Review Exercises for Chapter 12

For the first four Exercises, at the indicated point, compute the velocity vector, the acceleration vector, the speed, and the equation of the tangent line.

Question 12.75

\({\bf c}(t)=(t^3+1,e^{-t},\cos\, (\pi t/2)), \hbox{at } t=1\)

Question 12.76

\({\bf c}(t)=(t^2-1,\cos\, (t^2),t^4), \hbox{at }t=\sqrt{\pi}\)

Question 12.77

\({\bf c}(t)=(e^t,\sin t,\cos t), \hbox{at }t=0\)

Question 12.78

\({\bf c}(t)=\displaystyle\frac{t^2}{1+t^2}{\bf i}+t{\bf j}+ {\bf k}, \hbox{at }t=2\)

Question 12.79

Calculate the tangent and acceleration vectors for the helix \({\bf c}(t)=(\cos t,\sin t ,t)\) at \(t=\pi/4\).

Question 12.80

Calculate the tangent and acceleration vector for the cycloid \({\bf c}(t)=(t-\sin t,1-\cos t)\) at \(t=\pi/4\) and sketch.

Question 12.81

Let a particle of mass \(m\) move on the path \({\bf c}(t) = (t^2,\sin t,\cos t).\) Compute the force acting on the particle at \(t=0\).

Question 12.82

  • (a) Let \({\bf c}(t)\) be a path with \(\|{\bf c}(t)\|=\) constant; that is, the curve lies on a sphere. Show that \({\bf c}'(t)\) is orthogonal to \({\bf c}(t)\).
  • (b) Let \({\bf c}\) be a path whose speed is never zero. Show that \({\bf c}\) has constant speed if and only if the acceleration vector \({\bf c}''\) is always perpendicular to the velocity vector \({\bf c}'\).

Question 12.83

Let \(\textbf{c}(t)=(\cos t, \sin t, \sqrt3 t)\) be a path in \(\mathbb R^3\).

  • (a) Find the velocity and acceleration of this path.
  • (b) Find a parametrization for the tangent line to this path at \(t=0\).
  • (c) Find the arc length of this path for \(t\in [0, 2\pi]\).

Question 12.84

Express the arc length of the curve \(x^2 =y^3 =z^5\) between \(x=1\) and \(x=4\) as an integral, using a suitable parametrization.

Question 12.85

Find the arc length of \({\bf c}(t)=t{\bf i}+(\log t){\bf j}+2\sqrt{2t}{\bf k}\) for \(1\leq t\leq 2\).

Question 12.86

A particle is constrained to move around the unit circle in the \(xy\) plane according to the formula \((x,y,z)=(\cos\, (t^2),\sin \,(t^2),0),t\geq 0\).

  • (a) What are the velocity vector and speed of the particle as functions of \(t\)?
  • (b) At what point on the circle should the particle be released to hit a target at \((2,0,0)\)? (Be careful about which direction the particle is moving around the circle.)
  • (c) At what time \(t\) should the release take place? (Use the smallest \(t > 0\) that will work.)
  • (d) What are the velocity and speed at the time of release?
  • (e) At what time is the target hit?

Question 12.87

Write the curve described by the equations \(x-1=2y+1=3z+2\) in parametric form.

Question 12.88

Write the curve \(x=y^3=z^2+1\) in parametric form.

1Most scientists acknowledge that \({\bf F}=m{\bf a}\) is the single most important equation in all of science and engineering.

2For more information and history, consult S. Hildebrandt and A. J. Tromba, The Parsimonious Universe: Shape and Form in the Natural World, Springer-Verlag, New York/Berlin, 1995.

3For more information about Poincaré, see F. Diacu and P. Holmes, Celestial Encounters. The Origins of Chaos and Stability, Princeton University Press: Princeton, NJ, 1996.

4Several of these problems make use of the formula \[ \int\sqrt{x^2+a^2}{\,d} x={\textstyle\frac{1}{2}}\big[x\sqrt{x^2+a^2}+a^2\log \,(x+\sqrt{x^2+a^2})\big]+C \] from the table of integrals in the back of the book.