin once every 1 time unit.
c. The parametric equations \(x=t,\ y=t^2\) for \(t\ \geq\ 0\) describe the complete parabola \(y = x^2\).
d. The parametric equations \(x = \cos t,\ y = \sin t\) for \(-\pi/2\ \leq\ t\ \leq\ \pi/2\) describe a semicircle.
Questions & Answers
QUESTION:
Determine whether the following statements are true and give an explanation or counterexample.
a. The equations x = -cos t, y = -sin t, for \(0\ \leq\ t\ \leq\ 2\pi\) generate a circle in the clockwise direction.
b. An object following the parametric curve \(x = 2 \cos 2\pi t,\ y =2 \sin 2\pi t \) circles the origin once every 1 time unit.
c. The parametric equations \(x=t,\ y=t^2\) for \(t\ \geq\ 0\) describe the complete parabola \(y = x^2\).
d. The parametric equations \(x = \cos t,\ y = \sin t\) for \(-\pi/2\ \leq\ t\ \leq\ \pi/2\) describe a semicircle.
ANSWER:Solution 51EStep 1 of 2:In this problem we need to explain whether the given statements are true or false.1. The given statement is “ The equations x = -cos(t) , y = -sin(t) , for 0 t 2 generate a circle in the clockwise direction” is false .Because , = 1 = 1.Therefore , The equations x = -cos(t) , y = -sin(t) , for 0 t 2 generate a circle in the anticlockwise direction.The related graph is shown below; b) The given statement is “ An object following the parametric curve x = 2 cos 2t, y = 2 sin 2t circles the origin once every 1 lime unit” is true.Because , = 4 , since = 1. = 4.