Exercise · ZW-PARAMETRIC-CURVE-005 · ★☆☆ Basic
If \(\mathbf r(t)=\langle t^2,\,t^3\rangle\), what is \(\mathbf r'(1)\)?
Choose one option.
Show answer
- C. \(\langle2,3\rangle\)
Explanation
Differentiate component by component: \(\mathbf r'(t)=\langle2t,3t^2\rangle\), and at \(t=1\) this is \(\langle2,3\rangle\).
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-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?
- ZW-PARAMETRIC-CURVE-009 Let mathbf r_1(t)=langle t,t^2rangle and mathbf r_2(t)=langle t^2,t^4rangle, with t∈mathbb R. Which…