exercises

Sketch the curves that are the images of the paths in Exercises 1 to 4.

Question 12.4

\(x=\sin t, y= 4\cos t, \hbox{where } 0\leq t\leq 2\pi\)

Question 12.5

\(x=2\sin t, y=4\cos t, \hbox{where } 0\leq t\leq 2\pi\)

Question 12.6

\({\bf c}(t)=(2t-1,t+2,t)\)

Question 12.7

\({\bf c}(t)=(-t,2t,1/t), \hbox{where } 1\leq t\leq 3\)

Question 12.8

Consider the circle \(C\) of radius 2, centered at the origin.

  • (a) Find a parametrization for \(C\) inducing a counterclockwise orientation and starting at (2, 0).
  • (b) Find a parametrization for \(C\) inducing a clockwise orientation and starting at (0, 2).
  • (c) Find a parametrization for \(C\) if it is now centered at the point (4, 7).

Question 12.9

Give a parametrization for each of the following curves:

  • (a) The line passing through (1, 2, 3) and (\(-2\), 0, 7)
  • (b) The graph of \(f(x)=x^2\)
  • (c) The square with vertices (0, 0), (0, 1), (1, 1), and (1, 0) (Break it up into line segments.)
  • (d) The ellipse given by \(\frac{x^2}{9}+\frac{y^2}{25}=1\)

In Exercises 7 to 10, determine the velocity vector of the given path.

Question 12.10

\({\bf c}(t)=6t{\bf i}+3t^2{\bf j}+t^3{\bf k}\)

Question 12.11

\({\bf c}(t)=(\sin 3t){\bf i}+(\cos 3t){\bf j}+2t^{3/2}{\bf k}\)

Question 12.12

\({\bf c}(t)=(\cos^2t,3t-t^3,t)\)

Question 12.13

\({\bf c}(t)=(4e',6t^4,\cos t)\)

In Exercises 11 to 14, compute the tangent vectors to the given path.

Question 12.14

\({\bf c}(t)=(e^t,\cos t)\)

Question 12.15

\({\bf c}(t)=(3t^2,t^3)\)

Question 12.16

\({\bf c}(t)=(t\sin t, 4t)\)

Question 12.17

\({\bf c}(t)=(t^2, e^2)\)

Question 12.18

When is the velocity vector of a point on the rim of a rolling wheel horizontal? What is the speed at this point?

Question 12.19

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.

Question 12.20

\((\sin 3t,\cos 3t,2t^{5/2}); t=1\)

Question 12.21

\((\cos^2 t,3t-t^3,t); t=0\)

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 \(t_1\).

Question 12.22

\({\bf c}(t)=(t^2,t^3-4t,0)\), where \(t_0=2,t_1=3\)

Question 12.23

\({\bf c}(t)=(e^t,e^{-t},\cos t)\), where \(t_0=1,t_1=2\)

Question 12.24

\({\bf c}(t)=(4e^t, 6t^4, \cos t)\), where \(t_0=0, t_1=1\)

Question 12.25

\({\bf c}(t)=(\sin e^t,t,4-t^3)\), where \(t_0=1,t_1=2\)

Question 12.26

The position vector for a particle moving on a helix is \(\textbf{c}(t) = (\cos(t), \sin(t), t^2)\).

  • (a) Find the speed of the particle at time \(t_0=4\pi\).
  • (b) Is \(\textbf{c} '(t)\) ever orthogonal to \(\textbf{c}(t)\)?
  • (c) Find a parametrization for the tangent line to \(\textbf{c}(t)\) at \(t_0=4\pi\).
  • (d) Where will this line intersect the \(xy\) plane?

Question 12.27

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.

Question 12.28

Let \(\textbf{c}(t)=(t^3,t^2,2t)\) and \(f(x,y,z)=(x^2-y^2,2xy,z^2)\).

  • (a) Find \((f \circ \textbf{c})(t)\).
  • (b) Find a parametrization for the tangent line to the curve \(f \circ \textbf{c}\) at \(t=1\).