Sketch the curves that are the images of the paths in Exercises 1 to 4.
\(x=\sin t, y= 4\cos t, \hbox{where } 0\leq t\leq 2\pi\)
\(x=2\sin t, y=4\cos t, \hbox{where } 0\leq t\leq 2\pi\)
\({\bf c}(t)=(2t-1,t+2,t)\)
\({\bf c}(t)=(-t,2t,1/t), \hbox{where } 1\leq t\leq 3\)
Consider the circle \(C\) of radius 2, centered at the origin.
Give a parametrization for each of the following curves:
In Exercises 7 to 10, determine the velocity vector of the given path.
\({\bf c}(t)=6t{\bf i}+3t^2{\bf j}+t^3{\bf k}\)
\({\bf c}(t)=(\sin 3t){\bf i}+(\cos 3t){\bf j}+2t^{3/2}{\bf k}\)
\({\bf r}(t)=(\cos^2t,3t-t^3,t)\)
\({\bf r}(t)=(4e',6t^4,\cos t)\)
In Exercises 11 to 14, compute the tangent vectors to the given path.
\({\bf c}(t)=(e^t,\cos t)\)
\({\bf c}(t)=(3t^2,t^3)\)
\({\bf c}(t)=(t\sin t, 4t)\)
\({\bf c}(t)=(t^2, e^2)\)
When is the velocity vector of a point on the rim of a rolling wheel horizontal? What is the speed at this point?
If the position of a particle in space is \((6t,3t^2,t^3)\) at time \(t\), what is its velocity vector at \(t=0\)?
In Exercises 17 and 18, determine the equation of the tangent line to the given path at the specified value of t.
\((\sin 3t,\cos 3t,2t^{5/2}); t=1\)
\((\cos^2 t,3t-t^3,t); t=0\)
124
In Exercises 19 to 22, suppose that a particle following the given path \({\bf c}(t)\) flies off on a tangent at \(t=t_0\). Compute the position of the particle at the given time \(t_1\).
\({\bf c}(t)=(t^2,t^3-4t,0)\), where \(t_0=2,t_1=3\)
\({\bf c}(t)=(e^t,e^{-t},\cos t)\), where \(t_0=1,t_1=2\)
\({\bf c}(t)=(4e^t, 6t^4, \cos t)\), where \(t_0=0, t_1=1\)
\({\bf c}(t)=(\sin e^t,t,4-t^3)\), where \(t_0=1,t_1=2\)
The position vector for a particle moving on a helix is \(\textbf{c}(t) = (\cos(t), \sin(t), t^2)\).
Consider the spiral given by \(\textbf{c}(t)=(e^t \cos(t), e^t \sin(t))\). Show that the angle between \(\textbf{c}\) and \(\textbf{c}'\) is constant.
Let \(\textbf{c}(t)=(t^3,t^2,2t)\) and \(f(x,y,z)=(x^2-y^2,2xy,z^2)\).