Exercise · ZW-PARAMETRIC-CURVE-010 · ★★★ Advanced
For \(0\le t\le2\pi\), how many times does \(\mathbf r(t)=\langle\cos2t,\sin2t\rangle\) go around the unit circle?
Choose one option.
Show answer
- D. Twice
Explanation
The angle \(2t\) runs from \(0\) to \(4\pi\), that is, two full turns. The set of points is the same as for \(\langle\cos t,\sin t\rangle\), but it is traced at double speed.
Review the concept: Parametric curve
A parametric curve is the path traced by a point in space when its coordinates depend on a parameter \(t\) through a vector-valued function \(\mathbf r\).
\[\mathbf r(t)=\langle x(t),y(t),z(t)\rangle,\quad a\le t\le b\]
More practice problems: parametric curve
- ZW-PARAMETRIC-CURVE-001 Let mathbf r(t)=langle 2t, t^2rangle. Which point corresponds to t=3?
- ZW-PARAMETRIC-CURVE-002 Eliminate the parameter in x=t+1,\ y=2t. Which Cartesian equation results?
- ZW-PARAMETRIC-CURVE-003 What curve does mathbf r(t)=langlecos t,sin trangle trace for 0≤ t≤ 2π?
- ZW-PARAMETRIC-CURVE-004 Which space curve is mathbf r(t)=langlecos t,sin t,trangle?
- ZW-PARAMETRIC-CURVE-005 If mathbf r(t)=langle t^2, t^3rangle, what is mathbf r'(1)?
- ZW-PARAMETRIC-CURVE-006 What is the tangent line to mathbf r(t)=langle t, t^2rangle at t=1?
- ZW-PARAMETRIC-CURVE-007 Which Cartesian equation do the points of x=3cos t,\ y=2sin t satisfy?
- ZW-PARAMETRIC-CURVE-008 Which one parametrizes the segment from (1,2,0) to (3,0,4) with 0≤ t≤1?