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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.
\({\bf r}(t)=(\cos t){\bf i}+(\sin 2t){\bf j}, \,\hbox{at } t=0\)
\({\bf c}(t)=(t\,\sin t, t\,\cos t,\sqrt{3}t), \,\hbox{at } t=0\)
\({\bf r}(t)=\sqrt{2}t{\bf i}+e^t{\bf j}+e^{-t}{\bf k}, \,\hbox{at } t=0\)
\({\bf c}({t})=t{\bf i} + t {\bf j}+\frac{2}{3}t^{3/2}{\bf k}, \,\hbox{at } t=9\)
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.
\(\displaystyle\frac{d}{{\it dt}}[{\bf c}_1(t)+{\bf c}_2(t)]\)
\(\displaystyle\frac{d}{{\it dt}}[{\bf c}_1(t)\,{\cdot}\, {\bf c}_2(t)]\)
\(\displaystyle\frac{d}{{\it dt}}[{\bf c}_1(t)\times {\bf c}_2(t)]\)
\(\displaystyle\frac{d}{{\it dt}}\{{\bf c}_1(t)\,{\cdot}\, [2{\bf c}_2(t)+{\bf c}_1(t)]\}\)
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.
Prove the dot product rule.
Determine which of the following paths are regular:
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 \(\textbf{c}(t)\).
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.
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.
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.)
Show that, at a local maximum or minimum of \(\|{\bf r}(t)\|\), the vector \({\bf r}'(t)\) is perpendicular to \({\bf r}(t)\).
Find the path \({\bf c}\) such that \({\bf c}(0)=(0,-5,1)\) and \({\bf c}'(t)=(t,e^t,t^2)\).
Let \({\bf c}\) be a path in \({\mathbb R}^3\) with zero acceleration. Prove that \({\bf c}\) is a straight line or a point.
Find paths \({\bf c} (t)\) that represent the following curves or trajectories.
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