8.1: Vector-Valued Functions and Space Curves
( \newcommand{\kernel}{\mathrm{null}\,}\)
- Given the vector-valued function r(t)=(2t2+3,√3t−2), find r(2).
- Given the vector-valued function r(t)=secti+tantj, find r(π4).
- Sketch the curve of the vector-valued function r(t)=ti+(1−t2)j.
- Sketch the curve of the vector-valued function r(t)=(t,t2,t).
- Sketch the curve of the vector-valued function r(t)=(cost,sint). Be sure to indicate the orientation of the curve.
- Sketch the curve of the vector-valued function r(t)=sinti+costj+tk. Be sure to indicate the orientation of the curve.
- Find the domain of the vector-valued function r(t)=(1t−3,√t2−1).
- Find the domain of the vector-valued function r(t)=2eti+lntj+1t2+1k.
- Evaluate limt→0(2t+1,cost), if possible.
- Evaluate limt→2(t−2t2−4,et), if possible.
- Evaluate limt→∞(e−ti+3t2+2t−42−5t2j+arctan(t)k), if possible.
- Evaluate limt→0+(sintt,lnt,4), if possible.
- Express the vector-valued function r(t)=2ti−(3+t)j in Cartesian coordinates by eliminating the parameter t, then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
- Express the vector-valued function r(t)=(t3,t2) in Cartesian coordinates by eliminating the parameter t, then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
- Express the vector-valued function r(t)=1t(i+j) in Cartesian coordinates by eliminating the parameter t, then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
- Express the vector-valued function r(t)=(2sint,−3cost) in Cartesian coordinates by eliminating the parameter t, then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
- Express the vector-valued function r(t)=3sec2ti+tantj in Cartesian coordinates by eliminating the parameter t, then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
- Find a vector-valued function that traces the line from point P(1,0,1) to point Q(0,1,0).
- Find a vector-valued function that traces the parabola y=x2 from left-to-right.
- Find a vector-valued function that traces the ellipse 4x2+9y2=36 in the counterclockwise direction.
- Find a vector-valued function that traces the hyperbola x216−y2=1, where the right branch is oriented top-to-bottom.