Exercises for Section 12.2

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In the first four Exercises, find the velocity and acceleration vectors and the equation of the tangent line for each of the following curves, at the given value of t.

Question 12.30

\({\bf r}(t)=(\cos t){\bf i}+(\sin 2t){\bf j}, \,\hbox{at } t=0\)

Question 12.31

\({\bf c}(t)=(t\,\sin t, t\,\cos t,\sqrt{3}t), \,\hbox{at } t=0\)

Question 12.32

\({\bf r}(t)=\sqrt{2}t{\bf i}+e^t{\bf j}+e^{-t}{\bf k}, \,\hbox{at } t=0\)

Question 12.33

\({\bf c}({t})=t{\bf i} + t {\bf j}+\frac{2}{3}t^{3/2}{\bf k}, \,\hbox{at } t=9\)

In the next four Exercises, let \({\bf c}_1(t)=e^t{\bf i}+(\sin t){\bf j} + t^3{\bf k}\) and \({\bf c}_2(t)=e^{-t}{\bf i}+(\,\cos t){\bf j}-2t^3{\bf k}\). Find each of the stated derivatives in two different ways to verify the rules in the box preceding Example 1.

Question 12.34

\(\displaystyle\frac{d}{{\it dt}}[{\bf c}_1(t)+{\bf c}_2(t)]\)

Question 12.35

\(\displaystyle\frac{d}{{\it dt}}[{\bf c}_1(t)\,{\cdot}\, {\bf c}_2(t)]\)

Question 12.36

\(\displaystyle\frac{d}{{\it dt}}[{\bf c}_1(t)\times {\bf c}_2(t)]\)

Question 12.37

\(\displaystyle\frac{d}{{\it dt}}\{{\bf c}_1(t)\,{\cdot}\, [2{\bf c}_2(t)+{\bf c}_1(t)]\}\)

Question 12.38

Consider the helix given by \(\textbf{c}(t)=(a\cos t, a\sin t, bt)\). Show that the acceleration vector is always parallel to the \(xy\) plane.

Question 12.39

Prove the dot product rule.

Question 12.40

Determine which of the following paths are regular:

  • (a) \(\textbf{c}(t)=(\cos t, \sin t, t)\)
  • (b) \(\textbf{c}(t)=(t^3, t^5, \cos t)\)
  • (c) \(\textbf{c}(t)=(t^2, e^t, 3t+1)\)

Question 12.41

Let \(\textbf{v}\) and \(\textbf{a}\) denote the velocity and acceleration vectors of a particle moving on a path \(\textbf{c}\). Suppose the initial position of the particle is \(\textbf{c}(0)=(3, 4, 0)\), the initial velocity is \(\textbf{v}(0)=(1, 1, -2)\), and the acceleration function is \(\textbf{a}(t)=(0, 0, 6)\). Find \(\textbf{v}(t)\) and \(\textbf{c}(t)\).

Question 12.42

The acceleration, initial velocity, and initial position of a particle traveling through space are given by \[ \textbf{a}(t)=(2, -6, -4), \quad \textbf{v}(0)=(-5, 1, 3), \quad \textbf{r}(0)=(6, -2, 1). \] The particle’s trajectory intersects the \(yz\) plane exactly twice. Find these two intersection points.

Question 12.43

The acceleration, initial velocity, and initial position of a particle traveling through space are given by \[ \textbf{a}(t)=(-6, 2, 4), \quad \textbf{v}(0)=(2, -5, 1), \quad \textbf{r}(0)=(-3, 6, 2). \] The particle’s trajectory intersects the \(yz\) plane exactly twice. Find these two intersection points.

Question 12.44

Show that if the acceleration of an object is always perpendicular to the velocity, then the speed of the object is constant. (HINT: See Example 1.)

Question 12.45

Show that, at a local maximum or minimum of \(\|{\bf r}(t)\|\), the vector \({\bf r}'(t)\) is perpendicular to \({\bf r}(t)\).

Question 12.46

Find the path \({\bf c}\) such that \({\bf c}(0)=(0,-5,1)\) and \({\bf c}'(t)=(t,e^t,t^2)\).

Question 12.47

Let \({\bf c}\) be a path in \({\mathbb R}^3\) with zero acceleration. Prove that \({\bf c}\) is a straight line or a point.

Question 12.48

Find paths \({\bf c} (t)\) that represent the following curves or trajectories.

  • (a) \(\{(x,y)\mid y=e^x\}\)
  • (b) \(\{(x,y)\mid 4x^2+y^2=1\}\)
  • (c) A straight line in \({\mathbb R}^3\) passing through the origin and the point \((a,b,c)\)
  • (d) \(\{(x,y)\mid 9x^2+16 y^2=4\}\)

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